Chapter 5, Problem 5.88CP

Interpretation Introduction

Interpretation:

Using computer programming the value of Concentration profile and time required for given condition should be determined.

Concept introduction:

Equation used for calculating the diffusion is,

Fick's Law of diffusion: This law states that molar flux is directly proportional to concentration gradient. The law is stated as:

  $\text{J}\text{=}-\text{D}\frac{{\text{dC}}_{\text{A}}}{\text{dx}}$

Where,

J is the molar flux defined as the number of atoms passing per unit area per unit time.

D is the diffusion coefficient in ${\text{cm}}^{\text{2}}\text{/sec}$.

$\frac{{\text{dC}}_{\text{A}}}{\text{dx}}$ is the concentration gradient in $\frac{\text{atoms}}{{\text{cm}}^{\text{3}}\text{.cm}}$.

Factors affection diffusion are as follows:

1. Temperature
2. Diffusion coefficient

The following equation is stated as:

  $\text{D}={\text{D}}_{\text{0}}\exp \left(\frac{\text{-Q}}{\text{RT}}\right)$

Where,

Q is the activation energy in calorie/ mole.

R is universal gas constant in $\frac{\text{cal}}{\text{mole}\text{K}}$.

T is the absolute temperature in kelvin.

  ${\text{D}}_{\text{0}}$ is constant for given diffusion system.

Fick's Second law is applicable, which is used to define the non- steady state and diffusion of atoms. The equation is stated as:

  $\frac{c_s-c_x}{c_s-c_0}=erf\left(\frac{x}{2\sqrt{Dt}}\right)$

Where,

  $c_s$ is the constant concentration of the diffusing atoms at the surface of the material.

  $c_x$ is the concentration of diffusing atom at x having time t.

  $c_0$ is the initial concentration of the diffusing atom.

erf is the error function.