One mole of helium is initially at pressure P₁ and volume V₁. It expands isothermally to 4V₁, then adiabatically to 32V₁. Let W₁ be the work done during the isothermal part and W₂ the work done during the adiabatic part. If W₁/W₂=f ln 2, find f.
Physics · Thermodynamics
JEE Advanced 2020

Question

One mole of helium is initially at pressure P₁ and volume V₁. It expands isothermally to 4V₁, then adiabatically to 32V₁. Let W₁ be the work done during the isothermal part and W₂ the work done during the adiabatic part. If W₁/W₂=f ln 2, find f.

Physics · Thermodynamics

JEE Advanced 2020

G1 Physics · Accepted Answer

For helium, γ=5/3. Let T₁ be the temperature before the adiabatic expansion and T₂ the temperature after it. The isothermal work is W₁=RT ln4=2RT ln2. The adiabatic relation gives T₂=T₁/4, so W₂=Cᵥ(T₁−T₂)=(3R/2)(3T₁/4)=9RT₁/8. Hence f=16/9≈1.78.