For an ideal gas, an illustratio of three different paths `A(B+C)` and `(D+E)` from an initial state `P_(1), V_(1), T_(1)` to a final state `P_(2), V_(2),T_(1)` is shown in the given figure.

Path `A`represents a reversible isothermal expansion form `P_(1),V_(1)` to `P_(2),V_(2)`, Path `(B+C)` represents a reversible adiabatic expansion `(B)` from `P_(1),V_(1),T_(1)to P_(3),V_(2),T_(2)` followed by reversible heating the gas at constant volume `(C)`from `P_(3),V_(2),T_(2)` to `P_(2),V_(2),T_(1)`. Path `(D+E)` represents a reversible expansion at constant pressure `P_(1)(D)` from `P_(1),V_(1),T_(1)` to `P_(1),V_(2),T_(3)` followed by a reversible cooling at constant volume `V_(2)(E)` from `P_(1),V_(2),T_(3) to P_(2),V_(2),T_(1)`.
What is `q_(rev)`, for path `(D+E)`?

Options

A. `P(V_(2)-V_(1))`
B. `int_(T_(3))^(T_(1)) (C_(V)(T))/(T)dT`
C. `nRIn(V_(2))/(V_(1))`
D. `int_(T_(1))^(T_(3)) C_(V)(T)dT`

Explanation

`w =q = P_(1) (V_(2) -V_(1))`