Final answer:
Heat capacity of ideal gases can be understood through the first law of thermodynamics and the equipartition theorem. It reflects the distribution of energy across the molecules' degrees of freedom, with each degree of freedom contributing equally to the gas's heat capacity.
Step-by-step explanation:
The heat capacity of an ideal gas refers to how much heat energy is required to raise the temperature of a sample of the gas by one degree Celsius. According to the first law of thermodynamics, during an isothermal process, changes in internal energy (∆Eint) are zero and heat (Q) is equal to the work done (W).
During an adiabatic process, no heat is exchanged (Q = 0), so the change in internal energy is equal to the negative of the work done. In an isobaric process, the change in internal energy is the heat added minus the work done (∆Eint=Q-W), and in an isochoric process, there's no work done (W = 0), which makes the internal energy change equivalent to the heat added (∆Eint = Q).
For ideal gases, the molar heat capacity at constant pressure (Cp) is related to the molar heat capacity at constant volume (Cy) and the gas constant (R) by the formula Cp = Cy + R. This relationship arises because the total energy of an ideal gas is equally distributed among its degrees of freedom.
The molar heat capacity depends on the degrees of freedom (d) that a molecule can possess - translational, rotational, and, to a lesser extent, vibrational. Each degree of freedom contributes an equal share to the heat capacity. Thus, for monatomic gases, which have mostly translational degrees of freedom, the molar heat capacity is relatively low compared to polyatomic gases that also have rotational and perhaps vibrational degrees of freedom.
The equipartition theorem aids in understanding this: energy is partitioned equally among the degrees of freedom, so molar heat capacity is proportional to the number of degrees of freedom (d). For an ideal gas, we calculate the heat capacity with the volume held constant to avoid dealing with changes in volume or pressure with temperature, which aren't significant for solids or liquids but are for gases. Consequently, the molar heat capacities of ideal gases at room temperature can be understood and calculated through these principles.