Question

The temperature T of one mole of an ideal gas varies with its volume V as T=−αV3+βV2, where α and β are positive constants. The maximum pressure of the gas during this process is

A
αβ2R
B
β2R4α
C
(α+β)R2β2
D
α2R2β

Solution

The correct option is B β2R4αGiven, T=−αV3+βV2 .....(1) Ideal gas equation for 1 mole of an ideal gas is given by PV=RT .....(2) Using (2) in (1), we can write that P=(−αV2+βV)R ......(3) On differentiating both sides with respect to V, dPdV=(−2αV+β)R .....(4) To find the maxima, dPdV=0 and d2PdV2<0 From (4), we can say that, 0=(−2αV+β)R ⇒V=β2α ....(5) and d2PdV2=−2αR<0 Using (5) in (3), we get P=(−αβ24α2+β22α)R ⇒P=β2R4α Thus, option (b) is the correct answer.Physics

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