A gas mixture consists of 2 mole of oxygen and 4 mole of...

A gas mixture consists of 2 mole of oxygen and 4 mole of argon at temperature $\mathrm{T}$. Neglecting all vibrational modes, the total internal energy of the system is
(A) $11 \mathrm{RT}$
(B) $9 \mathrm{RT}$
(C) $15 \mathrm{RT}$
(D) $4 \mathrm{RT}$

Key Concepts

Equipartition Theorem

The equipartition theorem is a fundamental principle in statistical mechanics that states that, at thermal equilibrium, energy is equally distributed among all available degrees of freedom. Each quadratic degree of freedom contributes an equal amount of energy (typically ½ kT per particle, or ½ RT per mole) to the system’s total internal energy. This theorem is crucial for computing the internal energy of gases based on their molecular motion and structure.

Degrees of Freedom in Gases

Degrees of freedom refer to the independent ways in which a molecule can store energy, such as translational, rotational, and vibrational motions. For monatomic gases, there are typically three translational degrees of freedom, while diatomic gases have additional rotational degrees of freedom (usually two, ignoring vibrational contributions at moderate temperatures). The number of degrees of freedom directly affects the calculation of a gas's internal energy.

Internal Energy of Ideal Gases

The internal energy of an ideal gas is determined by the sum of the energy contributions from all its active degrees of freedom. Using the equipartition theorem, the energy per mole is calculated as (f/2)RT, where 'f' is the number of degrees of freedom and RT represents the product of the gas constant and absolute temperature. This approach allows one to compute the internal energy for both monatomic and diatomic gases under ideal conditions.

Gas Mixtures

When dealing with mixtures of ideal gases, the total internal energy of the system is the sum of the internal energies of the individual components. Each gas contributes according to its number of moles and specific degrees of freedom. This additive property is important for understanding the thermodynamic behavior of gas mixtures without interactions between different species.

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