One mole of a gas is in state A[P, V, T_A]. A small adiabatic process causes the state of the gas to change to B [P + dP, V + dV, T_B]. The changes dV & dP are infinitesimally small and dV is negative. An alternative process takes the gas from state A to B via A → C → B. A → C is isochoric and C → B is isobaric path. State at C is [P + dP, V, T_C]. Find $γ (= \frac{C_{P}}{C_{V}})$ of the gas in terms of T_A, T_B and T_C.

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$\frac{T_{B} - T_{A}}{T_{C} - T_{B}}$

$\frac{T_{C} - T_{B}}{T_{C} - T_{A}}$

$\frac{T_{C} - T_{A}}{T_{A} - T_{B}}$

$\frac{T_{C} - T_{A}}{T_{C} - T_{B}}$

answer is D.

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Detailed Solution

A → B (Adiabatic)
dW = PdV
dQ = 0
$∴$ dU = – dW = – PdV
A → C (Isochoric)
dW = 0
dU₁= dQ = C_V (T_C – T_A)
C → B (isobaric)
dU₂ = dQ – dW = – C_p (T_C – T_B) – Pdv
$∵$ dU = dU₁ + dU₂
$∴$ – PdV = C_V (T_C – T_A) – C_p(T_C – T_B) – PdV
$∴ \frac{C_{p}}{C_{V}} = \frac{T_{C} - T_{A}}{T_{C} - T_{B}}$