Temperature dependence of heat capacity:\nAt higher temperatures, how do molal heat capacities at constant pressure (Cp) of most gases change with temperature?

Introduction / Context:
Heat capacity Cp reflects the energy required to raise the temperature of a substance at constant pressure. For gases, Cp depends on the number of active degrees of freedom (translational, rotational, vibrational). As temperature rises, additional modes (especially vibrational) become thermally accessible, typically increasing Cp.

Given Data / Assumptions:

• Ideal-gas approximation within moderate to high temperature ranges.
• Most common diatomic and polyatomic gases considered.
• No dissociation at the temperatures discussed.

Concept / Approach:
According to statistical thermodynamics, vibrational modes are “frozen out” at low temperatures due to quantization. With increasing temperature, these modes become populated and contribute to internal energy and enthalpy changes, thus raising Cp. For monatomic gases, Cp is comparatively constant over wide ranges, but for diatomic and polyatomic gases, Cp generally increases with T, often nonlinearly and represented by NASA/Polynomial fits such as Cp/R = a + bT + cT^2 + ....

Step-by-Step Solution:

Verification / Alternative check:
Property tables show Cp(T) curves that rise with temperature for gases like CO2, N2, and H2O in the temperature ranges before dissociation becomes significant.

Why Other Options Are Wrong:

• “Decreases” or “unchanged” contradict empirical data for polyatomics.
• “Increases linearly” oversimplifies; the increase is typically nonlinear.
• “Oscillates” has no basis in equilibrium thermophysical behavior.

Common Pitfalls:
Assuming constant Cp in high-accuracy energy balances; use temperature-dependent Cp correlations when precision matters.

Final Answer:
Increases.