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.
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Asum
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Bdifference
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Cproduct
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Dratio
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Egeometric mean
Answer
Correct Answer: difference
Explanation
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:
Start from h = u + pv for ideal gases (with h = CpT, u = CvT).Differentiate: dh/dT − du/dT = vdp/dT + pdv/dT.For ideal gas, pv = R*T and manipulations lead to Cp − Cv = R.Conclude R equals the difference of the two specific heats.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