Question

One mole of a real gas is subjected to heating at constant volume from (P₁, V_1, T₁) state to (P₂, V_1, T₂) state. Then it is subjected to irreversible adiabatic compression against constant external pressure P₃ aim till system reaches final state (P_3, V_2, T₃). If the constant volume molar heat capacity of real gas is C_v. Find out correct expression for △H from state 1 to state 3.

(A) ​C_v (T₃ - T₁​) + ( ​P₃ V₁ - P₁ V₁)

(B) ​​C_v (T₂ - T₁​) + ( ​P₃ V₂ - P₁ V₁)

(C) ​ ​​C_v (T₂ - T₁​) + ( ​P₃ V₁ - P₁ V₁)

(D) ​​C_p (T₂ - T₁​) + ( ​P₃ V₁ - P₁ V₁)

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{Under}\mathrm{constant}\mathrm{volume}, \\ ∆\mathrm{H}=∆\mathrm{E}+∆\left(\mathrm{PV}\right) \\ \mathrm{Since},∆\mathrm{V}=0 \\ \mathrm{Therefore},∆\mathrm{H}=∆\mathrm{E}+\mathrm{V}∆\mathrm{P}={\mathrm{C}}_{\mathrm{V}}∆\mathrm{T}+{\mathrm{V}}_1{\mathrm{P}}_1+{\mathrm{V}}_2{\mathrm{P}}_2 \\ \mathrm{So},∆{\mathrm{H}}_{1-2}={\mathrm{C}}_{\mathrm{V}}({\mathrm{T}}_2-{\mathrm{T}}_1)+{\mathrm{V}}_1{\mathrm{P}}_1 \\ \mathrm{Under}\mathrm{irreversible}\mathrm{adiabtic}\mathrm{compression}, \\ \mathrm{q}=0 \\ \mathrm{and},∆\mathrm{E}=\mathrm{q}+\mathrm{w} \\ \mathrm{or},∆\mathrm{E}=\mathrm{w}={\mathrm{P}}_{\mathrm{ext}}∆\mathrm{V} \\ \mathrm{or},∆\mathrm{E}={\mathrm{P}}_3({\mathrm{V}}_2-{\mathrm{V}}_1) \\ \mathrm{So},∆{\mathrm{H}}_{2-3}=∆\mathrm{E}+∆\left(\mathrm{PV}\right)={\mathrm{P}}_3({\mathrm{V}}_2-{\mathrm{V}}_1)-{\mathrm{P}}_3{\mathrm{V}}_2=-{\mathrm{P}}_3{\mathrm{V}}_1 \\ \mathrm{Therefore},∆{\mathrm{H}}_{\mathrm{total}}=∆{\mathrm{H}}_{1-2}+∆{\mathrm{H}}_{2-3}={\mathrm{C}}_{\mathrm{V}}({\mathrm{T}}_2-{\mathrm{T}}_1)-({\mathrm{P}}_3{\mathrm{V}}_1-{\mathrm{P}}_1{\mathrm{V}}_1) \end{gathered}$$