The value $\gamma = \frac{C_P}{C_V}$ for hydrogen, helium, and another ideal diatomic gas $X$ (whose molecules

Question

The value $\gamma = \frac{C_P}{C_V}$ for hydrogen, helium, and another ideal diatomic gas $X$ (whose molecules are not rigid but have an additional vibrational mode), are respectively equal to:

Accepted Answer

1.  **Formula for $\gamma$:** The adiabatic index is given by $\gamma = 1 + \frac{2}{f}$, where $f$ is the number of degrees of freedom.
2.  **Hydrogen ($H_2$):** It is a diatomic gas. At ordinary temperatures, it behaves as a rigid rotator with $f = 3$ (translational) $+ 2$ (rotational) $= 5$. 
    $\gamma_{H_2} = 1 + \frac{2}{5} = \frac{7}{5}$.
3.  **Helium ($He$):** It is a monoatomic gas. It has only translational degrees of freedom, so $f = 3$.
    $\gamma_{He} = 1 + \frac{2}{3} = \frac{5}{3}$.
4.  **Gas $X$:** It is a diatomic gas with an additional vibrational mode. A vibrational mode contributes 2 degrees of freedom (potential + kinetic energy) per mode. Thus, $f = 3$ (translational) $+ 2$ (rotational) $+ 2$ (vibrational) $= 7$.
    $\gamma_{X} = 1 + \frac{2}{7} = \frac{9}{7}$.
5.  **Conclusion:** The respective values are $\frac{7}{5}, \frac{5}{3}, \frac{9}{7}$.

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