For an ideal gas, an illustration 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 `DeltaS` for path `A`?

Options

A. zero
B. `-nR ln. (V_(2))/(V_(1))`
C. `nRT_(1) ln. (V_(2))/(V_(1))`
D. `nRT_(1) ln .(V_(2))/(V_(1))`

Answer

D

Explanation

path (A) : isothermal `(Delta U = 0)`
`q_(rev) = w = nRT_(1) ln. (v_(2))/(v_(1))`
path `(B + C) : q_(rev) = 0 + (-nR ln.(v_(2))/(v_(1)))`
path `(D + E)`:
`(Delta S)_("sys") = nR ln.(v_(2))/(v_(1)) + .^(n)C_(V) ln. (T_(2))/(T_(1)) = nR ln. (v_(2))/(v_(1))`