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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 aluminum, given that the specific heat is \(4.60 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) at \(15 \mathrm{~K}\).

Short Answer

Expert verified
Answer: The estimated Debye temperature for aluminum is approximately 428.69 K.

Step by step solution

01

Combine Equations 1 and 2 to find the Debye temperature

First, we need to rewrite Equation 2 in terms of 𝜃D using Equation 1: \(C = \frac {12\pi^4RT^3}{5\theta_{\mathrm{D}}^3}\) Now, we can solve for 𝜃D: \(\theta_{\mathrm{D}}^3 = \frac {12\pi^4RT^3}{5C}\)
02

Plug in the values for aluminum and solve for 𝜃D

Now, plug in the given values for specific heat\(, C\), the gas constant \(R\), and the temperature \(T\). \(\theta_{\mathrm{D}}^3 = \frac {12\pi^4(8.314)\cdot(15)^3}{5(4.60)}\) Solve for 𝜃D: \(\theta_{\mathrm{D}} = \sqrt[3]{\frac {12\pi^4(8.314)\cdot(15)^3}{5(4.60)}}\) Calculate the result: \(\theta_{\mathrm{D}} \approx 428.69 \; \mathrm{K}\) So, the estimated Debye temperature for aluminum is \(428.69 \; \mathrm{K}\).

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