Assertion (A): The Einstein Specific heat ($C_V$), at low temperature is, $C_V = 3NK \left(\frac{h\nu}{KT}\right)^3 \exp\left(-\frac{h\nu}{KT}\right)$
Reason (R): The Einstein Specific heat, at high temperature is, $C_V = 3NK$
In the light of the above statements, choose the most appropriate answer from the options given below :
Einstein Specific Heat Analysis
This question evaluates the understanding of Einstein's model for the specific heat of solids, focusing on its behavior at low and high temperatures.
Assertion (A) Evaluation
Assertion (A) states the Einstein Specific Heat ($C_V$) at low temperature is:
$ C_V = 3NK \left(\frac{h\nu}{KT}\right)^3 \exp\left(-\frac{h\nu}{KT}\right) $
The low-temperature approximation for Einstein's model is characterized by an exponential cutoff. The correct formula derived from Einstein's model at low temperatures ($KT \ll h\nu$) is:
$ C_V \approx 3NK \left(\frac{h\nu}{KT}\right)^2 \exp\left(-\frac{h\nu}{KT}\right) $
Assertion (A) incorrectly uses a power of 3 instead of 2 in the pre-exponential factor. Therefore, Assertion (A) is incorrect.
Reason (R) Evaluation
Reason (R) states the Einstein Specific Heat ($C_V$) at high temperature is:
$ C_V = 3NK $
At high temperatures ($KT \gg h\nu$), the exponential term $\exp\left(-\frac{h\nu}{KT}\right)$ approaches 1, and the term $\left(\frac{h\nu}{KT}\right)$ becomes small. Applying this limit to the full Einstein model:
$ C_V = 3NK \left(\frac{h\nu}{KT}\right)^2 \frac{\exp\left(-\frac{h\nu}{KT}\right)}{\left(1 - \exp\left(-\frac{h\nu}{KT}\right)\right)^2} $
As $T \to \infty$, $h\nu/KT \to 0$. Using Taylor expansion $e^x \approx 1+x$ for small $x$, $1-\exp(-h\nu/KT) \approx h\nu/KT$. The expression simplifies towards $3NK$. This result matches the classical Dulong-Petit law.
Therefore, Reason (R) is correct.
Conclusion
Based on the analysis, Assertion (A) is incorrect, while Reason (R) is correct. This corresponds to Option D.
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