Equal volumes of monoatomic and diatomic gases at the same initial temperature and pressure are mixed. The rat

Question

Equal volumes of monoatomic and diatomic gases at the same initial temperature and pressure are mixed. The ratio of specific heats of the mixture ($C_p/C_v$) will be [AFMC 2002]:

Accepted Answer

To find the ratio of specific heats ($γ$) for a mixture, we use the values of molar heat capacities for monoatomic and diatomic gases.

1.  **Avogadro's Law:** According to the sources, equal volumes of gases at the same temperature and pressure contain an equal number of moles ($n$) . Thus, we assume $n_1$ (monoatomic) = $n_2$ (diatomic) = 1 mole.
2.  **Molar Heat Capacities:** From standard thermodynamic data :
    *   **Monoatomic gas:** $C_{v1} = \frac{3}{2}R$ and $C_{p1} = \frac{5}{2}R$.
    *   **Diatomic gas:** $C_{v2} = \frac{5}{2}R$ and $C_{p2} = \frac{7}{2}R$.
3.  **Heat Capacities of the Mixture:**
    *   $C_{v,mix} = \frac{n_1C_{v1} + n_2C_{v2}}{n_1 + n_2} = \frac{1.5R + 2.5R}{2} = 2R$.
    *   $C_{p,mix} = \frac{n_1C_{p1} + n_2C_{p2}}{n_1 + n_2} = \frac{2.5R + 3.5R}{2} = 3R$.
4.  **Ratio (γ):** The ratio of specific heats of the mixture is $\gamma_{mix} = \frac{C_{p,mix}}{C_{v,mix}} = \frac{3R}{2R} = 1.5$.

This calculation aligns with Option D.

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