A sample of an ideal gas initially having internal energy `U_(1)` is allowed to expand adiabatically performing work W. Heat Q is then supplied to it, keeping the volume constant at its new value, until the pressure raised to its original value. The internal energy is then `U_(3)` (see figure). find the increase in internal energy `(U_(3) - U_(1))`?

Options

A. `Q + W`
B. `Q - W`
C. `gamma W - Q`
D. `Q - gamma W`

Answer

B

Explanation

Process `1 rarr 2 ` is adiabatic
`Q_(12) = DeltaU_(12) + W_(12) = gt 0 = (U_2-U_1) + W`
`U_2 - U_1 = -W`
Process `2 rarr 3` is isochoric
`Q_(23) = Delta U_(23) + W_(23) implies Q = U_(3)-U_(2)`
`W_(23) = 0 :. V = consta nt`
`implies U_(3) -U_(2) = Q`
Now `(U_(3)-U_(2)) + (U_(2)-U_(1)) = U_(3)-U_(1)`
`implies Q _(-W) = U_(3) - U_(1)`
`implies U_3 = Q +U_(1) -W implies U_(3) -U_(1) = Q-W`.