Tru Physics

Fick’s Laws of Diffusion

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Introduction

Diffusion, the process through which particles spread out from a region of higher concentration to a region of lower concentration, is a fundamental transport phenomenon in physics. It is quantitatively described by Fick’s laws, named after the German physicist Adolf Fick who first stated them in 1855.

Fick’s First Law

Fick’s first law relates the diffusive flux to the concentration under the assumption of steady state. It states that the flux of particles J is proportional to the concentration gradient. Mathematically, this is expressed as:

J = -D \dfrac{\partial C}{\partial x}

where D is the diffusion coefficient (a positive constant), C is the concentration of particles, and \dfrac{\partial C}{\partial x} represents the concentration gradient.

Fick’s Second Law

Fick’s second law considers the time-dependent nature of diffusion, factoring in how the concentration changes over time. It states that the rate of change of concentration with respect to time is proportional to the second derivative of the concentration with respect to position. Mathematically, this is expressed as:

\dfrac{\partial C}{\partial t} = D \dfrac{\partial^2 C}{\partial x^2}

This is a partial differential equation known as the diffusion equation.

Determining the Diffusion Coefficient

The diffusion coefficient D depends on several factors including the temperature, the medium, and the nature of the particles. In dilute gases, D is described by the Chapman-Enskog equation:

D = \dfrac{3}{16} \dfrac{k_B T}{\pi m} \dfrac{1}{\sigma^2 \Omega_D}

where k_B is the Boltzmann constant, T is the temperature, m is the molar mass of the gas, \sigma is the collision diameter, and \Omega_D is the diffusion collision integral.

Diffusion in Multi-Dimensional Systems

In more than one dimension, Fick’s laws take a vector form. For example, in three dimensions, Fick’s first law becomes:

\vec{J} = -D \nabla C

and Fick’s second law becomes:

\dfrac{\partial C}{\partial t} = D \nabla^2 C

where \nabla is the gradient operator and \nabla^2 is the Laplacian operator.

Conclusion

Fick’s laws of diffusion provide the basic framework to understand the transport of particles in various fields, ranging from physics and engineering to biology and environmental science. They are at the core of phenomena such as heat conduction, osmosis, and the spread of pollutants. An in-depth understanding of these laws opens up avenues for studying complex systems and phenomena where diffusion plays a crucial role.

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