Given: $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$
Understanding Vacancy Concentration and Temperature
Vacancy concentration ($n_v$) in crystals increases with temperature ($T$). This dependence follows an Arrhenius-like relationship:
$n_v \propto e^{-E_f / (RT)}$
where $E_f$ is the enthalpy of vacancy formation and $R$ is the ideal gas constant.
Calculating Enthalpy of Vacancy Formation ($E_f$)
Given: Initial temperature $T_1 = 27 \text{ } ^\circ\text{C}$, final temperature $T_2 = 127 \text{ } ^\circ\text{C}$. The vacancy concentration doubles ($n_{v2} = 2 n_{v1}$). $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$.
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Convert Temperatures to Kelvin:
- $T_1 = 27 + 273.15 = 300.15 \text{ K}$
- $T_2 = 127 + 273.15 = 400.15 \text{ K}$
- Formulate Concentration Ratio: The ratio of vacancy concentrations relates to the absolute temperatures: $ \frac{n_{v2}}{n_{v1}} = e^{E_f \left(\frac{1}{T_1} - \frac{1}{T_2}\right)} $ Substitute the given doubling factor ($n_{v2} / n_{v1} = 2$): $ 2 = e^{E_f \left(\frac{1}{300.15 \text{ K}} - \frac{1}{400.15 \text{ K}}\right)} $
- Solve for $E_f$: Apply the natural logarithm to both sides: $ \ln(2) = E_f \left(\frac{1}{300.15 \text{ K}} - \frac{1}{400.15 \text{ K}}\right) $ Calculate the difference in inverse temperatures: $ \Delta\left(\frac{1}{T}\right) = \frac{1}{300.15 \text{ K}} - \frac{1}{400.15 \text{ K}} \approx 0.00083258 \text{ K}^{-1} $ Calculate $E_f$ in J/mol: $ E_f = \frac{\ln(2)}{\Delta\left(\frac{1}{T}\right)} \approx \frac{0.693147}{0.00083258 \text{ K}^{-1}} \approx 8325.9 \text{ J mol}^{-1} $
- Final Result: Convert $E_f$ to kJ/mol and round to two decimal places: $ E_f \approx \frac{8325.9}{1000} \text{ kJ mol}^{-1} \approx 8.33 \text{ kJ mol}^{-1} $ The calculated enthalpy of vacancy formation is $8.33 \text{ kJ mol}^{-1}$.
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.
