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 #51237, 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

$$v_{\max} = \frac{k_B T_D}{h} = \frac{1.38 \cdot 10^{-23} J / K \cdot 225 K}{6.62 \cdot 10^{-34} J \cdot s} = 4.7 \cdot 10^{12} \mathrm{~Hz}$$

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)

$$C_V(T) = 3 N k_B \left( \frac{3}{x_D^3} \int_0^{x_D} \frac{x^4 e^x}{(e^x - 1)^2} dx \right)$$

where $3N$ is number of normal modes; $x_D = T_D / T$.

Then

$$C_V(10) = 3 N k_B \left( \frac{3}{(225 / 10)^3} \int_0^{225 / 10} \frac{x^4 e^x}{(e^x - 1)^2} dx \right) = 0.0205 N k_B$$ $$C_V(500) = 3 N k_B \left( \frac{3}{(225 / 500)^3} \int_0^{225 / 500} \frac{x^4 e^x}{(e^x - 1)^2} dx \right) = 2.9698 N k_B$$

**Answer**: $v_{\max} = \frac{k_B T_D}{h} = 4.7 \cdot 10^{12} \mathrm{~Hz}$; $C_V(10) = 0.0205 N k_B$; $C_V(500) = 2.9698 N k_B$

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