Question 10.1: Express the molar heat capacity of an Einstein solid as a fu...
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).
Let us start from the molar entropy (10.13) of the crystal:
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)We can calculate the temperature as
T=\left(\frac{\partial s}{\partial u}\right)^{-1}=\frac{u_{0}}{3 R \ln \frac{u+u_{0}}{u}}Solving the above expression for u yields the molar internal energy:
u=\frac{u_{0}}{e^{\frac{u_{0}}{3 R T}-1}} .
Derivation of this function with respect to temperature directly gives the molar heat capacity:
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)}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.)
