Question

The specific heat capacity Cv (J K-1 mol-1) of copper at different temperatures varies as follows
T (K) 1.4 2 2.44 2.83 3.16 3.46 3.74
Cv(J K-1 mol-1 5.26 10.0 16.3 22.5 29.1 37.6 45.33
Calculate the value of γ in the equation Cv = γT + αT3 and the Debye temperature Td. Assume that the number of electrons per unit volume of copper is 8.5 x 1022 cm-3.

Answer 1

Answer on Question #75547, Physics / Molecular Physics | Thermodynamics

The specific heat capacity $\mathrm{Cv}$ ( $\mathrm{J} \, \mathrm{K}^{-1} \, \mathrm{mol}^{-1}$ ) of copper at different temperatures varies as follows

Calculate the value of $\gamma$ in the equation $C\nu = \gamma T + \alpha T^3$ and the Debye temperature $T_{\mathrm{d}}$ . Assume that the number of electrons per unit volume of copper is $8.5 \times 10^{22} \, \mathrm{cm}^{-3}$ .

Solution:

Plot $C_V / T$ against $T^2$ . The intercept at $T^2 = 0$ is the value of $\gamma$ .

Matlab code:

T = [1.4 2 2.44 2.83 3.16 3.46 3.74];
T2 = T.^2;
Cv = [5.26 10.0 16.3 22.5 29.1 37.6 45.33];
plot(T2,Cv./T,'*');
grid on;
coefs = polyfit(T2,Cv./T, 1);
intercept = coefs(2)
slope = coefs(1)
x1 = linspace(0,14);
y1 = polyval(coefs,x1);
hold on
plot(x1,y1,'r-');
xlabel('T^2'); ylabel('Cv/T');

Output:

intercept =

2.3387

slope =

0.7019

$\gamma = 2.3387 \approx 2.34 J K^{-2} \, \text{mol}^{-1}$

The slope of the line from the graph is

$$\alpha = 0.7019 = \frac{234Nk}{T_{D}^{3}},$$

where $N$ is the number of electrons, $k$ is the Boltzmann constant and $T_{D}$ is the Debye temperature.

The N is

$$N = \text{Concentration} \times \text{Molar volume} = (8.5 \times 10^{22} \text{cm}^{-3}) (7.1 \text{cm}^{3})$$

This gives

$$T_{D} = \sqrt[3]{\frac{234Nk}{\alpha}} = \sqrt[3]{\frac{234 \times (8.5 \times 10^{22} \times 7.1)(1.38 \times 10^{-23})}{0.7019}} = 14.05 \text{K}$$

Answer: $\gamma = 2.34 \text{J K}^{-2} \text{mol}^{-1}; T_{D} = 14.05 \text{K}$.

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