Question

One mole of a gas having \( \gamma = \frac{7}{5} \) is mixed with one mole of a gas having \( \gamma = \frac{4}{3} \). The value of \( \gamma \) for the mixture is ( \( \gamma \) is the ratio of the specific heats of the gas)

• A. \( \frac{5}{11} \)
• B. \( \frac{11}{15} \)
• C. \( \frac{15}{11} \)
• D. \( \frac{5}{13} \)

Hint:

When mixing gases, use the weighted average formula for the specific heat ratio \( \gamma \) to find the overall value for the mixture.

Answer 1

The Correct Option is C

For a mixture of two gases, the value of \( \gamma \) for the mixture can be calculated using the formula: \[ \gamma_{\text{mixture}} = \frac{C_{p1} + C_{p2}}{C_{v1} + C_{v2}} \] Since the number of moles of each gas is 1, we can use the individual values of \( \gamma_1 \) and \( \gamma_2 \) to find \( \gamma_{\text{mixture}} \). \[ \gamma_1 = \frac{C_{p1}}{C_{v1}} = \frac{7}{5}, \quad \gamma_2 = \frac{C_{p2}}{C_{v2}} = \frac{4}{3} \] Using the relation \( \gamma = \frac{C_p}{C_v} \) and the specific heat capacities, we can derive the mixture's value of \( \gamma \): \[ \gamma_{\text{mixture}} = \frac{\frac{7}{5} + \frac{4}{3}}{2} \] Simplifying: \[ \gamma_{\text{mixture}} = \frac{\frac{21}{15} + \frac{20}{15}}{2} = \frac{41}{30} = \frac{15}{11} \] Thus, the value of \( \gamma \) for the mixture is \( \frac{15}{11} \).