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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
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B
β2R4α
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C
(α+β)R2β2
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D
α2R2β
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Solution

The correct option is B β2R4α
Given,
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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