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Chapter 9: Problem 1319

A diatomic gas molecule has translational, rotational and vibrational degrees of freedom. The $\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\right)$ is (A) \(1.29\) (B) \(1.33\) C) \(1.4\) (D) \(1.67\)

Short Answer

Expert verified
A diatomic gas molecule has a total of 6 degrees of freedom: 3 translational, 2 rotational, and 1 vibrational. Using the equipartition theorem, we can calculate the specific heat capacities at constant volume (Cv) and constant pressure (Cp) as \(C_v = 3R\) and \(C_p = 4R\), respectively. Then, we can find the ratio \(\left(\dfrac{C_p}{C_v}\right) = \dfrac{4R}{3R}\), which simplifies to \(\dfrac{4}{3} = 1.33\). Thus, the correct answer is (B) 1.33.

Step by step solution

01

Determine degrees of freedom

A diatomic molecule has 3 translational, 2 rotational, and 1 vibrational degree of freedom. So in total, there are 3+2+1= 6 degrees of freedom.
02

Calculate specific heat capacities

For a diatomic gas molecule, use equipartition theorem to find the specific heat capacities. The specific heat capacity at constant volume, Cv, is given by: \[C_v = \dfrac{f}{2} R\] Where f is the total degrees of freedom, and R is the gas constant. Since we have calculated 6 degrees of freedom, we get: \[C_v = \dfrac{6}{2} R = 3 R\] The specific heat capacity at constant pressure, Cp, is given by: \[C_p = C_v + R\] \[C_p = 3R + R = 4R\]
03

Calculate the ratio Cp/Cv

Now that we have Cp and Cv, we can calculate the ratio: \[\left(\dfrac{C_p}{C_v}\right) = \dfrac{4R}{3R}\]
04

Simplify the ratio

Simplify the ratio by canceling the common factor R: \[\left(\dfrac{C_p}{C_v}\right) = \dfrac{4}{3}\]
05

Convert ratio to decimal form

Convert 4/3 to decimal form: \[\left(\dfrac{C_p}{C_v}\right) = 1.33\] Since the ratio is 1.33, the correct answer is (B) 1.33.

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