Ideal Gas Relation Between Specific Heats and Gas Constant For a calorically perfect gas, the gas constant R equals the difference between the specific heats at constant pressure and constant volume: R = Cp − Cv.

Introduction / Context:
Linking specific heats to the gas constant is foundational for ideal-gas property relations and for deriving many isentropic and energy equations used in thermodynamics and compressible flow.

Given Data / Assumptions:

• Ideal-gas behavior with constant specific heats (calorically perfect).
• Definitions: Cp is at constant pressure, Cv at constant volume.
• R is the specific gas constant (per unit mass).

Concept / Approach:

The Mayer relation gives Cp − Cv = R. Physically, Cp exceeds Cv because at constant pressure the gas must do boundary work during heating, requiring extra heat input per unit temperature rise compared to constant-volume heating.

Step-by-Step Solution:

Verification / Alternative check:

For air near room temperature: Cp ≈ 1.005 kJ/kg-K, Cv ≈ 0.718 kJ/kg-K, difference ≈ 0.287 kJ/kg-K, matching R for air.

Why Other Options Are Wrong:

Sum, product, or ratio do not appear in the Mayer relation and have no general meaning for R. Geometric mean is irrelevant.

Common Pitfalls:

Using molar quantities without noting units; mixing universal gas constant Ru with specific R (R = Ru/M).

Final Answer:

difference