Let 1 be the ratio of molar specific heat at constant pressure an… — Solution | JEE Physics

MCQEasyJEE 2023Degrees of Freedom & Law of Equipartition

Let $\gamma_1$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $\gamma_2$ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{\gamma_1}{\gamma_2}$ is:

A
$\frac{27}{35}$
B
$\frac{35}{27}$
C
$\frac{21}{25}$
D
$\frac{25}{21}$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: $\gamma_1$ is the ratio $\frac{C_p}{C_v}$ for a monoatomic gas and $\gamma_2$ is the similar ratio for a diatomic gas treated as a rigid rotator.

Find: The value of $\frac{\gamma_1}{\gamma_2}$ .

For a monoatomic gas:

\gamma_1 = \frac{C_p}{C_v} = \frac{5}{3}

For a diatomic gas at low temperatures:

\gamma_2 = \frac{C_p}{C_v} = \frac{7}{5}

Therefore,

\frac{\gamma_1}{\gamma_2} = \frac{5/3}{7/5} = \frac{5}{3} \cdot \frac{5}{7} = \frac{25}{21}

Hence, the value is $\frac{25}{21}$ . The solution states the correct option is B, although this value matches option D in the listed options.

Option Discrepancy Note

The worked solution gives:

\gamma_1 = \frac{5}{3}, \qquad \gamma_2 = \frac{7}{5}

So,

\frac{\gamma_1}{\gamma_2} = \frac{5/3}{7/5} = \frac{25}{21}

Among the provided options, $\frac{25}{21}$ is listed as option D. However, the solution explicitly marks the correct option as B. The recorded answer is B.

Common mistakes

Using the wrong heat capacity ratio for a diatomic gas. A rigid rotator diatomic gas has rotational degrees of freedom active, so use $\gamma = \frac{7}{5}$ , not the monoatomic value.
Multiplying the ratios incorrectly. In $\frac{\gamma_1}{\gamma_2} = \frac{5/3}{7/5}$ , division by a fraction means multiplying by its reciprocal, giving $\frac{25}{21}$ .
Trusting the option label without checking the computed value. Here the worked value is $\frac{25}{21}$ , which does not align with the marked label in the options list, so the numerical result must be verified carefully.

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