Question

Using Boltzmann equation of entropy, calculate residual entropy of a crystalline CH3D near 0°K.

Answer 1

The residual entropy of crystalline CH3D near 0°K is approximately
`\( k_B \ln(6) \),` where
`\( k_B \)` is the Boltzmann constant.

Step-by-step explanation:

The residual entropy of a crystalline substance at absolute zero temperature can be calculated using the Boltzmann equation of entropy:

`\[ S = k_B \ln(W) \]`

Where:

( S) = entropy

`\( k_B \)` = Boltzmann constant
`(\( 1.380649 * 10^(-23) \, \text{J/K} \))`

(W ) = number of microstates

For a molecule with three distinguishable hydrogen atoms and one deuterium atom (CH3D), at absolute zero, the molecule's rotation and vibration are limited due to its crystalline structure, resulting in six possible configurations with the same energy. Hence, the number of microstates (W ) at 0°K is 6.

Therefore, substituting the value of (W = 6 ) into the entropy equation:

`\[ S = k_B \ln(6) \]`

Calculating the natural logarithm of 6 using the Boltzmann constant yields the residual entropy of crystalline CH3D near 0°K.

The Boltzmann constant
`\( k_B \)` is a fundamental constant used to relate temperature to energy, and the natural logarithm provides the relationship between the number of microstates and entropy. In this case, the six possible configurations at absolute zero temperature contribute to a residual entropy of approximately
`\( k_B \ln(6) \)` for crystalline CH3D.