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Fick's laws of diffusion

Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, $D$. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

Fick's first law: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient.

Fick's second law: Prediction of change in concentration gradient with time due to diffusion.

A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called anomalous diffusion or non-Fickian diffusion.

History
In 1855, physiologist Adolf Fick first reported his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of Thomas Graham, which fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: Darcy's law (hydraulic flow), Ohm's law (charge transport), and Fourier's Law (heat transport).

Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible. Today, Fick's Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does not follow Fick's laws (which happens in cases of diffusion through porous media and diffusion of swelling penetrants, among others), it is referred to as non-Fickian.

Fick's first law
Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient. In one (spatial) dimension, the law can be written in various forms, where the most common form (see ) is in a molar basis:
 * $$J = -D \frac{d \varphi}{d x} $$

where
 * $J$ is the diffusion flux, of which the dimension is the amount of substance per unit area per unit time. $J$ measures the amount of substance that will flow through a unit area during a unit time interval.
 * $D$ is the diffusion coefficient or diffusivity. Its dimension is area per unit time.
 * $$ \frac{d \varphi}{d x} $$ is the concentration gradient
 * $φ$ (for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume.
 * $x$ is position, the dimension of which is length.

$D$ is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes–Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of $0.6 m^{2}/s$. For biological molecules the diffusion coefficients normally range from 10−10 to 10−11 m2/s.

In two or more dimensions we must use $∇$, the del or gradient operator, which generalises the first derivative, obtaining
 * $$ \mathbf{J}=- D\nabla \varphi$$

where $J$ denotes the diffusion flux vector.

The driving force for the one-dimensional diffusion is the quantity $−∂φ⁄∂x$, which for ideal mixtures is the concentration gradient.

Variations of the first law
Another form for the first law is to write it with the primary variable as mass fraction ($y_{i}$, given for example in kg/kg), then the equation changes to:
 * $$\mathbf{J}_i = -\frac{\rho D}{M_i}\nabla y_i $$

where
 * the index $i$ denotes the $i$th species,
 * $J_{i}$ is the diffusion flux vector of the $i$th species (for example in mol/m2-s),
 * $M_{i}$ is the molar mass of the $i$th species, and
 * $ρ$ is the mixture density (for example in kg/m3).

The $$\rho$$ is outside the gradient operator. This is because:
 * $$y_i = \frac{\rho_{si}}{\rho}$$

where $ρ_{si}$ is the partial density of the $i$th species.

Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written
 * $$J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}$$

where
 * the index $i$ denotes the $i$th species
 * $c$ is the concentration (mol/m3)
 * $R$ is the universal gas constant (J/K/mol)
 * $T$ is the absolute temperature (K)
 * $μ$ is the chemical potential (J/mol).

The driving force of Fick's law can be expressed as a fugacity difference:
 * $$J_i = - \frac{D}{RT} \frac{\partial f_i}{\partial x}$$

Fugacity $$ f_i $$ has Pa units. $$ f_i $$ is a partial pressure of component $i$ in a vapor $$ f_i^\text{G} $$ or liquid $$ f_i^\text{L} $$ phase. At vapor liquid equilibrium the evaporation flux is zero because $$ f_i^\text{G} = f_i^\text{L} $$.

Derivation of Fick's first law for gases
Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref. shows in detail how the diffusion equation from the kinetic theory of gases reduces to this version of Fick's law: $$ \mathbf{V_i}=- D\nabla \ln y_i ,$$ where $V_{i}$ is the diffusion velocity of species $i$. In terms of species flux this is $$\mathbf{J_i}=- \frac{\rho D}{M_i}\nabla y_i. $$

If, additionally, $$ \nabla \rho = 0$$, this reduces to the most common form of Fick's law, $$ \mathbf{J_i}=- D\nabla \varphi .$$

If (instead of or in addition to $$ \nabla \rho = 0$$) both species have the same molar mass, Fick's law becomes $$\mathbf{J_i}=- \frac{\rho D}{M_i}\nabla x_i, $$ where $$ x_i $$ is the mole fraction of species $i$.

Fick's second law
Fick's second law predicts how diffusion causes the concentration to change with respect to time. It is a partial differential equation which in one dimension reads:
 * $$\frac{\partial \varphi}{\partial t} = D\,\frac{\partial^2 \varphi}{\partial x^2}$$

where
 * $φ$ is the concentration in dimensions of $$[\mathsf{N}\mathsf{L}^{-3}]$$, example mol/m3; $φ = φ(x,t)$ is a function that depends on location $x$ and time $t$
 * $t$ is time, example s
 * $D$ is the diffusion coefficient in dimensions of $$[\mathsf{L}^2\mathsf{T}^{-1}]$$, example m2/s
 * $x$ is the position, example m

In two or more dimensions we must use the Laplacian $Δ = ∇^{2}$, which generalises the second derivative, obtaining the equation
 * $$\frac{\partial \varphi}{\partial t} = D\Delta \varphi$$

Fick's second law has the same mathematical form as the Heat equation and its fundamental solution is the same as the Heat kernel, except switching thermal conductivity $$k$$ with diffusion coefficient $$D$$: $$\varphi(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{x^2}{4Dt}\right).$$

Derivation of Fick's second law
Fick's second law can be derived from Fick's first law and the mass conservation in absence of any chemical reactions:
 * $$\frac{\partial \varphi}{\partial t} + \frac{\partial}{\partial x}J = 0

\Rightarrow\frac{\partial \varphi}{\partial t} -\frac{\partial}{\partial x}\left(D\frac{\partial}{\partial x}\varphi\right)\,=0$$

Assuming the diffusion coefficient $x$ to be a constant, one can exchange the orders of the differentiation and multiply by the constant:
 * $$\frac{\partial}{\partial x}\left(D\frac{\partial}{\partial x} \varphi\right) = D\frac{\partial}{\partial x} \frac{\partial}{\partial x} \varphi = D\frac{\partial^2\varphi}{\partial x^2}$$

and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions Fick's second law becomes
 * $$\frac{\partial \varphi}{\partial t} = D\,\nabla^2\varphi,$$

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields
 * $$\frac{\partial \varphi}{\partial t} = \nabla \cdot (D\,\nabla\varphi).$$

An important example is the case where $Δx$ is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant $t$, the solution for the concentration will be a linear change of concentrations along $x$. In two or more dimensions we obtain
 * $$ \nabla^2\varphi = 0,$$

which is Laplace's equation, the solutions to which are referred to by mathematicians as harmonic functions.

Example solutions and generalization
Fick's second law is a special case of the convection–diffusion equation in which there is no advective flux and no net volumetric source. It can be derived from the continuity equation:
 * $$ \frac{\partial \varphi}{\partial t} + \nabla\cdot\mathbf{j} = R, $$

where $Δt$ is the total flux and $J$ is a net volumetric source for $N(x,t)$. The only source of flux in this situation is assumed to be diffusive flux:
 * $$\mathbf{j}_{\text{diffusion}} = -D \nabla \varphi$$

Plugging the definition of diffusive flux to the continuity equation and assuming there is no source ($x + Δx$), we arrive at Fick's second law:
 * $$\frac{\partial \varphi}{\partial t} = D\frac{\partial^2 \varphi}{\partial x^2}$$

If flux were the result of both diffusive flux and advective flux, the convection–diffusion equation is the result.

Example solution 1: constant concentration source and diffusion length
A simple case of diffusion with time $a$ in one dimension (taken as the $D$-axis) from a boundary located at position $Δt$, where the concentration is maintained at a value $(Δx)^{2}$ is
 * $$n \left(x,t \right)=n_0 \operatorname{erfc} \left( \frac{x}{2\sqrt{Dt}}\right) .$$

where $(Δx)^{2}⁄2Δt$ is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space – that is, the corrosion product layer – is semi-infinite, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is infinite (lasting both through the layer with $Δx$, $φ$ and that with $j$, $φ$), then the solution is amended only with coefficient $D$ in front of $R = 0$ (as the diffusion now occurs in both directions). This case is valid when some solution with concentration $x = 0$ is put in contact with a layer of pure solvent. (Bokstein, 2005) The length $n_{0}$ is called the diffusion length and provides a measure of how far the concentration has propagated in the $D$-direction by diffusion in time $x$ (Bird, 1976).

As a quick approximation of the error function, the first two terms of the Taylor series can be used:
 * $$n(x,t)=n_0 \left[ 1 - 2 \left(\frac{x}{2\sqrt{Dt\pi}}\right) \right] $$

If $R$ is time-dependent, the diffusion length becomes
 * $$ 2\sqrt{\int_0^t D( \tau ) \,d\tau}. $$

This idea is useful for estimating a diffusion length over a heating and cooling cycle, where $t$ varies with temperature.

Example solution 2: Brownian particle and mean squared displacement
Another simple case of diffusion is the Brownian motion of one particle. The particle's Mean squared displacement from its original position is: $$\text{MSD} \equiv \langle (\mathbf{x}-\mathbf{x_0})^2\rangle=2nDt$$ where $$n$$ is the dimension of the particle's Brownian motion. For example, the diffusion of a molecule across a cell membrane 8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a eukaryotic cell is a 3-D diffusion. For a cylindrical cactus, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion.

The square root of MSD, $$\sqrt{2nDt}$$, is often used as a characterization of how far has the particle moved after time $$t$$ has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is a Maxwell-Boltzmann distribution.

Generalizations

 * In non-homogeneous media, the diffusion coefficient varies in space, $erfc$. This dependence does not affect Fick's first law but the second law changes: $$\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=D(x) \Delta \varphi(x,t)+\sum_{i=1}^3 \frac{\partial D(x)}{\partial x_i} \frac{\partial \varphi(x,t)}{\partial x_i}$$
 * In anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor $n(x, 0) = 0$. Fick's first law changes to $$J=-D \nabla \varphi ,$$ it is the product of a tensor and a vector: $$ J_i=-\sum_{j=1}^3 D_{ij} \frac{\partial \varphi}{\partial x_j}.$$ For the diffusion equation this formula gives $$\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D \nabla \varphi(x,t)\bigr)=\sum_{i=1}^3\sum_{j=1}^3D_{ij} \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}. $$ The symmetric matrix of diffusion coefficients $x > 0$ should be positive definite. It is needed to make the right hand side operator elliptic.
 * For inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in $$\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=\sum_{i,j=1}^3\left(D_{ij}(x) \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}+ \frac{\partial D_{ij}(x)}{\partial x_i } \frac{\partial \varphi(x,t)}{\partial x_i}\right). $$
 * The approach based on Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components: $$\frac{\partial \varphi_i}{\partial t} = \sum_j \nabla\cdot\left(D_{ij} \frac{\varphi_i}{\varphi_j} \nabla \, \varphi_j\right) .$$ where $x$ are concentrations of the components and $1⁄2$ is the matrix of coefficients. Here, indices $x$ and $t$ are related to the various components and not to the space coordinates.

The Chapman–Enskog formulae for diffusion in gases include exactly the same terms. These physical models of diffusion are different from the test models $n(x, 0) = n_{0}$ which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.

For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example $x ≤ 0$, where $n_{0}$ refer to the components and $n_{0}$ correspond to the space coordinates.

Applications
Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, nuclear materials, plasma physics, and semiconductor doping processes. The theory of voltammetric methods is based on solutions of Fick's equation. On the other hand, in some cases a "Fickian (another common approximation of the transport equation is that of the diffusion theory) " description is inadequate. For example, in polymer science and food science a more general approach is required to describe transport of components in materials undergoing a glass transition. One more general framework is the Maxwell–Stefan diffusion equations of multi-component mass transfer, from which Fick's law can be obtained as a limiting case, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell–Stefan equations are used. See also non-diagonal coupled transport processes (Onsager relationship).

Fick's flow in liquids
When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular random walks take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.

In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a tautology, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing the fluctuating hydrodynamics equations.

Sorption rate and collision frequency of diluted solute
Adsorption, absorption, and collision of molecules, particles, and surfaces are important problems in many fields. These fundamental processes regulate chemical, biological, and environmental reactions. Their rate can be calculated using the diffusion constant and Fick's laws of diffusion especially when these interactions happen in diluted solutions.

Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the Stokes–Einstein equation. In the ultrashort time limit, in the order of the diffusion time a2/D, where a is the particle radius, the diffusion is described by the Langevin equation. At a longer time, the Langevin equation merges into the Stokes–Einstein equation. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the fluctuation-dissipation theorem based on the Langevin equation in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:
 * $$ D(t) = \mu \, k_{\rm B} T\left(1-e^{-t/(m\mu)}\right) $$

where (all in SI units) For a single molecule such as organic molecules or biomolecules (e.g. proteins) in water, the exponential term is negligible due to the small product of mμ in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute.
 * kB is the Boltzmann constant
 * T is the absolute temperature
 * μ is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory)
 * m is the mass of the particle
 * t is time.

The adsorption or absorption rate of a dilute solute to a surface or interface in a (gas or liquid) solution can be calculated using Fick's laws of diffusion. The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in the simulated molecular diffusion in the first section of this page:
 * $$ \Gamma= 2AC_b\sqrt{\frac{Dt}{\pi}}$$

The equation is named after American chemists Irving Langmuir and Vincent Schaefer.
 * $D$ is the surface area (m2).
 * $$C_b$$ is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m3).
 * $D$ is diffusion coefficient of the adsorber (m2/s).
 * $φ_{i}$ is elapsed time (s).
 * $$ \Gamma $$ is the accumulated number of molecules in unit # molecules adsorbed during the time $$t$$.

Briefly as explained in, the concentration gradient profile near a newly created (from $$t=0$$) absorptive surface (placed at $$x=0$$) in a once uniform bulk solution is solved in the above sections from Fick's equation,
 * $$ \frac{\partial C}{\partial x} = \frac{C_b}{\sqrt{\pi Dt}}\text{exp}(-\frac{x^2}{4Dt}) $$

The concentration gradient at the subsurface at $$x = 0$$ is simplified to the pre-exponential factor of the distribution
 * $D_{ij}$ is the number concentration of adsorber molecules at $$ x, t $$ (#/m3).
 * $$ (\frac{\partial C}{\partial x}) _{x = 0} = \frac{C_b}{\sqrt{\pi Dt}} $$

And the rate of diffusion (flux) across area $$A$$ of the plane is
 * $$ (\frac{\partial \Gamma }{\partial t}) _{x = 0} = -\frac{DAC_b}{\sqrt{\pi Dt}} $$

Integrating over time,
 * $$ \Gamma = \int_0^t (\frac{\partial \Gamma}{\partial t}) _{x = 0} = 2AC_b\sqrt{\frac{Dt}{\pi}} $$

The Langmuir–Schaefer equation can be extended to the Ward–Tordai Equation to account for the "back-diffusion" of rejected molecules from the surface:
 * $$ \Gamma= 2A{C_\text{b}}\sqrt{\frac{Dt}{\pi}} - A\sqrt{\frac{D}{\pi}}\int_0^\sqrt{t}\frac{C(\tau)}{\sqrt{t-\tau}} \, d\tau $$

where $$C_b$$ is the bulk concentration, $$C$$ is the sub-surface concentration (which is a function of time depending on the reaction model of the adsorption), and $$\tau$$ is a dummy variable.

Monte Carlo simulations show that these two equations work to predict the adsorption rate of systems that form predictable concentration gradients near the surface but have troubles for systems without or with unpredictable concentration gradients, such as typical biosensing systems or when flow and convection are significant.

A brief history of diffusive adsorption is shown in the right figure. A noticeable challenge of understanding the diffusive adsorption at the single-molecule level is the fractal nature of diffusion. Most computer simulations pick a time step for diffusion which ignores the fact that there are self-similar finer diffusion events (fractal) within each step. Simulating the fractal diffusion shows that a factor of two corrections should be introduced for the result of a fixed time-step adsorption simulation, bringing it to be consistent with the above two equations.

A more problematic result of the above equations is they predict the lower limit of adsorption under ideal situations but is very difficult to predict the actual adsorption rates. The equations are derived at the long-time-limit condition when a stable concentration gradient has been formed near the surface. But real adsorption is often done much faster than this infinite time limit, i.e., the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have predicted for low or none energy barrier adsorption (unless there is a significant adsorption energy barrier that slows down the absorption significantly), for example, thousands to millions time faster in the self-assembly of monolayers at the water-air or water-substrate interfaces. As such, it is necessary to calculate the evolution of the concentration gradient near the surface and find out a proper time to stop the imagined infinite evolution for practical applications. While it is hard to predict when to stop but it is reasonably easy to calculate the shortest time that matters, the critical time when the first nearest neighbor from the substrate surface feels the building-up of the concentration gradient. This yields the upper limit of the adsorption rate under an ideal situation when there are no other factors than diffusion that affect the absorber dynamics:
 * $$ \langle r \rangle = \frac{4}{\pi}Ac_b^{4/3}D$$

 * $$ \langle r \rangle $$ is the adsorption rate assuming under adsorption energy barrier-free situation, in unit #/s.
 * $$ A $$ is the area of the surface of interest on an "infinite and flat" substrate (m2).
 * $$ C_b $$ is the concentration of the absorber molecule in the bulk solution (#/m3).
 * $$ D $$ is the diffusion constant of the absorber (solute) in the solution (m2/s) defined with Fick's law.
 * Dimensional analysis of these units is satisfied.

This equation can be used to predict the initial adsorption rate of any system; It can be used to predict the steady-state adsorption rate of a typical biosensing system when the binding site is just a very small fraction of the substrate surface and a near-surface concentration gradient is never formed; It can also be used to predict the adsorption rate of molecules on the surface when there is a significant flow to push the concentration gradient very shallowly in the sub-surface.

This critical time is significantly different from the first passenger arriving time or the mean free-path time. Using the average first-passenger time and Fick's law of diffusion to estimate the average binding rate will significantly over-estimate the concentration gradient because the first passenger usually comes from many layers of neighbors away from the target, thus its arriving time is significantly longer than the nearest neighbor diffusion time. Using the mean free path time plus the Langmuir equation will cause an artificial concentration gradient between the initial location of the first passenger and the target surface because the other neighbor layers have no change yet, thus significantly lower estimate the actual binding time, i.e., the actual first passenger arriving time itself, the inverse of the above rate, is difficult to calculate. If the system can be simplified to 1D diffusion, then the average first passenger time can be calculated using the same nearest neighbor critical diffusion time for the first neighbor distance to be the MSD,
 * $$L = \sqrt{2Dt} $$

In this critical time, it is unlikely the first passenger has arrived and adsorbed. But it sets the speed of the layers of neighbors to arrive. At this speed with a concentration gradient that stops around the first neighbor layer, the gradient does not project virtually in the longer time when the actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem,
 * $$L~=C_b^{-1/3} $$ (unit m) is the average nearest neighbor distance approximated as cubic packing, where $$C_b$$ is the solute concentration in the bulk solution (unit # molecule / m3).
 * $$D$$ is the diffusion coefficient defined by Fick's equation (unit m2/s).
 * $$t$$ is the critical time (unit s).
 * $$  =a/t= 2aC_b^{2/3}D $$

 * $$ a$$ is a factor of converting the 3D diffusive adsorption problem into a 1D diffusion problem whose value depends on the system, e.g., a fraction of adsorption area $$A$$ over solute nearest neighbor sphere surface area $$4 \pi L^2 /4$$ assuming cubic packing each unit has 8 neighbors shared with other units. This example fraction converges the result to the 3D diffusive adsorption solution shown above with a slight difference in pre-factor due to different packing assumptions and ignoring other neighbors.

When the area of interest is the size of a molecule (specifically, a long cylindrical molecule such as DNA), the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule (random orientation) hit one side of the other. The diffusion constant need to be updated to the relative diffusion constant between two diffusing molecules. This estimation is especially useful in studying the interaction between a small molecule and a larger molecule such as a protein. The effective diffusion constant is dominated by the smaller one whose diffusion constant can be used instead.

The above hitting rate equation is also useful to predict the kinetics of molecular self-assembly on a surface. Molecules are randomly oriented in the bulk solution. Assuming 1/6 of the molecules has the right orientation to the surface binding sites, i.e. 1/2 of the z-direction in x, y, z three dimensions, thus the concentration of interest is just 1/6 of the bulk concentration. Put this value into the equation one should be able to calculate the theoretical adsorption kinetic curve using the Langmuir adsorption model. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry.

The bimolecular collision frequency related to many reactions including protein coagulation/aggregation is initially described by Smoluchowski coagulation equation proposed by Marian Smoluchowski in a seminal 1916 publication, derived from Brownian motion and Fick's laws of diffusion. Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target molecule, e.g. B is the target molecule holding fixed relatively, and A is the moving molecule that creates a concentration gradient near the target molecule B due to the coagulation reaction between A and B. Smoluchowski calculated the collision frequency between A and B in the solution with unit #/s/m3:
 * $$ Z_{AB} = 4{\pi}RD_rC_AC_B$$

where,
 * $$R$$ is the radius of the collision
 * $$D_r = D_A + D_B$$ is the relative diffusion constant between A and B (m2/s)
 * $$C_A$$ and $$C_B$$ are number concentrations of A and B respectively (#/m3).

The reaction order of this bimolecular reaction is 2 which is the analogy to the result from collision theory by replacing the moving speed of the molecule with diffusive flux. In the collision theory, the traveling time between A and B is proportional to the distance which is a similar relationship for the diffusion case if the flux is fixed.

However, under a practical condition, the concentration gradient near the target molecule is evolving over time with the molecular flux evolving as well, and on average the flux is much bigger than the infinite time limit flux Smoluchowski has proposed. Before the first passenger arrival time, Fick's equation predicts a concentration gradient over time which does not build up yet in reality. Thus, this Smoluchowski frequency represents the lower limit of the real collision frequency.

In 2022, Chen calculates the upper limit of the collision frequency between A and B in a solution assuming the bulk concentration of the moving molecule is fixed after the first nearest neighbor of the target molecule. Thus the concentration gradient evolution stops at the first nearest neighbor layer given a stop-time to calculate the actual flux. He named this the critical time and derived the diffusive collision frequency in unit #/s/m3:
 * $$ Z_{AB} = \frac{8}{\pi}{\sigma} D_rC_AC_B\sqrt[3]{C_A+C_B} $$

where,
 * $${\sigma}$$ is the area of the cross-section of the collision (m2).
 * $$D_r = D_A + D_B$$ is the relative diffusion constant between A and B (m2/s).
 * $$C_A$$ and $$C_B$$ are number concentrations of A and B respectively (#/m3).

This equation assumes the upper limit of a diffusive collision frequency between A and B is when the first neighbor layer starts to feel the evolution of the concentration gradient, whose reaction order is $i$ instead of 2. Both the Smoluchowski equation and the JChen equation satisfy dimensional checks with SI units. But the former is dependent on the radius and the latter is on the area of the collision sphere. From dimensional analysis, there will be an equation dependent on the volume of the collision sphere but eventually, all equations should converge to the same numerical rate of the collision that can be measured experimentally. The actual reaction order for a bimolecular unit reaction could be between 2 and $j$, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules.

Biological perspective
The first law gives rise to the following formula:
 * $$\text{flux} = {-P \left(c_2 - c_1\right)}$$

in which
 * $A$ is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
 * $2√Dt$ is the difference in concentration of the gas across the membrane for the direction of flow (from $D = D(x)$ to $D_{ji} = D_{ij}$).

Fick's first law is also important in radiation transfer equations. However, in this context, it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

Under the condition of a diluted solution when diffusion takes control, the membrane permeability mentioned in the above section can be theoretically calculated for the solute using the equation mentioned in the last section (use with particular care because the equation is derived for dense solutes, while biological molecules are not denser than water. Also this equation assumes ideal concentration gradient forms near the membrane and evolves over time):
 * $$ P= 2A_p\eta_{tm} \sqrt{ D/(\pi t)}$$

where
 * $$A_P$$ is the total area of the pores on the membrane (unit m2).
 * $$\eta_{tm}$$ transmembrane efficiency (unitless), which can be calculated from the stochastic theory of chromatography.
 * D is the diffusion constant of the solute unit m2⋅s−1.
 * t is time unit s.
 * c2, c1 concentration should use unit mol m−3, so flux unit becomes mol s−1.

The flux is decay over the square root of time because a concentration gradient builds up near the membrane over time under ideal conditions. When there is flow and convection, the flux can be significantly different than the equation predicts and show an effective time t with a fixed value, which makes the flux stable instead of decay over time. A critical time has been estimated under idealized flow conditions when there is no gradient formed. This strategy is adopted in biology such as blood circulation.

Semiconductor fabrication applications
The semiconductor is a collective term for a series of devices. It mainly includes three categories：two-terminal devices, three-terminal devices, and four-terminal devices. The combination of the semiconductors is called an integrated circuit.

The relationship between Fick's law and semiconductors: the principle of the semiconductor is transferring chemicals or dopants from a layer to a layer. Fick's law can be used to control and predict the diffusion by knowing how much the concentration of the dopants or chemicals move per meter and second through mathematics.

Therefore, different types and levels of semiconductors can be fabricated.

Integrated circuit fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law.

CVD method of fabricate semiconductor
The wafer is a kind of semiconductor whose silicon substrate is coated with a layer of CVD-created polymer chain and films. This film contains n-type and p-type dopants and takes responsibility for dopant conductions. The principle of CVD relies on the gas phase and gas-solid chemical reaction to create thin films.

The viscous flow regime of CVD is driven by a pressure gradient. CVD also includes a diffusion component distinct from the surface diffusion of adatoms. In CVD, reactants and products must also diffuse through a boundary layer of stagnant gas that exists next to the substrate. The total number of steps required for CVD film growth are gas phase diffusion of reactants through the boundary layer, adsorption and surface diffusion of adatoms, reactions on the substrate, and gas phase diffusion of products away through the boundary layer.

The velocity profile for gas flow is: $$\delta(x) = \left( \frac{5x}{\mathrm{Re}^{1/2}} \right) \mathrm{Re}=\frac{v\rho L}{\eta}$$ where
 * $$\delta$$ is the thickness
 * $$\mathrm{Re}$$ is the Reynolds number
 * $D$ is the length of the substrate
 * $D_{ij}$ at any surface
 * $$\eta$$ is viscosity
 * $$\rho$$ is density.

Integrated the $t$ from $∂_{t}φ_{i} = Σ_{j} D_{ij} Δφ_{j}$ to $C$, it gives the average thickness: $$\delta = \frac{10L}{3\mathrm{Re}^{1/2}}$$

To keep the reaction balanced, reactants must diffuse through the stagnant boundary layer to reach the substrate. So a thin boundary layer is desirable. According to the equations, increasing vo would result in more wasted reactants. The reactants will not reach the substrate uniformly if the flow becomes turbulent. Another option is to switch to a new carrier gas with lower viscosity or density.

The Fick's first law describes diffusion through the boundary layer. As a function of pressure (P) and temperature (T) in a gas, diffusion is determined.

$$D = D_0 \left(\frac{P_0}{P}\right) \left(\frac{T}{T_0}\right)^{3/2}$$ where
 * $$P_0$$ is the standard pressure
 * $$T_0$$ is the standard temperature
 * $$D_0$$ is the standard diffusitivity.

The equation tells that increasing the temperature or decreasing the pressure can increase the diffusivity.

Fick's first law predicts the flux of the reactants to the substrate and product away from the substrate: $$J = -D_i \left ( \frac{dc_i}{dx} \right )$$ where
 * $$x$$ is the thickness $$\delta$$
 * $$dc_i$$ is the first reactant's concentration.

In ideal gas law $$PV = nRT$$, the concentration of the gas is expressed by partial pressure.

$$J = - D_i \left ( \frac{P_i-P_0}{\delta RT} \right )$$ where
 * $$R$$ is the gas constant
 * $$\frac{P_i-P_0}{\delta}$$ is the partial pressure gradient.

As a result, Fick's first law tells us we can use a partial pressure gradient to control the diffusivity and control the growth of thin films of semiconductors.

In many realistic situations, the simple Fick's law is not an adequate formulation for the semiconductor problem. It only applies to certain conditions, for example, given the semiconductor boundary conditions: constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate).

Invalidity of Fickian diffusion
Even though Fickian diffusion has been used to model diffusion processes in semiconductor manufacturing (including CVD reactors) in early days, it often fails to validate the diffusion in advanced semiconductor nodes (< 90 nm). This mostly stems from the inability of Fickian diffusion to model diffusion processes accurately at molecular level and smaller. In advanced semiconductor manufacturing, it is important to understand the movement at atomic scales, which is failed by continuum diffusion. Today, most semiconductor manufacturers use random walk to study and model diffusion processes. This allows us to study the effects of diffusion in a discrete manner to understand the movement of individual atoms, molecules, plasma etc.

In such a process, the movements of diffusing species (atoms, molecules, plasma etc.) are treated as a discrete entity, following a random walk through the CVD reactor, boundary layer, material structures etc. Sometimes, the movements might follow a biased-random walk depending on the processing conditions. Statistical analysis is done to understand variation/stochasticity arising from the random walk of the species, which in-turn affects the overall process and electrical variations.

Food production and cooking
The formulation of Fick's first law can explain a variety of complex phenomena in the context of food and cooking: Diffusion of molecules such as ethylene promotes plant growth and ripening, salt and sugar molecules promotes meat brining and marinating, and water molecules promote dehydration. Fick's first law can also be used to predict the changing moisture profiles across a spaghetti noodle as it hydrates during cooking. These phenomena are all about the spontaneous movement of particles of solutes driven by the concentration gradient. In different situations, there is different diffusivity which is a constant.

By controlling the concentration gradient, the cooking time, shape of the food, and salting can be controlled.

title: Fick's laws of diffusion


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Far East

The Far East is the geographical region that encompasses the easternmost portion of the Asian continent, including East, North and Southeast Asia.

The term first came into use in European geopolitical discourse in the 15th century, particularly the British, denoting the Far East as the "farthest" of the three "Easts", beyond the Near East and the Middle East. Likewise, during the Qing dynasty of the 19th and early 20th centuries, the term "Tàixī (泰西)" – i.e., anything further west than the Arab world – was used to refer to the Western countries.

Since the mid-20th century, the term has mostly gone out of use for the region in international mass media outlets due to its perceived eurocentric connotations. North Asia is sometimes excluded due to cultural and ethnic differences.

Popularization
Among Western Europeans, prior to the colonial era, Far East referred to anything further east than the Middle East. In the 16th century, King John III of Portugal called India a "rich and interesting country in the Far East (Extremo Oriente)." The term was popularized during the period of the British Empire as a blanket term for lands to the east of British India.

In pre-World War I European geopolitics, Near East referred to the relatively nearby lands of the Ottoman Empire, Middle East denoted north-western Southern Asian region and Central Asia, and the Far East meant countries along the western Pacific Ocean and eastern Indian Ocean. Many European languages have analogous terms, such as the French (Extrême-Orient), Spanish (Extremo Oriente), Portuguese (Extremo Oriente), Italian (Estremo Oriente), German (Ferner Osten), Polish (Daleki Wschód), Norwegian (Det fjerne Østen) and Dutch (Verre Oosten).

Cultural and geographic meaning
Significantly, the term evokes cultural as well as geographic separation; the Far East is not just geographically distant, but also culturally exotic. It never refers, for instance, to the culturally Western nations of Australia and New Zealand, which lie even farther to the east of Europe than East Asia itself. This combination of cultural and geographic subjectivity was well illustrated in 1939 by Robert Menzies, a Prime Minister of Australia. Reflecting on his country's geopolitical situation with the onset of war, Menzies commented that: "The problems of the Pacific are different. What Great Britain calls the Far East is to us the near north."

Far East, in its usual sense, is comparable to terms such as the Orient (Latin for "East"), Eastern world, or simply the East, all of which may refer, broadly, to East and South-East Asia in general. Occasionally, albeit more in the past, the Russian Far East and South Asia have been deemed to be part of the Far East.

Commenting on such terms, John K. Fairbank and Edwin O. Reischauer (both professors of East Asian Studies at Harvard University) wrote, in East Asia: The Great Tradition: "When Europeans traveled far to the east to reach Cathay, Japan and the Indies, they naturally gave those distant regions the general name 'Far East.' Americans who reached China, Japan and Southeast Asia by sail and steam across the Pacific could, with equal logic, have called that area the 'Far West.' For the people who live in that part of the world, however, it is neither 'East' nor 'West' and certainly not 'Far.' A more generally acceptable term for the area is 'East Asia,' which is geographically more precise and does not imply the outdated notion that Europe is the center of the civilized world."

Today, the term remains in the names of some longstanding institutions, including the Far Eastern Federal University in Vladivostok, Far Eastern University in Manila, the Far East University in South Korea, and Far East, the periodical magazine of the Missionary Society of St. Columban. Furthermore, the United States and United Kingdom have historically incorporated Far East in the names of several military units and commands in the region, such as the British Royal Navy's Far East Fleet, for instance.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Far East


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Fawlty Towers

Fawlty Towers is a British television sitcom written by John Cleese and Connie Booth, originally broadcast on BBC Two in 1975 and 1979. Two series of six episodes each were made. The show was ranked first on a list of the 100 Greatest British Television Programmes drawn up by the British Film Institute in 2000 and, in 2019, it was named the greatest ever British TV sitcom by a panel of comedy experts compiled by the Radio Times.

The series is set in Fawlty Towers, a dysfunctional fictional hotel in the English seaside town of Torquay in Devon. The plots centre on the tense, rude and put-upon owner Basil Fawlty (Cleese), his bossy wife Sybil (Prunella Scales), the sensible chambermaid Polly (Booth) who often is the peacemaker and voice of reason, and the hapless and English-challenged Spanish waiter Manuel (Andrew Sachs). They show their attempts to run the hotel amidst farcical situations and an array of demanding and eccentric guests and tradespeople. The idea of the show came from Cleese after he stayed at the Gleneagles Hotel in Torquay, Devon, in 1970 (along with the rest of the Monty Python troupe), where he encountered the eccentric hotel owner Donald Sinclair. Stuffy and snobbish, Sinclair treated guests as though they were a hindrance to his running of the hotel (a waitress who worked for him stated "it was as if he didn't want the guests to be there"). Sinclair was the inspiration for Cleese's character Basil Fawlty.

In 1976 and 1980, Fawlty Towers won the British Academy Television Award for Best Scripted Comedy. In 1980, Cleese received the British Academy Television Award for Best Entertainment Performance, and, in a 2001 poll conducted by Channel 4, Basil Fawlty was ranked second on their list of the 100 Greatest TV Characters. The popularity of Fawlty Towers has endured, and it is often re-broadcast. The BBC profile for the series states that "the British sitcom by which all other British sitcoms must be judged, Fawlty Towers withstands multiple viewings, is eminently quotable ('don't mention the war') and stands up to this day as a jewel in the BBC's comedy crown."

A sequel series starring Cleese and his daughter Camilla was in development in February 2023. Cleese subsequently confirmed to GB News that the sequel series, unlike the original series, would not be broadcast on the BBC.

Origins
In May 1970, the Monty Python comedy group stayed at the now demolished Gleneagles Hotel in Torquay, Devon while filming on location in Paignton. John Cleese was fascinated with the behaviour of the owner, Donald Sinclair, later describing him as "the rudest man I've ever come across in my life". Among such behaviour by Sinclair was his criticism of Terry Gilliam's "too American" table etiquette and tossing Eric Idle's briefcase out of a window "in case it contained a bomb". Asked why anyone would want to bomb the hotel, Sinclair replied, "We've had a lot of staff problems". Michael Palin states Sinclair "seemed to view us as a colossal inconvenience". Rosemary Harrison, a waitress at the Gleneagles under Sinclair, described him as "bonkers" and lacking in hospitality, deeming him wholly unsuitable for a hotel proprietor. "It was as if he didn't want the guests to be there." Cleese and his then-wife Connie Booth stayed on at the hotel after filming, furthering their research of its owner. Demolished in 2015, the building was replaced by a new retirement home named Sachs Lodge in memory of Andrew Sachs who played Manuel in the sitcom and who died in 2016.

Cleese was a writer on the 1970s British TV sitcom Doctor in the House for London Weekend Television. An early prototype of the character that became known as Basil Fawlty was developed in an episode ("No Ill Feeling") of the third Doctor series (titled Doctor at Large). In this edition, the main character checks into a small-town hotel, his very presence seemingly winding up the aggressive and incompetent manager (played by Timothy Bateson) with a domineering wife. The show was broadcast on 30 May 1971.

Cleese said in 2008 that the first Fawlty Towers script he and Booth wrote was rejected by the BBC. At a 30th anniversary event honouring the show, Cleese said, "Connie and I wrote that first episode and we sent it in to Jimmy Gilbert, [the executive], whose job it was to assess the quality of the writing, said, (and I can quote [his note to me] fairly accurately,) 'This is full of clichéd situations and stereotypical characters and I cannot see it as being anything other than a disaster.' And Jimmy himself said, 'You're going to have to get them out of the hotel, John. You can't do the whole thing in the hotel.' Whereas, of course, it's in the hotel that the whole pressure cooker builds up."

Cleese was paid £6,000 for 43 weeks of work and supplemented his income by appearing in television advertisements. He states, "I have to thank the advertising industry for making this possible. Connie and I used to spend six weeks writing each episode and we didn't make a lot of money out of it. If it hadn't been for the commercials I wouldn't have been able to afford to spend so much time on the script."

Production
Although the series is set in Torquay, no part of it was shot in South West England. For the exterior filming, the Wooburn Grange Country Club in Wooburn Green, Buckinghamshire was used instead of a hotel. In several episodes of the series (notably "The Kipper and the Corpse", "The Anniversary", and "Basil the Rat"), the entrance gate at the bottom of the drive states the real name of the location. This listed building later served for a short time as a nightclub named "Basil's" after the series ended, before being destroyed by a fire in March 1991. The remnants of the building were demolished and a housing estate was built on the site. Few traces of the original site exist today.

Other location filming was done mostly around Harrow: firstly the 'damn good thrashing' scene in "Gourmet Night" in which Basil loses his temper and attacks his broken-down car with a tree branch. It was filmed at the T-junction of Lapstone Gardens and Mentmore Close (51.5811°N, -0.3091°W). Secondly the episode "The Germans", the opening shot is of Northwick Park Hospital. Thirdly "Gourmet Night"'s exterior of André's restaurant was at Preston Road (51.5734°N, -0.2944°W). It is now a Chinese and Indian restaurant called Wings, next to a launderette. Both Cleese and Booth were keen on every script being perfect, and some episodes took four months and required as many as ten drafts until they were satisfied.

Cleese said one of the reasons the series worked so well was the quality of the scripts and the care taken over the editing. He told a TV interviewer that while the average BBC half-hour comedy script had 65 pages, the ones for Fawlty Towers had between 135 and 140 pages. "We literally did twice as many camera cuts - average shows got 200, we used to have 400. So there was an enormous amount in there. The other thing is that they were very well constructed," he said. Once an episode was in the can, the editing process started. "We did anything between 20 and 25 hours editing each show. Almost every minute you see up on the screen, we spent one hour editing and it was only by doing that you could just tighten it up, just tighten it there and take out a line of dialogue, sometimes take out a repetition, they'll then lose two lines of dialogue there. That's what really got the pace on it."

The theme music was composed by Dennis Wilson. It was recorded by the highly respected Aeolian Quartet, who were asked by director John Davis to perform the piece badly, although in the end they did not.

Plot directions and examples
The series focuses on the exploits and misadventures of short-fused hotelier Basil Fawlty and his acerbic wife Sybil, as well as their employees: waiter Manuel, Polly Sherman, and, in the second series, chef Terry. The episodes typically revolve around Basil's efforts to "raise the tone" of his hotel and his increasing frustration at numerous complications and mistakes, both his own and those of others, which prevent him from doing so.

Much of the humour comes from Basil's overly aggressive manner, engaging in angry but witty arguments with guests, staff and, in particular, Sybil, whom he addresses (in a faux-romantic way) with insults such as "that golfing puff adder", "my little piranha fish" and "my little nest of vipers". Despite this, Basil frequently feels intimidated, Sybil being able to cow him at any time, usually with a short, sharp cry of "Basil!" At the end of some episodes, Basil succeeds in annoying (or at least bemusing) the guests and frequently gets his comeuppance.

The plots occasionally are intricate and always farcical, involving coincidences, misunderstandings, cross-purposes and meetings both missed and accidental. The innuendo of the bedroom farce is sometimes present (often to the disgust of the socially conservative Basil) but it is his eccentricity, not his lust, that drives the plots. The events test to the breaking point what little patience Basil has, sometimes causing him to have a near breakdown by the end of the episode.

The guests at the hotel typically are comic foils to Basil's anger and outbursts. Guest characters in each episode provide different characteristics (working class, promiscuous, foreign) that he cannot stand. Requests both reasonable and impossible test his temper. Even the afflicted annoy him, for example in the episode "Communication Problems", revolving around the havoc caused by the frequent misunderstandings between the staff and the hard-of-hearing Mrs. Richards. Near the end, Basil pretends to faint just at the mention of her name. This episode is typical of the show's careful weaving of humorous situations through comedy cross-talk. The show also uses mild black humour at times, notably when Basil is forced to hide a dead body ("2 dead, 25 to go.") and in his comments about Sybil ("Did you ever see that film, How to Murder Your Wife? ... Awfully good. I saw it six times.") and to Mrs Richards, ("May I suggest that you consider moving to a hotel closer to the sea? Or preferably in it.").

Basil's physical outbursts are primarily directed at Manuel, an emotional but largely innocent Spaniard whose confused English vocabulary causes him to make elementary mistakes. At times, Basil beats Manuel with a frying pan and smacks his forehead with a spoon. The violence towards Manuel caused rare negative criticism of the show. Sybil and Polly, on the other hand, are more patient and understanding towards Manuel; everyone's usual excuse to guests for his behaviour is, "He's from Barcelona"; Manuel even once used the excuse for himself.

Basil longs for a touch of class, sometimes playing recordings of classical music. In the first episode he is playing music by Brahms when Sybil remarks, after pestering him asking to do different tasks: "You could have them both done by now if you hadn't spent the whole morning skulking in there listening to that racket." Basil replies, with exasperation, "Racket?? That's Brahms! Brahms' Third Racket!" Basil often displays blatant snobbishness as he attempts to climb the social ladder, frequently expressing disdain for the "riff-raff", "cretins" and "yobbos" that he believes regularly populate his hotel. His desperation is readily apparent as he makes increasingly hopeless manoeuvres and painful faux pas in trying to curry favour with those he perceives as having superior social status. Yet he finds himself forced to serve those individuals that are "beneath" him. As such, Basil's efforts tend to be counter-productive, with guests leaving the hotel in disgust and his marriage (and sanity) stretching to breaking point.

Basil Fawlty
Basil Fawlty, played by John Cleese, is a cynical and snobbish misanthrope who is desperate to belong to a higher social class. He sees a successful hotel as a means of achieving this, yet his job forces him to be polite to people he despises.

He is intimidated by his wife Sybil Fawlty. He yearns to stand up to her, but his plans frequently conflict with her demands. She is often verbally abusive (describing him as "an ageing, brilliantined stick insect") but although he towers over her, he often finds himself on the receiving end of her temper, verbally and physically (as in "The Builders"), and it is only on one occasion when Sybil mistakenly believes he is stalking an attractive Australian guest that he finally snaps and stands up to her.

Basil usually turns to Manuel or Polly to help him with his schemes, while trying his best to keep Sybil from discovering them. However, Basil occasionally laments the time when there was passion in their relationship, now seemingly lost. Also, it appears he still does care for her and remains loyal to her, and actively resists the flirtations of a French guest in one episode. The penultimate episode, "The Anniversary", is about his efforts to put together a surprise anniversary party involving their closest friends. Things go wrong as Basil pretends the anniversary date does not remind him of anything though he pretends to have a stab at it by reeling off a list of random anniversaries, starting with the Battle of Agincourt, for which he receives a slap from Sybil, who becomes increasingly frustrated and angry. He continues guessing even after Sybil is out of earshot, and mentions other anniversaries (none of which happened on 17 April), including the Battle of Trafalgar and Yom Kippur, just to enhance the surprise. Sybil believes he really has forgotten, and leaves in a huff. In an interview in the DVD box set, Cleese claims this episode deliberately takes a slightly different tone from the others, fleshing out their otherwise inexplicable status as a couple.

In keeping with the lack of explanation about the marriage, not much is revealed of the characters' back-stories. It is known that Basil served in the British Army and saw action in the Korean War, possibly as part of his National Service. (John Cleese himself was only 13 when the Korean War ended, making the character of Basil at least five or six years older than he.) Basil exaggerates this period of his life, proclaiming to strangers, "I killed four men." To this Sybil jokes that "He was in the Catering Corps. He used to poison them." Basil is often seen wearing regimental and old-boy style ties, perhaps spuriously, one of which is in the colours of the Army Catering Corps. He also claims to have sustained a shrapnel injury to his leg; it tends to flare up at suspiciously convenient times. The only person towards whom Basil consistently exhibits tolerance and good manners is the old and senile Major Gowen, a veteran of one of the world wars (which one is never specified, though he once mentions to Mrs Peignoir that he was in France in 1918) who permanently resides at the hotel. When interacting with Manuel, Basil displays a rudimentary knowledge of Spanish (Basil states that he "learned classical Spanish, not the strange dialect he [Manuel] seems to have picked up"); this knowledge is also ridiculed, as in the first episode in which a guest, whom Basil has immediately dismissed as working-class, communicates fluently with Manuel in Spanish after Basil is unable to do so.

Cleese described Basil as thinking that "he could run a first-rate hotel if he didn't have all the guests getting in the way" and as being "an absolutely awful human being" but says that in comedy if an awful person makes people laugh they unaccountably feel affectionate towards him. Indeed, he is not entirely unsympathetic. The "Hotel Inspectors" and "Gourmet Night" episodes feature guests who are shown to be deeply annoying, with constant and unreasonable demands. In "Gourmet Night" the chef gets drunk and is unable to cook dinner, leaving Basil to scramble in an attempt to salvage the evening. Much of the time, Basil is an unfortunate victim of circumstance.

Sybil Fawlty
Sybil Fawlty, played by Prunella Scales, is Basil's wife. Energetic and petite, she prefers a working wardrobe of tight skirt-suits in shiny fabrics and sports a tower of permed hair augmented with hairpieces and wigs necessitating the use of overnight curlers. She often is a more effective manager of the hotel, making sure Basil gets certain jobs done or stays out of the way when she is handling difficult guests. Typically when Basil is on the verge of a meltdown due to a crisis (usually of his own making), it is Sybil who steps in to clear up the mess and bring some sense to the situation. Despite this, she rarely participates directly in the running of the hotel. During busy check-in sessions or meal times, while everyone else is busy working, Sybil is frequently talking on the phone to one of her friends with her phrase "Oohhh, I knoooooooow" or chatting to customers. She has a distinctive conversational tone and braying laugh, which Basil compares to "someone machine-gunning a seal". Being his wife, she is the only regular character who refers to Basil by his first name. When she barks his name at him, he flinchingly freezes in his tracks.

Basil refers to her by a number of epithets, occasionally to her face, including "that golfing puff-adder", "the dragon", "toxic midget", "the sabre-toothed tart", "my little kommandant", "my little piranha fish", "my little nest of vipers" and "you rancorous, coiffured old sow". Despite these nasty nicknames, Basil is terrified of her. The 1979 episode "The Psychiatrist" contains the only time he loses patience and snaps at her (Basil: "Shut up, I'm fed up." Sybil: "Oh, you've done it now.").

Prunella Scales speculated in an interview for The Complete Fawlty Towers DVD box set that Sybil married Basil because his origins were of a higher social class than hers.

Polly Sherman
Polly Sherman, played by Connie Booth, is a waitress and general assistant at the hotel with artistic aspirations. She is the most competent of the staff and the voice of sanity during chaotic moments, but is frequently embroiled in ridiculous masquerades as she loyally attempts to aid Basil in trying to cover up a mistake or keep something from Sybil.

In "The Anniversary" she snaps and refuses to help Basil out when he wants her to impersonate Sybil in the semi-darkness of her bedroom in front of the Fawltys' friends, Basil having dug himself into a hole by claiming Sybil was ill instead of admitting she had stormed out earlier in annoyance with him. Polly finally agrees, but only on condition that Basil lends her money to purchase a car, which he has previously refused to do.

Polly generally is good-natured but sometimes shows her frustration, and has odd moments of malice. In "The Kipper and the Corpse", the pampered Shih Tzu dog of an elderly guest bites Polly and Manuel. As revenge, Polly laces the dog's sausages with black pepper and Tabasco sauce ("bangers à la bang"), making it ill and eventually killing it.

Despite her part-time employment (during meal times), Polly frequently is saddled with many other duties, including as manager in "The Germans" when Sybil and Basil are incapacitated. In the first series, Polly is said to be an art student who, according to Basil, has spent three years at college. In "Gourmet Night", she is seen drawing a sketch (presumably of Manuel), which everyone but Basil immediately recognises and she sells it to the chef for 50p. Polly is not referred to as a student in the second series, although in both series she is shown to have a flair for languages, displaying ability in both Spanish and German. In "The Germans", Basil alludes to Polly's polyglot inclination by saying that she does her work "while learning two Oriental languages". Like Manuel, she has a room of her own at the hotel.

Manuel
Manuel, a waiter played by Andrew Sachs, is a well-meaning but disorganised and confused Spaniard from Barcelona with a poor grasp of the English language and customs. He is verbally and physically abused by his boss. When told what to do, he often responds, "¿Qué?" ("What?"). Manuel's character is used to demonstrate Basil's instinctive lack of sensitivity and tolerance. Every episode involves Basil becoming enraged at Manuel's confusion at his boss's bizarre demands and even basic requests. Manuel is afraid of Fawlty's quick temper and violent assaults, yet often expresses his appreciation for being given employment. He is relentlessly enthusiastic and is proud of what little English he knows.

During the series, Sachs was seriously injured twice. Cleese describes using a real metal pan to knock Manuel unconscious in "The Wedding Party", although he would have preferred to use a rubber one. The original producer and director, John Howard Davies, said that he made Basil use a metal one and that he was responsible for most of the violence on the show, which he felt was essential to the type of comical farce they were creating. Later, when Sachs's clothes were treated to give off smoke after he escaped the burning kitchen in "The Germans", the corrosive chemicals ate through them and gave Sachs severe burns.

Manuel's exaggerated Spanish accent is part of the humour of the show. In fact, Sachs's original language was German; he emigrated to Britain as a child.

The character's nationality was switched to Italian (and the name to Paolo) for the Spanish dub of the show, while in Catalonia and France, Manuel is a Mexican.

Other regular characters and themes

 * Terry Hughes, played by Brian Hall, is the hotel chef throughout the second series. A sly, somewhat shifty Cockney, he is nonetheless a competent chef ("I 'ave been to catering school!"). His cooking methods are occasionally somewhat casual, which frustrates and worries the neurotic Basil; and he also has limited concern for food and kitchen hygiene, claiming that "What the eye don't see, the chef gets away with" and that "the better the kitchen, the filthier it is." He used to work in Dorchester (not at The Dorchester, as a guest wrongly infers). In "The Anniversary" Terry and Manuel come to blows since Terry doesn't like anyone overshadowing him as a cook, so he proceeds to sabotage the paella Manuel is making for Basil and Sybil, leading to fisticuffs at the end of the episode. Cleese himself told Hall to portray Terry as if he were on the run from the police.
 * Major Gowen, played by Ballard Berkeley, is a slightly senile, amiable old soldier who is a permanent resident of the hotel. He is one of the few guests whom Basil seems to like. This is because he has the establishment status that Basil craves. He usually wears the Royal Artillery jagged-striped tie, and once mentions to Mrs. Peignoir being in France in 1918. He often is introduced as their "oldest resident" and in the episode "Waldorf Salad" Basil reveals that the Major has lived there for seven years. He enjoys talking about the world outside, especially the cricket scores and workers' strikes (the frequent strikes at British Leyland during the time of the series' original transmission were often mentioned), and is always on the lookout for the newspaper. In the episode "The Germans" he shows he has trouble forgiving the Germans because of the wars. The best he can say is that German women make good card players. In the same episode, he also demonstrates his outdated racial attitudes when he comments about the ethnic difference between "wogs" and "niggers". Despite his good intentions, the Major can cause Basil's plans to go awry, notably in the episode "Communication Problems" in which Basil tries his best to keep secret from Sybil the money he won in a bet.
 * Miss Tibbs and Miss Gatsby, played by Gilly Flower and Renee Roberts, are the other two permanent residents. Seemingly inseparable, these sweet-natured, dotty spinsters appear to have taken a fancy to Basil, feeling that they need to take care of him. In response, Basil vacillates between superficial charm and blunt rudeness during his conversations with them.
 * Audrey is Sybil's lifelong best friend, and is mostly acknowledged during gossipy telephone calls. Talking with her is a refuge for Sybil. When times get tough for Audrey, who has a dysfunctional relationship with her husband George, Sybil will offer solutions and guidance, often resulting in the catchphrase "Ohhh, I knowwww..." when she tries to commiserate with Audrey's problems. In Audrey's one on-screen appearance, in "The Anniversary", she is played by actress Christine Shaw. Basil tells Major Gowen that he thinks she is a "dreadful woman".
 * A running gag throughout the two series is the rearranged letters of the "Fawlty Towers" hotel sign which is shown at the beginning of every episode except "The Germans", when a hospital exterior is used as an establishing shot. In series one, the letters slowly fall from the sign due to lack of maintenance. In series two, the letters are re-arranged into a series of deliberate anagrams. The paperboy, though rarely seen, is revealed at the beginning of "The Psychiatrist" to be the prankster who rearranges the letters on the sign to sometimes crude phrases.
 * Terence Conoley appears in two episodes as entirely different characters. In "A Touch of Class" he plays Mr. Wareing, and in "Waldorf Salad" he portrays Mr. Johnston.

Episodes

The first episode of Fawlty Towers was recorded as a pilot on 24 December 1974, the rest of the series being recorded later in 1975. It was then originally broadcast on 19 September. The 12th and final episode was first shown on 25 October 1979. The first series was directed by John Howard Davies, the second by Bob Spiers. Both had their premieres on BBC Two.

When originally transmitted, the individual episodes had no on-screen titles. The ones in common currency were first used for the VHS release of the series in the 1980s. There were working titles, such as "USA" for "Waldorf Salad", "Death" for "The Kipper and the Corpse" and "Rat" for "Basil the Rat", which have been printed in some programme guides. In addition, some of the early BBC audio releases of episodes on vinyl and cassette included other variations, such as "Mrs. Richards" and "The Rat" for "Communication Problems" and "Basil the Rat" respectively.

In 2022, a “lost” scene cut from the episode “The Anniversary” (that went unfilmed) was uncovered as part of a script copy, featuring Basil climbing out his bedroom window to avoid sex with a drunken Sybil, who had wanted to make up.

It has long been rumoured that a 13th episode of the series was written and filmed, but never progressed further than a rough cut. Lars Holger Holm, author of the book Fawlty Towers: A Worshipper's Companion, has made detailed claims about the episode's content, but he provides no concrete evidence of its existence.

On the subject of whether more episodes would be produced, Cleese said (in an interview for the complete DVD box set, which was republished in the book Fawlty Towers Fully Booked) that he once had the genesis of a feature-length special—possibly sometime during the mid-1990s. The plot, never fleshed out beyond his initial idea, would have revolved around the chaos that a now-retired Basil typically caused as he and Sybil flew to Barcelona to visit their former employee Manuel and his family. Of the idea, Cleese said:

We had an idea for a plot which I loved. Basil was finally invited to Spain to meet Manuel's family. He gets to Heathrow and then spends about 14 frustrating hours waiting for the flight. Finally, on the plane, a terrorist pulls a gun and tries to hijack the thing. Basil is so angry he overcomes the terrorist, and when the pilot says, "We have to fly back to Heathrow" Basil says, "No, fly us to Spain or I'll shoot you." He arrives in Spain, is immediately arrested, and spends the entire holiday in a Spanish jail. He is released just in time to go back on the plane with Sybil.

It was very funny, but I couldn't do it at the time. Making "Fawlty Towers" work at 90 minutes was a very difficult proposition. You can build up the comedy for 30 minutes, but at that length there has to be a trough and another peak. It doesn't interest me. I don't want to do it.

Cleese also may have been reluctant because of Connie Booth's unwillingness to be involved. She had practically retreated from public life after the show finished (and had been initially unwilling to collaborate on a second series, which explains the four-year gap between productions).

The decision by Cleese and Booth to quit before a third series has often been lauded as it ensured the show's successful status would not be weakened with later, lower-quality work. Subsequently, it has inspired the makers of other shows to do likewise. Ricky Gervais and Stephen Merchant refused to make a third series of either The Office or Extras, citing Fawlty Towers'  short lifespan. Rik Mayall, Ben Elton and Lise Mayer, the writers behind The Young Ones, which also ran for only two series (each with six episodes), used this explanation as well. Victoria Wood also indicated this influenced her decision to limit dinnerladies to 16 episodes over two series.

The origins, background and eventual cancellation of the series were later humorously referenced in 1987's The Secret Policeman's Third Ball in a sketch in which Hugh Laurie and Stephen Fry present Cleese—whom they comically misname "Jim Cleese"—with a Dick Emery Lifetime Achievement Award ("Silver Dick") for his contributions to comedy, then launch into a comical series of questions regarding the show, including Cleese's marriage and divorce from Booth, innocently ridiculing Cleese and reducing him to tears, to a point at which he gets on his knees and crawls off the stage while crying.

Series 2 (1979)
The second series was transmitted three-and-a-half years later, with the first episode being broadcast on 19 February 1979. Due to an industrial dispute at the BBC, which resulted in a strike, the final episode was not completed until well after the others, being finally shown as a one-off instalment on 25 October 1979. The cancelled episode on 19 March was replaced with a repeat of "Gourmet Night" from series 1. In the second series the anagrams were created by Ian McClane, Bob Spier's assistant floor manager. However, the only one which is actually a true anagram for the hotel's name is "Flowery Twats", created for "The Anniversary"

Contemporary
At first, the series was not held in particularly high esteem. The Daily Mirror's review of the show in 1975 had the headline "Long John Short On Jokes". One critic of the show was Richard Ingrams, then television reviewer for The Spectator, who wrote a caustic piece condemning the programme. Cleese got his revenge by naming one of the guests in the second series "Mr. Ingrams", who is caught in his room with a blow-up doll. Eventually, though, as the series began to gain popularity, critical acclaim followed. Clive James writing in The Observer said the second episode had him "retching with laughter."

Retrospective
On Rotten Tomatoes, Fawlty Towers has an aggregate score of 100% based on 13 critic reviews. The website's consensus reads: "Fawlty Towers looms large over British comedy with John Cleese's impeccably hapless performance and an endless array of exuberant slapstick—making for a supremely stimulating chuckler."

In an interview for the "TV Characters" edition of Channel 4's "talking heads" strand 100 Greatest (in which Basil placed second, between Homer Simpson and Edmund Blackadder), TV critic A. A. Gill theorised that the initially muted response may have been caused by Cleese seemingly ditching his label as a comic revolutionary—earned through his years with Monty Python—to do something more traditional.

In a list of the 100 Greatest British Television Programmes drawn up by the British Film Institute in 2000, voted for by industry professionals, Fawlty Towers was placed first. It was also voted fifth in the "Britain's Best Sitcom" poll in 2004, and second only to Frasier in The Ultimate Sitcom poll of comedy writers in January 2006. Basil Fawlty came top of the Britain's Funniest Comedy Character poll, held by Five on 14 May 2006. In 1997, "The Germans" was ranked No. 12 on TV Guide's 100 Greatest Episodes of All Time. Named in Empire magazine's 2016 list of the greatest TV shows of all time, the entry states,

"One of British TV's greatest ever sitcoms, the central question of Fawlty Towers—why Basil Fawlty, the world's least hospitable man would go into hospitality in the first place – remains tantalisingly unanswered across 12 kipper-serving, Siberian hamster-hunting, German-baiting episodes. A straight zero on TripAdvisor, the very layout of Fawlty Towers itself offers comedy gold as Basil (John Cleese), his wife Sybil (Prunella Scales), waitress Polly (Connie Booth) and poor, benighted Manuel (Andrew Sachs) manoeuvre themselves (and the odd corpse) around its dowdy interior without ruining anyone's stay. Basil, needless to say, fails. Often and hilariously."

Awards and accolades
Three British Academy Television Awards (BAFTAs) were awarded to people for their involvement with the series. Both of the series were awarded the BAFTA in the category Best Scripted Comedy, the first being won by John Howard Davies in 1976, and the second by Douglas Argent and Bob Spiers in 1980. In 1980, Cleese received the BAFTA for Best Entertainment Performance.

In a list drawn up by the British Film Institute in 2000, voted by industry professionals, Fawlty Towers was named the best British television series of all time.

Legacy

John Lennon was a fan of the show. He said in 1980: "I love Fawlty Towers. I'd like to be in that. [It's] the greatest show I've seen in years... what a masterpiece, a beautiful thing." Kate Bush stated, "I still think Fawlty Towers is the best sitcom ever." Filmmaker Martin Scorsese has remarked he is a great fan of Fawlty Towers and named "The Germans" as his favourite episode. He described the scene with Basil impersonating Hitler as "so tasteless, it's hilarious".

Remakes, adaptations and reunions
Three attempted remakes of Fawlty Towers were started for the American market, with two making it into production. The first, Chateau Snavely, starring Harvey Korman and Betty White, was produced by ABC for a pilot in 1978, but the transfer from coastal hotel to highway motel proved too much and the series was never produced. The second, also by ABC, was Amanda's, starring Bea Arthur, notable for switching the sexes of its Basil and Sybil equivalents. It also failed to attract a major audience and was dropped after ten episodes had been aired, despite 13 completed episodes. A third remake, called Payne (produced by and starring John Larroquette), aired in 1999, but was cancelled shortly afterwards. Nine episodes were produced, of which eight aired on American television (though the complete run was broadcast overseas). A German pilot was made in 2001, named Zum letzten Kliff (To the last cliff), but no further episodes were made after its first series.

Popular American sitcoms 3rd Rock from the Sun and Cheers (in both of which Cleese made guest appearances) have cited Fawlty Towers as an inspiration, especially regarding its depiction of a dysfunctional workplace "family". Arthur Mathews and Graham Linehan have cited Fawlty as a major influence on their sitcom Father Ted. Guest House on Pakistan's PTV also resembled the series.

Several of the characters have made other appearances, as spinoffs or in small cameo roles. In 1981, in character as Manuel, Sachs recorded his own version of the Joe Dolce cod-Italian song "Shaddap You Face" (with the B-side "Waiter, There's a Spanish Flea in My Soup") but the record was not released because Dolce took out an injunction: he was about to issue his version in Britain. Sachs also portrayed a Manuel-like character in a series of British TV advertisements for life insurance. Gilly Flower and Renee Roberts, who played elderly ladies Miss Tibbs and Miss Gatsby in the series, reprised their roles in a 1983 episode of Only Fools and Horses. In 2006, Cleese played Basil for the first time in 27 years, for an unofficial England 2006 World Cup song, "Don't Mention the World Cup", taking its name from the phrase, "Don't mention the war," which Basil used in the episode "The Germans". In 2007, Cleese and Sachs reprised their roles for a six-episode corporate business video for the Norwegian oil company Statoil. In the video, Fawlty is running a restaurant called "Basil's Brasserie" while Manuel owns a Michelin-starred restaurant in London.

In November 2007, Scales returned to play Sybil in a series of sketches for the BBC's annual Children in Need charity telethon. The character was seen taking over the management of the eponymous hotel from the BBC drama series Hotel Babylon, interacting with characters from that programme as well as other 1970s sitcom characters. The character of Sybil was used by Cleese's permission. In 2007, the Los Angeles Film School produced seven episodes of Fawlty Tower Oxnard, starring Robert Romanus as Basil.

At a 2009 reunion event for the Gold channel as Fawlty Towers: Re-Opened, Cleese said that the cast would never make another episode of the series because they are "too old and tired" and expectations would be too high.

In 2016, Cleese reprised his role as Basil in a series of TV adverts for High Street optician chain Specsavers. The same year, Cleese and Booth reunited to create and co-write the official theatrical adaptation of Fawlty Towers, which premiered in Melbourne at the Comedy Theatre. It was critically well received, subsequently embarking on a successful tour of Australia. Cleese was intimately involved in the creation of the stage version from the beginning, including casting. He visited Australia to promote, as well as oversee its success. Melbourne was chosen to premiere the adaptation due to the series' enduring popularity in Australia, and also as it had become a popular international test market for large-scale theatrical productions in recent years, having recently been the city where the revised Love Never Dies and the new King Kong also premiered. Cleese also noted he did not believe the London press would give the adaptation fair, unbiased reviews, so he deliberately chose to premiere it elsewhere. On 2 February 2024, it was announced that the 2016 stage show based on Fawlty Towers would launch at London's Apollo Theatre later that year.

A sequel series was announced in February 2023, written by and starring Cleese and his daughter Camilla. The show is under development at Castle Rock Entertainment, with Matthew George, Rob Reiner, Michele Reiner and Derrick Rossi as executive producers. The premise sees Basil (Cleese) trying to operate Fawlty Towers with help from his long-lost daughter (to be played by Camilla) and adjusting to the modern world. Cleese said in a July 2023 interview that his daughter had only "half an idea" for the series, and that the production company had been too hasty in announcing the revival. Cleese said that the series would "probably" be set in the Caribbean.

Overseas
In 1977 and 1978 alone, the original TV show was sold to 45 stations in 17 countries and was the BBC's best-selling overseas programme for that year. Fawlty Towers became a huge success in almost all countries in which it aired. Although it initially was a flop in Spain, largely because of the portrayal of the Spanish waiter Manuel, it was successfully resold with the Manuel character's nationality changed to Italian except in Spain's Catalan region where Manuel was Mexican. To show how badly it translated, Clive James picked up a clip containing Manuel's "¿Qué?" phrase to show on Clive James on Television in 1982. The series also briefly was broadcast in Italy in the 1990s on the satellite channel Canal Jimmy, in the original English with Italian subtitles.

In Australia, the show originally was broadcast on ABC Television, the first series in 1976 and the second series in 1980. The show then was sold to the Seven Network where it has been repeated numerous times.

Audio releases
Four albums were released by BBC Records on vinyl LP and cassette. These consisted of the original television soundtracks, and from the second album onwards had additional voice-over from Sachs (in character as Manuel) describing scenes which relied on visual humour.

The first album, simply titled Fawlty Towers, was released in 1979 and contained audio from "Communication Problems" (as "Mrs Richards") and "Hotel Inspectors". The second album, Second Sitting, was released in 1981 and contained audio from "Basil the Rat" (as "The Rat") and "The Builders". Both of these first two albums reached the Top 30 of the UK Albums Chart.

At Your Service was released in 1982, and contained audio from "The Kipper and the Corpse" (as "Death") and "The Germans" (as "Fire Drill"). Finally, A La Carte was released in 1983, and contained audio from "Waldorf Salad" (as "The Americans") and "Gourmet Night".

The albums were re-released as double-cassette packs under the titles Fawlty Towers 1 and Fawlty Towers 2 in 1988. The remaining four episodes did not get an audio-only release until 1994 on audio cassette as Fawlty Towers 3.

The first CD release of the audio versions was in a box set in 2003, titled Fawlty Towers—The Collector's Edition, which included spoken introductions to each episode by Cleese, and an interview with Scales and Sachs.

The four vinyl records were reissued in a limited edition box set, along with the remaining four episodes on vinyl for the first time, for Record Store Day in 2021.

Home media
Fawlty Towers was originally released by BBC Video in 1984, with three episodes on each of four tapes. Each was edited with the credits from all three episodes put at the end of the tape. A LaserDisc containing all episodes spliced together as a continuous episode was released in the U.S. on 23 June 1993. It was re-released in 1994, unedited but digitally remastered. It also was re-released in 1997 with a special interview with Cleese. Fawlty Towers—The Complete Series was released on DVD on 16 October 2001, available in regions 1, 2 and 4. A "Collector's Edition" is available in region 2.

The original DVD contained a slightly edited version of "The Kipper and the Corpse", in which Basil's line "Is it your legs?" (said to Mr Lehman when asking why he wants breakfast in bed) is missing. This line was restored in subsequent remasters.

Series one was released on UMD Video for PSP. In July 2009, BBC America announced a DVD reissue, released on 20 October 2009. Titled Fawlty Towers Remastered: Special Edition, contains commentary by Cleese on every episode as well as remastered video and audio. All episodes are available as streamed video-on-demand via Britbox, Netflix and Amazon Prime Video. Additionally, both series are available for download on iTunes. In 2021 all episodes were made available on the BBC iPlayer.

Computer game
A Fawlty Towers game was released on PC in 2000 and featured a number of interactive games, desktop-customizing content, and clips from the show.

Books
The original scripts were released in a hardback book by Methuen, The Complete Fawlty Towers, in 1988.

title: Fawlty Towers


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False friend

In linguistics, a false friend is a word in a different language that looks or sounds similar to a word in a given language, but differs significantly in meaning. Examples of false friends include English embarrassed and Spanish embarazado 'pregnant'; English parents versus Portuguese parentes and Italian parenti (both meaning 'relatives'); English demand and French demander 'ask'; and English gift, German Gift 'poison', and Norwegian gift 'married'.

The term was introduced by a French book, Les faux amis: ou, Les trahisons du vocabulaire anglais (False friends, or, the betrayals of English vocabulary), published in 1928.

As well as producing completely false friends, the use of loanwords often results in the use of a word in a restricted context, which may then develop new meanings not found in the original language. For example, angst means 'fear' in a general sense (as well as 'anxiety') in German, but when it was borrowed into English in the context of psychology, its meaning was restricted to a particular type of fear described as "a neurotic feeling of anxiety and depression". Also, gymnasium meant both 'a place of education' and 'a place for exercise' in Latin, but its meaning became restricted to the former in German and to the latter in English, making the expressions into false friends in those languages as well as in Ancient Greek, where it started out as 'a place for naked exercise'.

Definition and origin
False friends are bilingual homophones or bilingual homographs, i.e., words in two or more languages that look similar (homographs) or sound similar (homophones), but differ significantly in meaning.

The origin of the term is as a shortened version of the expression "false friend of a translator", the English translation of a French expression (faux amis du traducteur) introduced by Maxime Kœssler and Jules Derocquigny in their 1928 book, with a sequel, Autres Mots anglais perfides.

Causes
From the etymological point of view, false friends can be created in several ways.

Shared etymology

If language A borrowed a word from language B, or both borrowed the word from a third language or inherited it from a common ancestor, and later the word shifted in meaning or acquired additional meanings in at least one of these languages, a native speaker of one language will face a false friend when learning the other. Sometimes, presumably both senses were present in the common ancestor language, but the cognate words took on different restricted senses in Language A and Language B.

In loanwords
Actual, which in English is usually a synonym of real, has a different meaning in other European languages, in which it means 'current' or 'up-to-date', and has the logical derivative as a verb, meaning 'to make current' or 'to update'. Actualise (or 'actualize') in English means 'to make a reality of'.

The Italian word confetti 'sugared almonds' has acquired a new meaning in English, French and Dutch; in Italian, the corresponding word is coriandoli.

English and Spanish, both of which have borrowed from Ancient Greek and Latin, have multiple false friends, such as: English and Japanese also have diverse false friends, many of them being wasei-eigo and gairaigo words.

In native words
The word friend itself has cognates in the other Germanic languages, but the Scandinavian ones (like Swedish frände, Danish frænde) predominantly mean 'relative'. The original Proto-Germanic word meant simply 'someone whom one cares for' and could therefore refer to both a friend and a relative, but it lost various degrees of the 'friend' sense in the Scandinavian languages, while it mostly lost the sense of 'relative' in English (the plural friends is still, rarely, used for "kinsfolk", as in the Scottish proverb Friends agree best at a distance, quoted in 1721).

The Estonian and Finnish languages are related, which gives rise to false friends such as swapped forms for south and south-west:

Or Estonian vaim 'spirit; ghost' and Finnish vaimo 'wife'; or Estonian huvitav 'interesting' and Finnish huvittava 'amusing' or Estonian koristaja 'a cleaner' and Finnish koristaja 'a decorator'.

A high level of lexical similarity exists between German and Dutch, but shifts in meaning of words with a shared etymology have in some instances resulted in 'bi-directional false friends':

The meanings could diverge significantly. For example, the Proto-Malayo-Polynesian word *qayam "domesticated animal" became specialized in descendant languages: Malay/Indonesian ayam 'chicken', Cebuano ayam 'dog', and Gaddang ayam 'pig'.

Homonyms
In Swedish, the word rolig means 'fun': ett roligt skämt 'a funny joke', while in the closely related languages Danish and Norwegian it means 'calm' (as in "he was calm despite all the commotion around him"). However, the Swedish original meaning of 'calm' is retained in some related words such as ro 'calmness', and orolig 'worrisome, anxious', literally 'un-calm'. The Danish and Norwegian word semester means term (as in school term), but the Swedish word semester means holiday. The Danish word frokost means lunch, while the Norwegian word frokost and the Swedish word frukost both mean breakfast.

Pseudo-anglicisms
Pseudo-anglicisms are new words formed from English morphemes independently from an analogous English construct and with a different intended meaning.

Japanese is replete with pseudo-anglicisms, known as wasei-eigo 'Japan-made English'.

Semantic change
In bilingual situations, false friends often result in a semantic change—a real new meaning that is then commonly used in a language. For example, the Portuguese humoroso 'capricious' changed its meaning in American Portuguese to 'humorous', owing to the English surface-cognate humorous.

The American Italian fattoria lost its original meaning, "farm", in favor of "factory", owing to the phonetically similar surface-cognate English factory (cf. Standard Italian fabbrica, 'factory'). Instead of the original fattoria, the phonetic adaptation American Italian farma became the new signifier for "farm" (Weinreich 1963: 49; see "one-to-one correlation between signifiers and referents").

Due to the closeness between Italian terra rossa 'red soil' and Portuguese terra roxa 'purple soil', Italian farmers in Brazil used terra roxa to describe a type of soil similar to the red Mediterranean soil. The actual Portuguese word for "red" is vermelha. Nevertheless, terra roxa and terra vermelha are still used interchangeably in Brazilian agriculture.

Quebec French is also known for shifting the meanings of some words toward those of their English cognates, but such words are considered false friends in European French. For example, éventuellement is commonly used as "eventually" in Quebec but means "perhaps" in Europe.

This phenomenon is analyzed by Ghil'ad Zuckermann as "(incestuous) phono-semantic matching".

title: False friend


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False cognate

False cognates are pairs of words that seem to be cognates because of similar sounds and meaning, but have different etymologies; they can be within the same language or from different languages, even within the same family. For example, the English word dog and the Mbabaram word dog have exactly the same meaning and very similar pronunciations, but by complete coincidence. Likewise, English much and Spanish mucho came by their similar meanings via completely different Proto-Indo-European roots, and same for English have and Spanish haber. This is different from false friends, which are similar-sounding words with different meanings, and may or may not be cognates.

Even though false cognates lack a common root, there may still be an indirect connection between them (for example by phono-semantic matching or folk etymology).

Phenomenon
The term "false cognate" is sometimes misused to refer to false friends, but the two phenomena are distinct. False friends occur when two words in different languages or dialects look similar, but have different meanings. While some false friends are also false cognates, many are genuine cognates (see False friends § Causes). For example, English pretend and French prétendre are false friends, but not false cognates, as they have the same origin.

"Mama and papa" type
The basic kinship terms mama and papa comprise a special case of false cognates.

Examples
Note: Some etymologies may be simplified to avoid overly long descriptions.

False cognates used in the coinage of new words
The coincidental similarity between false cognates can sometimes be used in the creation of new words (neologization). For example, the Hebrew word דַּל dal ("poor") (which is a false cognate of the phono-semantically similar English word dull) is used in the new Israeli Hebrew expression אין רגע דל en rega dal (literally "There is no poor moment") as a phono-semantic matching for the English expression Never a dull moment.

Similarly, the Hebrew word דיבוב dibúv ("speech, inducing someone to speak"), which is a false cognate of (and thus etymologically unrelated to) the phono-semantically similar English word dubbing, is then used in the Israeli phono-semantic matching for dubbing. The result is that in Modern Hebrew, דיבוב dibúv means "dubbing".

title: False cognate