(a) How much heat does it take to increase the temperature...

(a) How much heat does it take to increase the temperature of 2.50 mol of a diatomic ideal gas by 50.0 $\mathrm{K}$ near room temperature if the gas is held at constant volume? (b) What is the answer to the question in part (a) if the gas is monatomic rather than diatomic?

Key Concepts

Molar Heat Capacity

The molar heat capacity at constant volume (Cv) quantifies the amount of heat required per mole per unit temperature change. For ideal gases, its value depends on the degrees of freedom available. Typically, for a monatomic ideal gas, Cv is 3/2R, and for a diatomic ideal gas (at temperatures where vibrational modes are inactive), Cv is taken as 5/2R due to the addition of rotational degrees of freedom.

Degrees of Freedom in Ideal Gases

The degrees of freedom are the number of independent ways in which a gas molecule can store energy. Monatomic gases, being single-atom particles, have only translational degrees of freedom, whereas diatomic gases have additional rotational degrees of freedom. This difference affects their molar heat capacities, with diatomic gases generally having a higher Cv than monatomic gases under similar conditions.

Ideal Gas Behavior

Ideal gas behavior is a useful approximation where the gas particles do not interact except through elastic collisions and occupy negligible volume. This assumption allows using the equipartition theorem to relate the internal energy of the gas to its temperature and the number of degrees of freedom, simplifying the calculation of quantities like heat transfer in processes such as heating at constant volume.

Heat Transfer in Constant Volume Processes

In a constant volume process, the work done is zero (since work is given by P?V and ?V is zero), so the heat added to the system directly increases its internal energy. The relation used here is Q = ?U = nCv?T, where Q is the heat added, n is the number of moles, Cv is the molar heat capacity at constant volume, and ?T is the change in temperature.

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