(III) The specific heat per mole of potassium at low...

(III) The specific heat per mole of potassium at low temperatures is given by $C_{V}=a T+b T^{3}, \quad$ where $a=2.08 \mathrm{mJ} / \mathrm{mol} \cdot \mathrm{K}^{2}$ and $b=2.57 \mathrm{mJ} / \mathrm{mol} \cdot \mathrm{K}^{4} .$ Determine (by integration) the entropy change of 0.15 $\mathrm{mol}$ of potassium when its temperature is lowered from 3.0 $\mathrm{K}$ to 1.0 $\mathrm{K}$ .

Key Concepts

Integration for Entropy Change

In thermodynamics, the change in entropy when a system undergoes a temperature change can be determined by integrating the specific heat capacity divided by temperature over the interval of interest. This relationship, expressed as ?S = ?(C/T)dT, is fundamental for calculating entropy changes in systems where the heat capacity varies with temperature.

Low Temperature Behavior of Solids

At low temperatures, the specific heat of solids often exhibits contributions from different sources, such as electrons and phonons. In metals, the electronic contribution typically varies linearly with temperature while the phonon contribution follows a cubic temperature dependence. This combined behavior, reflected in the equation C = aT + bT^3, provides insights into the microscopic interactions and excitations within the material.

Specific Heat Capacity

Specific heat capacity is a thermodynamic property that measures the amount of heat required to change a substance's temperature by a given amount. It can be expressed per mole or per unit mass and plays a crucial role in understanding how energy is stored in a material, especially when studying its temperature-dependent behavior.

Entropy

Entropy is a measure of the disorder or randomness in a system and is a central concept in the second law of thermodynamics. It quantifies the number of microscopic configurations corresponding to a macroscopic state, and changes in entropy are often calculated when a system undergoes temperature or phase changes.

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