Question

The Debye temperature for silver is 225 K. Calculate the highest possible frequency for
lattice vibrations in silver and its molar heat capacity at 10 K and 500 K.

Answer 1

Answer on Question #51562, Physics, Solid State Physics

The Debye temperature for silver is $225\,\mathrm{K}$. Calculate the highest possible frequency for lattice vibrations in silver and its molar heat capacity at $10\,\mathrm{K}$ and $500\,\mathrm{K}$.

Solution:

The maximum frequency of vibration of the atoms of solid bodies is given by Eq.(1)

$$f_{D} = \frac{k_{B} T_{D}}{h} = \frac{1.38 \cdot 10^{-23} J / K \cdot 225\,K}{6.62 \cdot 10^{-34} J \cdot s} = 4.7 \cdot 10^{12} \, \mathrm{Hz}$$

where $k_{B} = 1.38 \cdot 10^{-23} J / K$ is the Boltzmann constant; $h = 6.62 \cdot 10^{-34} J \cdot s$ is the Planck constant; $T_{D}$ is the Debye temperature.

The molar heat capacity is given by Eq.(2)

$$C_{V}(T) = 9 N k_{B} \left(\frac{T}{T_{D}}\right)^{3} \int_{0}^{T_{D}/T} \frac{\xi^{4} e^{\xi}}{\left(e^{\xi} - 1\right)^{2}} d\xi$$

where $N$ is the number of atoms in a solid body.

Then

$$C_{V}(T = 10) = 9 \left(\frac{10}{225}\right)^{3} 6.02 \cdot 10^{23} \cdot 1.38 \cdot 10^{-23} J / K \left(\int_{0}^{225/10} \frac{\xi^{4} e^{\xi}}{\left(e^{\xi} - 1\right)^{2}} d\xi\right) = 0.17 J / K$$ $$C_{V}(T = 500) = 9 \left(\frac{500}{225}\right)^{3} 6.02 \cdot 10^{23} \cdot 1.38 \cdot 10^{-23} J / K \left(\int_{0}^{225/500} \frac{\xi^{4} e^{\xi}}{\left(e^{\xi} - 1\right)^{2}} d\xi\right) = 26.67 J / K$$

**Answer**: $f_{D} = \frac{k_{B} T_{D}}{h} = 4.7 \cdot 10^{12} \, \mathrm{Hz}$; $C_{V}(T = 10) = 0.17 J / K$; $C_{V}(T = 500) = 26.67 J / K$

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