A gaseous mixture contains equal number of hydrogen and...

A gaseous mixture contains equal number of hydrogen and nitrogen molecules. Specific heat measurements on this mixture at temperature below $150 \mathrm{~K}$ would indicate the value of $\gamma=C_{p} / C_{V}$ for the mixture as
(a) $3 / 2$
(b) $4 / 3$
(c) $5 / 3$
(d) $7 / 5$

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Key Concepts

Degrees of Freedom

Degrees of freedom in gas molecules denote the independent ways in which a molecule can store energy (e.g., through translational, rotational, and vibrational motions). At sufficiently low temperatures, vibrational modes are generally frozen due to their high energy gaps, leaving only translational and rotational degrees of freedom active in diatomic molecules.

Specific Heat Capacities

The specific heats at constant volume (Cv) and constant pressure (Cp) are directly related to the number of active degrees of freedom in a molecule. For ideal gases, each degree of freedom contributes a fixed amount of energy, so for diatomic molecules with five active degrees of freedom (three translational and two rotational), Cv is (5/2)R and Cp is (7/2)R.

Heat Capacity Ratio (gamma)

The heat capacity ratio, gamma, defined as Cp/Cv, is a crucial parameter in thermodynamics that characterizes the behavior of gases during adiabatic processes. For diatomic gases with five degrees of freedom, gamma is calculated as (7/2)R divided by (5/2)R, resulting in a value of 7/5.

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