Heat capacities of gases:\nAt which absolute temperature condition can Cp equal Cv for a substance?

Introduction / Context:
For ideal gases, Cp − Cv = R holds independent of temperature, implying Cp ≠ Cv at ordinary temperatures. However, heat capacities are thermodynamic response functions that trend to zero as temperature approaches absolute zero (Third Law behavior for real substances). The question asks under what limiting condition Cp could equal Cv.

Given Data / Assumptions:

• Cp and Cv are heat capacities at constant pressure and volume, respectively.
• For an ideal gas, Cp − Cv = R > 0 at finite temperature.
• Third Law: as T → 0 K, heat capacities of real substances approach 0.

Concept / Approach:
While Cp − Cv = R is a strict ideal-gas relation, real materials at the absolute zero limit have vanishing heat capacities. In that limit, both Cp and Cv tend to 0 and therefore become equal in the limit. Thus, the only correct selection consistent with thermodynamics is 0 K (absolute zero). At finite temperatures, Cp exceeds Cv for gases because expansion work at constant pressure requires additional energy input compared to constant volume heating.

Step-by-Step Solution:

Verification / Alternative check:
Statistical thermodynamics shows accessible degrees of freedom freeze out at very low T, driving heat capacities toward zero. This reconciles the limiting equality.

Why Other Options Are Wrong:

• 0 °C, 0 °F, 0 °R: finite temperatures where Cp still exceeds Cv for gases.
• 'Never': incorrect because in the T → 0 K limit, both approach zero and become equal.

Common Pitfalls:
Applying Cp − Cv = R outside its domain (ideal gases at finite T); ignoring the Third Law limit; conflating ideal-gas identities with real-substance low-temperature behavior.

Final Answer:
0 K (absolute zero)