Two moles of an ideal monoatomic gas, initially at pressure `p_1` and volume `V_1`, undergo an adiabatic compression until its volume is `V_2`. Then the gas is given heat Q at constant volume `V_2`.
(i) Sketch the complete process on a p-V diagram.
(b) Find the total work done by the gas, the total change in its internal energy and the final temperature of the gas. [Give your answer in terms of `p_1,V_1,V_2, Q and R`]

Answer

(i) `(##ALN_PHY_C07(I)_E01_440_A01##)` (ii) `W = (3)/(2) P_(1)V_(1)[1-((V_(1))/(V_(2)))^(2//3)] , DeltaU = (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3) - 1] , T= (Q)/(3R) + (P_(1)V_(2))/(2R) ((V_(1))/(V_(2)))^(2//3)`

Explanation

The `P-V` diagram for the complete process will be as follows

Process `A rarr B` is adiabatic compression and
Process `B rarrC` is isochoric.
(b) (i) Total work done by the gas process `A-B: W_(AB) = (P_(A)V_(A)-P_(B)V_(B))/(gamma-1) = (P_(i)V_(i) - P_(f)V_(f))/(gamma-1) = (P_(1)V_(1)-P_(2)V_(2))/((5)/(3)-1)`
=`(P_(1)V_(1)-P_(1)((V_(1))/(V_(2)))^(gamma)V_(2))/(2//3)[{:(P_(1)V_(1)^(gamma)= P_(2)V_(2)^(gamma)),(thereforeP_(2)=P_(1)((V_(1))/(V_(2)))^(gamma)):}]`
=`(3)/(2)P_(1)V_(1)[1-((V_(1))/(V_(2)))^(gamma-1)] = (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(5//3-1)-1]`
=`-(3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3) - 1]`
Process `B-C : W_(BC) = 0` (V= constant)
`therefore W_(Total) = W_(AB) + W_(BC) = -(3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3)-1]`
(ii) Total change in internal energy
Process `A-B: Q_(AB) = 0` (Process is adiabatic)
`therefore DeltaU_(AB) = -W_(AB) = (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3)-1]`
Process `B-C: W_(BC) = 0`
`therefore DeltaU_(BC) = Q_(BC) = Q` (Given)
`therefore DeltaU_(Total)= DeltaU_(AB) + DeltaU_(BC) = (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3)-1] + Q`
(iii) Final temperature of the gas
`DeltaU_(Total) = nC_(V)DeltaT = 2((R)/(gamma-1))(T_(C) - T_(A))`
`therefore (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3)-1] + Q = (2R)/(5//3-1)(T_(C) - (P_(A)V_(A))/(2R))`
`implies (3)/(2)P_(1)V_(1)[((V_(1))/(V_(2)))^(2//3)-1] + Q = 3R (T_(C) - (P_(1)V_(1))/(2R))`
`therefore T_(C) = (Q)/(3R) + (P_(1)V_(1))/(2R)((V_(1))/(V_(2))^(2//3) = T_(final)`