(II) At low temperature the specific heat of diamond varies...

(II) At low temperature the specific heat of diamond varies with absolute temperature $T$ according to the Debye equation $C_{V}=1.88 \times 10^{3}\left(T / T_{\mathrm{D}}\right)^{3} \mathrm{J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1}$ where the Debye temperature for diamond is $T_{\mathrm{D}}=2230 \mathrm{K}$ . Use a spreadsheet and numerical integration to determine the entropy change of 1.00 $\mathrm{mol}$ of diamond when it is heated at constant volume from 4 $\mathrm{K}$ to 40 $\mathrm{K}$ . Your result should agree within 2$\%$ of the result obtained by integrating the expression for $d S .\left[$Hint$: d S=n C_{V} d T / T,$ where $n$ is the number \right. of moles.

Key Concepts

Debye Model

The Debye model explains the behavior of the specific heat in crystalline solids at low temperatures by considering the quantized vibrational modes of the lattice. It predicts that the specific heat varies as T³ at temperatures well below the Debye temperature, which is a significant divergence from classical predictions. This model is fundamental in understanding thermal properties of insulators and semiconductors at low temperatures.

Low-Temperature Specific Heat

At low temperatures, the specific heat of a solid does not remain constant but follows a temperature-dependent trend, often described by a power law. In the context of the Debye model, the T³ dependency is crucial for accurately calculating how much the internal energy changes with temperature. This concept is important in thermodynamics when assessing heat capacities of materials as a function of temperature.

Entropy Change through Integration

Entropy is a measure of the disorder or randomness in a system, and its change is directly related to heat capacity and temperature changes. The formula dS = nCV dT/T is used to calculate the change in entropy, where integrating the ratio of the heat capacity to temperature over a temperature range provides the total entropy change. This method is a key concept in statistical mechanics and thermodynamics for linking microscopic properties to macroscopic observables.

Constant Volume Process

A constant volume process in thermodynamics implies that no work is done by the system due to volume change, simplifying the relationship between heat and internal energy. When a substance is heated at constant volume, the heat added goes entirely into increasing the internal energy, making the calculation of entropy change more straightforward. This condition is important to isolate thermal effects from mechanical work.

Numerical Integration

Numerical integration is a computational technique used to approximate the value of definite integrals when an analytical solution is difficult or impossible to obtain. In the context of thermodynamics, numerical integration is often employed to calculate physical quantities such as entropy change from temperature-dependent heat capacity data, using tools like spreadsheets to perform the stepwise summation across discrete temperature intervals.

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