ANS:B - RT dlnf

The chemical potential (𝜇μ) for a real gas can be expressed in terms of the partial molar volume (𝑉ˉVˉ), which can be further related to pressure and fugacity. The correct expression for the chemical potential of a real gas is: 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓𝑃∘)μ=μ∘+RTln(P∘f​) Where:

• 𝜇∘μ∘ is the standard chemical potential,
• 𝑅R is the gas constant,
• 𝑇T is the temperature,
• 𝑓f is the fugacity of the gas,
• 𝑃∘P∘ is the reference pressure.

So, in terms of the change in fugacity (𝑑𝑓df), the expression for the chemical potential would be: 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓+𝑑𝑓𝑃∘)μ=μ∘+RTln(P∘f+df​) 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓𝑃∘+𝑑𝑓𝑃∘)μ=μ∘+RTln(P∘f​+P∘df​) 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓𝑃∘)+𝑅𝑇ln⁡(1+𝑑𝑓𝑓)μ=μ∘+RTln(P∘f​)+RTln(1+fdf​) 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓𝑃∘)+𝑅𝑇(𝑑𝑓𝑓)μ=μ∘+RTln(P∘f​)+RT(fdf​) In the limit of infinitesimal changes, 𝑑𝑓df, the last term becomes 𝑑(ln⁡𝑓)d(lnf). So, the correct expression for the chemical potential is: 𝜇=𝜇∘+𝑅𝑇ln⁡(𝑓𝑃∘)+𝑅𝑇(𝑑(ln⁡𝑓))μ=μ∘+RTln(P∘f​)+RT(d(lnf)) 𝜇=𝑅𝑇ln⁡𝑓+constantμ=RTlnf+constant So, the correct option is: 𝑅𝑇 𝑑(ln⁡𝑓)RTd(lnf).