Thermodynamics

JEE Main level, single correct, medium

60 of 190

Question 60 of 190

An ideal gas at pressure PP and temperature TT is expanding such that PT3=PT^3 = constant. The coefficient of volume expansion of the gas is _______.

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Answer: (C) 4T\dfrac{4}{T}

Given the process equation for the ideal gas:

PT3=constantPT^3 = \text{constant}

From the ideal gas equation, PV=nRTPV = nRT, which gives P=nRTVP = \dfrac{nRT}{V}.

Substituting PP into the process equation:

(nRTV)T3=constant\left(\dfrac{nRT}{V}\right) T^3 = \text{constant}

T4V=constant\dfrac{T^4}{V} = \text{constant}

V=kT4V = k T^4

Differentiating both sides with respect to temperature TT:

dVdT=4kT3\dfrac{dV}{dT} = 4k T^3

The coefficient of volume expansion γ\gamma is defined as:

γ=1VdVdT\gamma = \dfrac{1}{V} \dfrac{dV}{dT}

Substituting the values of VV and dVdT\dfrac{dV}{dT}:

γ=1kT4(4kT3)=4T\gamma = \dfrac{1}{k T^4} (4k T^3) = \dfrac{4}{T}

Answer: 4T\dfrac{4}{T}

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