Given: The interdiffusion coefficient for copper in aluminium at 500°C and 600°C are $4\times10^{-14}$ m$^2$s$^{-1}$ and $8\times10^{-13}$ m$^2$s$^{-1}$.
Diffusion Couple Time Calculation
To achieve the same concentration profile in a diffusion couple, the total amount of diffusion must be equivalent. The extent of diffusion is directly related to the product of the interdiffusion coefficient ($D$) and the time ($t$).
Fundamental Relationship
For a given concentration profile to be replicated at two different temperatures ($T_1$ and $T_2$), the quantity $Dt$ must remain constant:
$D_1 t_1 = D_2 t_2$
Where:
- $D_1$ = interdiffusion coefficient at temperature $T_1$
- $t_1$ = time elapsed at temperature $T_1$
- $D_2$ = interdiffusion coefficient at temperature $T_2$
- $t_2$ = time elapsed at temperature $T_2$ (the value we need to find)
Provided Data
The problem provides the following information:
- Temperature $T_1 = 600^\circ\text{C}$
- Time $t_1 = 10$ hours
- Interdiffusion coefficient $D_1$ at $600^\circ\text{C} = 8 \times 10^{-13} \text{ m}^2\text{s}^{-1}$
- Temperature $T_2 = 500^\circ\text{C}$
- Interdiffusion coefficient $D_2$ at $500^\circ\text{C} = 4 \times 10^{-14} \text{ m}^2\text{s}^{-1}$
Calculating Time ($t_2$)
Rearrange the fundamental relationship to solve for $t_2$:
$t_2 = \frac{D_1 t_1}{D_2}$
Substitute the given values into the equation:
$t_2 = \frac{(8 \times 10^{-13} \text{ m}^2\text{s}^{-1}) \times (10 \text{ hours})}{(4 \times 10^{-14} \text{ m}^2\text{s}^{-1})}$
Simplify the expression:
$t_2 = \frac{80 \times 10^{-13}}{4 \times 10^{-14}} \text{ hours}$
$t_2 = 20 \times 10^{(-13 - (-14))} \text{ hours}$
$t_2 = 20 \times 10^{1} \text{ hours}$
$t_2 = 200 \text{ hours}$
Result Interpretation
The calculated time required to achieve the same concentration profile at 500°C is 200 hours. This result is consistent with the provided answer range of 180 to 220 hours.
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.
