It takes $10 \ hours$ to homogenize an alloy at $1273 \ K$. The time required (in hours) to achieve the same extent of homogenization at $1373 \ K$ is ________. Given: Diffusivity, $D_{1373 \ K} = 10^{-18}m^2 \ s^{-1}$ and $D_{1273 \ K} = 10^{-19}m^2 \ s^{-1}$
Homogenization Time Calculation
The extent of homogenization in an alloy is dependent on the diffusion process, which is influenced by temperature. For a given extent of homogenization, the product of diffusivity ($D$) and time ($t$) is often considered constant, as the characteristic diffusion length ($L$) achieved is proportional to $\sqrt{D \times t}$. If the same extent of homogenization is required, then $L$ must be the same, leading to the relationship $D_1 t_1 = D_2 t_2$.
Given Information
- Time at $1273 \ K$, $t_1 = 10 \ hours$
- Diffusivity at $1273 \ K$, $D_1 = 10^{-19} \ m^2 s^{-1}$
- Diffusivity at $1373 \ K$, $D_2 = 10^{-18} \ m^2 s^{-1}$
- Target temperature, $T_2 = 1373 \ K$
Calculating Homogenization Time
First, convert the initial time $t_1$ to seconds for consistency with the diffusivity units:
$t_1 = 10 \ hours \times \frac{3600 \ seconds}{1 \ hour} = 36000 \ seconds$
Using the relationship $D_1 t_1 = D_2 t_2$, we can solve for the time $t_2$ required at $1373 \ K$:
$t_2 = t_1 \times \frac{D_1}{D_2}
Substitute the given values:
$t_2 = 36000 \ s \times \frac{10^{-19} \ m^2 s^{-1}}{10^{-18} \ m^2 s^{-1}}
$t_2 = 36000 \ s \times 10^{-1}
$t_2 = 3600 \ s
Now, convert the calculated time $t_2$ back into hours:
$t_2 = 3600 \ s \times \frac{1 \ hour}{3600 \ seconds} = 1 \ hour$
Therefore, the time required to achieve the same extent of homogenization at $1373 \ K$ is $1 \ hour$.
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.
