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 `(A)`?

Options

A. `P (V_(2)-V_(1))`
B. `-nRT_(1)In(V_(2))/(V_(1))`
C. `-nR In(V_(2))/(V_(1))`
D. `-nRT_(1)In(V_(2))/(V_(1))`

Explanation

According to the first law of thermodynamics
`DeltaU = q -w`
In isothermal process, `DeltaU = 0`
`:.q = w`
or `w = nRT_(1) In(V_(2))/(V_(1))`