Show that the heat capacity of a system can be written as...

Show that the heat capacity of a system can be written as $C=T d^{2}(k T \ln Z) / d T^{2}$, where $Z$ is the partition function $Z=\sum_{i} g_{i} e^{-\beta E_{i}}$. [Hint: Use the results of Problem $15.52 .$

Key Concepts

Partition Function

The partition function is a fundamental concept in statistical mechanics that aggregates information about all accessible microstates of a system. It is defined as a sum over states, with each term weighted by the Boltzmann factor exp(?E/(kT)). This function not only normalizes the probability distribution over states but also encodes all thermodynamic properties of the system, since many quantities can be derived from its logarithm and its temperature derivatives.

Heat Capacity

Heat capacity is a thermodynamic quantity that measures how much the energy of a system changes with temperature. In statistical mechanics, it is often expressed as the derivative of the average energy with respect to temperature. Alternatively, through appropriate thermodynamic identities, it can be derived from the second derivative of the logarithm of the partition function with respect to temperature, thereby linking microscopic statistical properties to macroscopic thermal responses.

Thermodynamic Derivatives

Thermodynamic derivatives refer to the use of mathematical differentiation of thermodynamic potentials, such as the free energy, to obtain observable physical quantities. In the context of statistical mechanics, taking derivatives of kT ln(Z) with respect to temperature yields expressions for quantities like internal energy and heat capacity. This approach highlights the deep relationship between microscopic state probabilities captured by the partition function and the macroscopic properties of the system.

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