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 gas mixture is determined by the mole fractions and individual adiabatic exponents of its constituent gases.

Provided Data: Monoatomic gas: n₁ = 3 moles, γ₁ = \( \frac{5}{3} \). Diatomic gas: n₂ = 2 moles, γ₂ = \( \frac{7}{5} \).

Total Moles Calculation: The total number of moles n is the sum of individual moles: n = n₁ + n₂ = 3 + 2 = 5.

Formula for Mixture Adiabatic Exponent: The adiabatic exponent for the mixture, γ_mixture, is calculated as:

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

Value Substitution:

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

Final Result: The calculated average adiabatic exponent is:

\[ \gamma_{\text{mixture}} = \frac{39}{25} = 1.56 \]