If 1.0 mol of a monatomic gas has a total internal energy of 5.0 ×10^3 J at a certain temperatu

step 1 We have a number of mole equals to 1 for the monoatomic gas and number of moles for the diatomic

If 1.0 mol of a monatomic gas has a total internal energy of...

If 1.0 mol of a monatomic gas has a total internal energy of $5.0 \times 10^{3} \mathrm{~J}$ at a certain temperature, what is the total internal energy of $1.0 \mathrm{~mol}$ of a diatomic gas at the same temperature?

Key Concepts

Equipartition Theorem

The equipartition theorem states that in thermal equilibrium every degree of freedom that appears quadratically in the energy contributes an average energy of (1/2)kT per molecule (or (1/2)RT per mole) to the system’s internal energy. This principle is fundamental in understanding how energy is distributed among the translational, rotational, and vibrational modes of gas molecules.

Degrees of Freedom

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 three translational degrees of freedom, while diatomic gases typically have additional rotational degrees of freedom (and vibrational degrees that may be inactive at lower temperatures), leading to a higher total energy storage at a given temperature.

Molecular Structure and Energy Storage

The molecular structure of a gas influences its internal energy because different types of motion contribute to the overall energy. Monatomic gases, consisting of single atoms, have only translational motion, whereas diatomic gases have both translational and rotational motion. This difference results in a higher internal energy for diatomic gases at the same temperature compared to monatomic gases.

Internal Energy of Gases

The internal energy of a gas is the total energy contained within it due to the motion and interactions of its molecules. In ideal gases, this energy is purely a function of temperature and the number of degrees of freedom available to the molecules. Understanding how internal energy is calculated for different types of gases is crucial in thermodynamics and helps explain differences in energy content under identical thermal conditions.

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