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Chapter 19: Problem 5

The constant \(A\) in Equation \(19.2\) is \(12 \pi^{4} R / 5 \theta_{\mathrm{D}}^{3}\), where \(R\) is the gas constant and \(\theta_{\mathrm{D}}\) is the Debye temperature \((\mathrm{K})\). Estimate \(\theta_{\mathrm{D}}\) for copper, given that the specific heat is \(0.78 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) at \(10 \mathrm{~K}\).

Short Answer

Expert verified
Question: Estimate the Debye temperature for copper, given its specific heat at 10 K is 0.78 J·kg⁻¹·K⁻¹. Answer: The estimated Debye temperature for copper is approximately 345 K.

Step by step solution

01

Write down the given values and the formula for the constant A

We are given the specific heat of copper at 10 K: \(c_{v} = 0.78 \, \mathrm{J} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1}\). The constant A in Equation 19.2 is given by \(A = \frac{12 \pi^{4} R}{5 \theta_{\mathrm{D}}^{3}}\).
02

Find the relationship between specific heat and the constant A

According to the Debye theory, the specific heat at a given temperature T is related to the constant A by \(c_{v} = AT^{3}\).
03

Substitute the given values into the equation relating specific heat and the constant A

We know the specific heat \(c_v\) at \(T = 10\,K\). We can substitute these values into the equation \(c_{v} = AT^3\): \(0.78 = A(10)^3\)
04

Calculate the value of A

Solve the equation for A: \(A = \frac{0.78}{(10)^3} = \frac{0.78}{1000} = 7.8 \times 10^{-4}\)
05

Substitute the value of A into the equation for A in terms of the Debye temperature

Now we have the value of A, we can substitute it into the equation \(A = \frac{12 \pi^{4} R}{5 \theta_{\mathrm{D}}^3}\) to find the value of \(\theta_{\mathrm{D}}\): \(7.8 \times 10^{-4} = \frac{12 \pi^{4} R}{5 \theta_{\mathrm{D}}^3}\)
06

Rearrange the equation to solve for the Debye temperature

Rearranging the equation to solve for \(\theta_{\mathrm{D}}\), we get: \(\theta_{\mathrm{D}} = \left(\frac{12 \pi^{4} R}{5 (7.8 \times 10^{-4}) }\right)^\frac{1}{3}\)
07

Calculate the Debye temperature

To estimate the Debye temperature, we need the gas constant R. The gas constant R is approximately \(8.314 \, \mathrm{J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1}\): \(\theta_{\mathrm{D}} = \left(\frac{12 \pi^{4} (8.314)}{5 (7.8 \times 10^{-4}) }\right)^\frac{1}{3} = 345 \, \mathrm{K}\) Thus, the estimated Debye temperature for copper is approximately 345 K.

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