Reversible adiabatic (isentropic) ideal-gas relation: For a reversible adiabatic process between states 1 and 2 of an ideal gas, which expression correctly gives T1/T2?

Introduction / Context:
Isentropic (reversible adiabatic) relations connect temperature, pressure, and volume changes of ideal gases and appear in nozzle, compressor, and turbine analyses. Choosing the correct exponent and arrangement is critical for accurate calculations.

Given Data / Assumptions:

• Ideal gas with constant specific heats; γ = cp/cv.
• Process is adiabatic and reversible (isentropic).
• States 1 and 2 with (p1, v1, T1) and (p2, v2, T2).

Concept / Approach:
For an ideal gas under isentropic change, p * v^γ = constant and T * v^(γ − 1) = constant. From the latter, T1 * v1^(γ − 1) = T2 * v2^(γ − 1). Rearranging gives T1/T2 = (v2/v1)^(γ − 1). Alternative correct forms include T1/T2 = (p1/p2)^((γ − 1)/γ) and T2/T1 = (v1/v2)^(γ − 1). Only the volume-based expression in the offered list matches the correct relation exactly as written.

Step-by-Step Solution:

Verification / Alternative check:
Combine pv^γ = const with ideal-gas law to obtain Tp^((1 − γ)/γ) = const, which is equivalent and yields the same temperature ratio when expressed in pressure terms.

Why Other Options Are Wrong:

• (v1/v2)^(γ − 1): Reciprocal of the correct ratio.
• (p1/p2)^(γ − 1): Wrong exponent; should be ((γ − 1)/γ).
• (p2/p1)^((γ − 1)/γ): Reciprocal in pressure form.

Common Pitfalls:
Confusing exponents γ − 1 and (γ − 1)/γ; flipping state indices inadvertently.

Final Answer:
T1/T2 = (v2/v1)^(γ - 1)