If for a gas (R)/(Cv)=0.67, this gas is made up of molecules which are: (a) Monatomic (b) Diatom

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If for a gas (R)/(Cv)=0.67, this gas is made up of molecules...

Key Concepts

Relationship Between Heat Capacities and Molecular Structure

This concept connects the measured ratios of the gas constant to the specific heat capacity at constant volume (R/C_v) with the nature of the gas molecules. Since monatomic gases possess only translational degrees of freedom, they have lower specific heat capacities compared to diatomic or polyatomic gases which have additional rotational and vibrational modes. By analyzing R/C_v, one can infer details about the molecular composition and complexity of the gas.

Equipartition Theorem

The equipartition theorem states that at thermal equilibrium, energy is distributed equally among all available degrees of freedom, with each degree contributing a fixed amount (typically 1/2 times the Boltzmann constant per molecule, or 1/2 R per mole) to the internal energy. This principle is used to relate the specific heat capacities of a gas to its molecular structure, and it serves as a foundation for understanding how different types of gases (monatomic, diatomic, polyatomic) store thermal energy.

Specific Heat Capacity at Constant Volume (C_v)

This concept refers to the amount of energy required to raise the temperature of a unit amount of a gas by one degree while keeping the volume constant. It is directly related to the degrees of freedom of the gas molecules, where each degree of freedom contributes a specific amount of energy according to the equipartition theorem. Understanding C_v helps in characterizing the energy storage capacity of the gas's internal modes of motion, such as translational, rotational, and vibrational.

Degrees of Freedom

Degrees of freedom are the independent ways in which the molecules of a gas can store energy. In gases, these include translational motions (common to all gases) and additional modes like rotational and vibrational motions that become significant depending on the molecular structure. Monatomic gases, for example, have three translational degrees of freedom, while diatomic and polyatomic gases have extra rotational and vibrational degrees, which results in higher values of C_v.

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