Ideal Gas Mixture Adiabatic Index Calculation

This solution details the calculation of the adiabatic index ($\gamma = \frac{C_p}{C_v}$) for a mixture of two ideal gases.

Given Information

• Gas 1: Moles $n_1 = 2$, Adiabatic index $\gamma_1 = \frac{5}{3}$
• Gas 2: Moles $n_2 = 3$, Adiabatic index $\gamma_2 = \frac{4}{3}$

Adiabatic Index Formula for Mixture

For a mixture of ideal gases, the adiabatic index ($\gamma_{\text{mix}}$) can be calculated using the individual moles ($n_i$) and adiabatic indices ($\gamma_i$) of the components. The formula is derived from the mixing of total internal energy and total enthalpy:

$ \gamma_{\text{mix}} = \frac{\sum_{i} n_i \frac{\gamma_i}{\gamma_i-1}}{\sum_{i} n_i \frac{1}{\gamma_i-1}} $

This formula relies on the relationships $C_{v,i} = \frac{R}{\gamma_i-1}$ and $C_{p,i} = \frac{\gamma_i R}{\gamma_i-1}$ for each ideal gas component $i$, where $R$ is the ideal gas constant.

Calculation Steps

1. Calculate $\gamma_i - 1$ for each gas:
   • Gas 1: $\gamma_1 - 1 = \frac{5}{3} - 1 = \frac{2}{3}$
   • Gas 2: $\gamma_2 - 1 = \frac{4}{3} - 1 = \frac{1}{3}$
2. Calculate the terms $n_i / (\gamma_i - 1)$:
   • Gas 1: $\frac{n_1}{\gamma_1 - 1} = \frac{2}{2/3} = 2 \times \frac{3}{2} = 3$
   • Gas 2: $\frac{n_2}{\gamma_2 - 1} = \frac{3}{1/3} = 3 \times 3 = 9$
3. Calculate the sum of the denominator terms: $ \sum n_i \frac{1}{\gamma_i-1} = 3 + 9 = 12 $
4. Calculate the terms $n_i \gamma_i / (\gamma_i - 1)$:
   • Gas 1: $\frac{n_1 \gamma_1}{\gamma_1 - 1} = \frac{2 \times (5/3)}{2/3} = \frac{10/3}{2/3} = 5$
   • Gas 2: $\frac{n_2 \gamma_2}{\gamma_2 - 1} = \frac{3 \times (4/3)}{1/3} = \frac{4}{1/3} = 12$
5. Calculate the sum of the numerator terms: $ \sum n_i \frac{\gamma_i}{\gamma_i-1} = 5 + 12 = 17 $
6. Compute the mixture adiabatic index $\gamma_{\text{mix}}$: $ \gamma_{\text{mix}} = \frac{17}{12} $

Final Result

The calculated value is:

This value is approximately 1.42.