Gas constant R – relation to specific heats For a calorically perfect ideal gas, the gas constant R is equal to which of the following combinations of the mass-specific heat capacities?

Introduction / Context:The link between the gas constant R and the specific heats cp and cv is central to ideal-gas thermodynamics. Remembering the correct algebraic relation prevents frequent mistakes in property calculations and cycle analysis.

Given Data / Assumptions:

• Ideal gas with constant specific heats (calorically perfect).
• Mass-specific heats cp and cv.
• Definition of gamma: γ = cp / cv.

Concept / Approach:

For ideal gases, enthalpy h = cp * T and internal energy u = cv * T. The enthalpy–internal energy relation is h − u = R * T, which rearranges to (cp − cv) * T = R * T. Cancel T to obtain the fundamental identity R = cp − cv. This holds on both a per–unit-mass and per–mole basis (with appropriate R values).

Step-by-Step Solution:

Verification / Alternative check:

Measured values for air at 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/ratio/square do not satisfy the identity derived from h − u and the ideal-gas equation of state.

Common Pitfalls:

Confusing γ = cp / cv with R = cp − cv; both appear often but represent distinct relations.

Final Answer:

difference of two specific heats