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

Path (A) represents a reversible isothernal expanison from `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)` to `P_(1) , V_(2), T_(3)` followed by reversible cooling at a constant volume `V_(2)` (E) from `P_(1), V_(2), T_(3)` to `P_(2), V_(2), T_(1)`
What is `Delta S` for path `(D + E)` ?

Options

A. zero
B. `underset(T_(1))overset(T_(2))int (C_(V)(T))/(T) dT`
C. `-nR ln. (V_(2))/(V_(1))`
D. `nR 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))`