At very low temperatures the molar heat capacity of rock...

At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's $T^{3}$ law: $$ C=k \frac{T^{3}}{\theta^{3}} $$ where $k=1940 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}$ and $\theta=281 \mathrm{~K}$. (a) How much heat is required to raise the temperature of $1.50 \mathrm{~mol}$ of rock salt from $10.0 \mathrm{~K}$ to $40.0 \mathrm{~K} ?$ (Hint: Use Eq. (17.18) in the form $d Q=n C d T$ and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at $40.0 \mathrm{~K} ?$

At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's $T^{3}$ law:
$$
C=k \frac{T^{3}}{\theta^{3}}
$$
where $k=1940 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}$ and $\theta=281 \mathrm{~K}$. (a) How much heat is required to raise the temperature of $1.50 \mathrm{~mol}$ of rock salt from $10.0 \mathrm{~K}$ to $40.0 \mathrm{~K} ?$ (Hint: Use Eq. (17.18) in the form $d Q=n C d T$ and integrate.)
(b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at $40.0 \mathrm{~K} ?$

Key Concepts

Debye's T³ Law

Debye's T³ Law is a theoretical prediction that at sufficiently low temperatures, the molar heat capacity of a crystalline solid varies as the cube of temperature. This stems from the phonon contributions to the thermal properties of solids, where low-frequency vibrational modes dominate and result in a T³ dependence, contrasting with the classical Dulong-Petit law valid at high temperatures.

Molar Heat Capacity

Molar heat capacity is a physical property that measures the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin. It provides insight into the energy storage and lattice dynamics of materials, and it is especially important in thermodynamics and solid-state physics for characterizing heat transfer processes.

Integration of Heat Capacity

The integration of heat capacity is used to calculate the total amount of heat required to change the temperature of a substance over a finite temperature interval. This process involves integrating the differential heat dQ = nC dT, where C may be a function of temperature, to account for the variation of heat capacity with temperature in calculating energy changes.

Average and Instantaneous Molar Heat Capacity

The average molar heat capacity is defined as the total heat added divided by the product of the number of moles and the temperature change over an interval. In contrast, the instantaneous or true molar heat capacity at a specific temperature is given by the value of the heat capacity function at that temperature. Understanding this distinction is crucial when dealing with temperature-dependent heat capacities in thermodynamic processes.

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