A thin layer of material B (of total amount m) is plated on the end faces of two long rods of material A. These are then joined together on the plated side (see the figure below) and heated to a high temperature. Assuming the diffusion coefficient of B in A is D, the composition profile $C_B$ along the rod axis x after a time t is described by 
The problem involves the diffusion of material B into material A along the rod axis x. The diffusion profile is described by a specific equation, which needs to be chosen from the given options.
Concept Explanation
In physical metallurgy, the diffusion of atoms in solids can often be described by Fick's second law of diffusion. When analyzing diffusion along a semi-infinite solid, the Gaussian error function or an exponential function is often used to describe the concentration profile.
Given Information
- A layer of material B of total amount m is plated at the junction of two rods of material A.
- The diffusion coefficient of B in A is D.
- We need to find the composition profile \(C_B\) along the rod axis x after a time t.
Solution Approach
The Gaussian solution to Fick's second law for a concentration profile is given by:
\(C_B(x,t) = \frac{m}{2\sqrt{\pi Dt}} \exp \left[ -\frac{x^2}{4Dt} \right]\)
This equation describes how a point source of material B, initially concentrated at the junction, spreads out into the material A over time. The concentration decreases exponentially with the square of the distance from the origin, and the spreading depends on the diffusion coefficient \(D\) and the time \(t\).
Conclusion
The correct option, which matches the diffusion behavior described, is:
\(C_B = \frac{m}{2\sqrt{\pi Dt}} \exp \left[ -\frac{x^2}{4Dt} \right]\)
The exponential form is correct for modeling diffusion from a localized source in an infinite medium. The error function (erf) solutions or linear t dependence do not fit this scenario.
Important Questions from Diffusion Fick's Second Law Concentration Profile
During carburizing of a steel, the surface concentration is kept constant at 1.4 wt.% carbon. Diffusivity of carbon for the steel at 950 $^\circ$C is $6.25 \times 10^{-11}$ m$^2$/s. At 950 $^\circ$C, the time required to carburize the steel with an initial composition of 0.2 wt.% carbon to 0.8859 wt.% carbon at a depth of 0.2 mm is ______________ seconds (approximate to the nearest integer).
Use the nearest value of the error function from the table given below for your calculation.
z erf (z) 0.3 0.3268 0.4 0.4284 0.5 0.5205 - What is the depth (in $µm$) from the surface of the specimen at which a composition of 0.4 wt.% C is obtained after carburizing at $870^\circ C$ for 10 h?
For self-diffusion in polycrystalline copper with a lattice diffusion coefficient $D_L$, grain boundary diffusion coefficient $D_{GB}$, and surface diffusion coefficient $D_S$, the correct relationship is
- The concentration $C$ of a solute (in units of atoms$\cdot\text{mm}^{-3}$) in a solid along $x$direction (for $x > 0$) follows the expression
$C = a_1x^2 + a_2x$
where $x$ is in mm, $a_1$ and $a_2$ are in units of atoms$\cdot\text{mm}^{-5}$ and atoms$\cdot\text{mm}^{-4}$,respectively. Assuming $a_1= a_2= 1$, the magnitude of flux at $x = 2 \text{ mm}$ is________ $\times 10^{-3} \text{ atoms} \cdot \text{mm}^{-2} \cdot \text{s}^{-1}$ (answer rounded off to the nearest integer).
Given: diffusion coefficient of the solute in the solid is $3 \times 10^{-3} \text{ mm}^2 \cdot \text{s}^{-1}$. - Determine the correctness or otherwise of the following Assertion [a] and the Reason [r]
Assertion [a]: The rate of homogenization in a dilute substitutional solid solution of B in A is controlled by the diffusivity of B.
Reason [r]: Atomic migration cannot occur along dislocations and grain boundaries.
