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Chapter 19: Problem 4

For aluminum, the heat capacity at constant volume \(C_{v}\) at \(30 \mathrm{~K}\) is \(0.81 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\), and the Debye temperature is \(375 \mathrm{~K}\). Estimate the specific heat (a) at \(50 \mathrm{~K}\) and (b) at \(425 \mathrm{~K}\).

Short Answer

Expert verified
Based on the Debye Model, estimate the specific heat of aluminum at 50 K and 425 K using the given values of heat capacity at constant volume, Debye temperature, and temperature.

Step by step solution

01

Understand the Debye Model formula for specific heat

The Debye model formula for estimating specific heat \(c_v(T)\) at a certain temperature \(T\) is given by: \[c_v(T) = 9Nk \left ( \frac{T}{\Theta_D} \right )^3 \int_{0}^{\frac{\Theta_D}{T}} \frac{x^4e^x}{(e^x - 1)^2} dx \] Here, \(N\) represents the number of atoms per mole, \(k\) is the Boltzmann constant, \(T\) stands for the temperature we want to find the specific heat at, and \(\Theta_D\) denotes the Debye temperature of the material.
02

Calculate the constant terms

In this formula, the constant terms are \(9Nk\) and \(\Theta_D\). We are given the Debye temperature of aluminum, \(\Theta_D = 375 \, \text{K}\). To find the value of \(9Nk\), we can use the given value of \(C_v\) at \(30 \, \text{K}\): \(C_v(30\, \text{K}) = 0.81\, \text{J/mol.K} = 9Nk \left ( \frac{30}{375} \right )^3 \int_{0}^{\frac{375}{30}} \frac{x^4e^x}{(e^x - 1)^2} dx\) We can use this equation to calculate the value of \(9Nk\).
03

Evaluate the constant terms

To find the value of \(9Nk\), we can solve the equation derived in Step 2: \(0.81 = 9Nk \left ( \frac{1}{12.5} \right )^3 \int_{0}^{12.5} \frac{x^4e^x}{(e^x - 1)^2} dx\) After solving for \(9Nk\), we get: \(9Nk = 0.81 \cdot \left ( \frac{1}{\int_{0}^{12.5} \frac{x^4e^x}{(e^x - 1)^2} dx}\right )\)
04

Estimate the specific heat at 50 K

Now, we can use the Debye Model formula and the derived value of \(9Nk\) to estimate the specific heat of aluminum at \(50 \, \text{K}\): \(c_v(50\, \text{K}) = 9Nk \left ( \frac{50}{375} \right )^3 \int_{0}^{\frac{375}{50}} \frac{x^4e^x}{(e^x - 1)^2} dx\)
05

Estimate the specific heat at 425 K

Similarly, we can use the Debye Model formula and the derived value of \(9Nk\) to estimate the specific heat of aluminum at \(425 \, \text{K}\): \(c_v(425\, \text{K}) = 9Nk \left ( \frac{425}{375} \right )^3 \int_{0}^{\frac{375}{425}} \frac{x^4e^x}{(e^x - 1)^2} dx\) To summarize, we analyzed the given problem, used the Debye Model formula for specific heat estimation, and derived the values needed to estimate specific heat at the given temperatures. The specific heat can be calculated with the derived equations at \(50 \, \text{K}\) and \(425 \, \text{K}\).

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