Express the molar heat capacity of an Einstein solid as a function of temperature.
Calculate the T → 0 and T → ∞ limits. Draw a plot of the molar heat capacity as a function of temperature between 0 K and 10 K (use the reference value of $u_{0}$ = 100 J).

Question

Express the molar heat capacity of an Einstein solid as a function of temperature.
Calculate the T → 0 and T → ∞ limits. Draw a plot of the molar heat capacity as a function of temperature between 0 K and 10 K (use the reference value of [latex]u_{0}[/latex] = 100 J).

Accepted Answer

Let us start from the molar entropy (10.13) of the crystal:

[latex]s=3 R \ln \left(1+\frac{u}{u_{0}}\right)+3 R \frac{u}{u_{0}} \ln \left(1+\frac{u_{0}}{u}\right)[/latex]

We can calculate the temperature as

[latex]T=\left(\frac{\partial s}{\partial u}\right)^{-1}=\frac{u_{0}}{3 R \ln \frac{u+u_{0}}{u}}[/latex]

Solving the above expression for u yields the molar internal energy:

[latex]u=\frac{u_{0}}{e^{\frac{u_{0}}{3 R T}-1}}[/latex] .

Derivation of this function with respect to temperature directly gives the molar heat capacity:

[latex]c_{V}=\frac{\partial u}{\partial T}=\frac{u_{0}^{2} e^{\frac{u_{0}}{3 R T}}}{3 R T^{2}\left(e^{\frac{u_{0}}{3 R T}}-1\right)}[/latex]

The limits are in accordance with experimental data; the T → 0 limit is zero, the T → ∞ limit is 3R (complying with the Dulong–Petit rule). The plot of the function from 0 to 10 K is the one below. (Note that the exponential rise is not in accordance with experimental data.)

Question 10.1: Express the molar heat capacity of an Einstein solid as a fu...

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