Ideal-gas specific heats – identities connecting cp, cv, γ, and R Which of the following relations correctly connect specific heats at constant pressure and volume with the gas constant and γ = cp/cv?

Introduction / Context:
For ideal gases with temperature-independent specific heats over a range, cp, cv, γ, and the gas constant R are linked by simple identities used throughout thermodynamics and gas dynamics.

Given Data / Assumptions:

• Ideal-gas behaviour.
• Specific heats cp and cv are taken as constants over the temperature interval.
• γ is defined as cp/cv.

Concept / Approach:
From the ideal-gas relations and definitions, two cornerstone identities follow: cp − cv = R and cp/cv = γ. Combining them yields convenient forms for cp and cv in terms of R and γ: cp = γR/(γ − 1) and cv = R/(γ − 1).

Step-by-Step Solution:
Start with cp − cv = R.Use γ = cp/cv ⇒ cp = γcv.Substitute into cp − cv = R → γcv − cv = R → cv(γ − 1) = R.Solve: cv = R/(γ − 1) and cp = γR/(γ − 1).

Verification / Alternative check:
Dimensional check: cp, cv, and R share units J/(kg·K) (for specific values); formulas are dimensionally consistent. Plugging back reproduces cp − cv = R and γ = cp/cv.

Why Other Options Are Wrong:

• (a) is true by definition but incomplete alone; the question asks for relations connecting to R as well.
• (e) cp − cv = γR is incorrect; it would imply cp/cv varies improperly with R.

Common Pitfalls:
Confusing molar vs. specific (per mass) quantities; the same relations hold for both as long as R is the corresponding gas constant (Rmolar or Rspecific).

Final Answer:
Both (b) and (c)