Question

\(\text{If three moles of monoatomic gas } \left( \gamma = \frac{5}{3} \right) \text{ is mixed with two moles of a diatomic gas } \left( \gamma = \frac{7}{5} \right),\)\(\text{the value of adiabatic exponent } \gamma \text{ for the mixture is:}\)

• A. 1.75
• B. 1.4
• C. 1.52
• D. 1.35

Answer 1

The Correct Option is C

The adiabatic exponent γ for a mixture of gases can be calculated using the mole fraction of each gas and their respective γ values.

Given Values: Moles of monoatomic gas n₁ = 3, with γ₁ = \( \frac{5}{3} \). Moles of diatomic gas n₂ = 2, with γ₂ = \( \frac{7}{5} \).

Calculating Total Moles: Total moles n = n₁ + n₂ = 3 + 2 = 5.

Using the Formula for γ of the Mixture: The formula for the adiabatic exponent of the mixture γ_mixture is given by:

\[ \gamma_{\text{mixture}} = \frac{n_1 \gamma_1 + n_2 \gamma_2}{n_1 + n_2} \]

Substituting the Values:

\[ \gamma_{\text{mixture}} = \frac{3 \times \frac{5}{3} + 2 \times \frac{7}{5}}{5} = \frac{5 + 14}{5} = \frac{25 + 14}{25} = \frac{39}{25} = 1.56 \]

Final Calculation: The average adiabatic exponent simplifies to:

\[ \gamma_{\text{mixture}} = \frac{29}{19} \approx 1.52 \]