One mole of a monoatomic ideal gas is taken through the cycle shown in Fig:
`AtoB`: adiabatic expansion
`BtoC`: cooling at constant volume
`CtoD`: adiabatic compression
`DtoA`: heating at constant volume

The pressure and temperature at A,B,etc. are denoted by `P_A, T_A,P_B,T_B`etc. respectively. Given that `T_A=1000K`, `P_B=(2//3)P_A and P_C=(1//3)P_A`, calculate the following quantities:
(i) The work done by the gas in the process `AtoB`
(ii) The heat lost by the gas in the process `BtoC`.
(iii) The temperature `T_D`. [Given :`(2//3)^(2//5)=0.85`]

Explanation

a. The work done in adiabatic process AB is given by
`W_(1) (R (T_(A) - T_(B)))/(gamma - 1) = (R (T_(A) - T_(B)))/((5//3) - 1)`
(as for monatomic gas `gamma = 5//3`)
or `= (3)/(2) R (T_(A) - T_(B))`
For adiabatic change, we have
`P_(A)^(gamma - 1) T_(A)^(gamma) = P_(B)^(gamma - 1) T_(B)^(-gamma)`
or ` ((P_(A))/(P_(B))) = ((T_(A))/(T_(B)))` or `((I_(A))/(I_(B))) = ((P_(A))/(P_(B)))^(2//5)`
or `((T_(A))/(T_(B))) = ((3)/(2))^(2//3)`
or `T_(B) = T_(A) ((2)/(3))^(2//3)`
`= 1000 xx 0.85 = 850 K`
From Eq. (i)
`W_(1) = (3)/(2) xx 8.31 xx (1000 - 850) = 1869.83 J`
b. Heat lost by the gas in process BC is given by
`C_(V) (T_(B) - T_(C )) = ((R )/(gamma - 1)) (T_(B) - T_(C )) = ((3)/(2)) R (T_(B) - T_(C ))`
Process BC is under constant volume , hence
`((P_(B))/(T_(B))) = ((P_(C ))/(T_(C )))` or `T_(C ) = ((P_(C ))/ (P_(B))) xx T_(B)`
or `T_(C ) = ((P_(A)//3))/(2P_(A)//3) T_(B) = ((T)/(2)) = 425 K`
From Eq. (ii) we get
Heat lost `= ((3)/(2)) xx 8.31 xx (850 - 425) = 5279.63 J`
c. For path AB
`P_(A)^(gamma - 1) T_(A)^(-gamma) = P_(B)^(gamma - 1) T_(B)^(-gamma)`
or `((P_(A))/(P_(B)))^(gamma - 1) = ((T_(A))/(T_(B)))^(gamma)`
For path BC
`((P_(B))/(T_(B))) = ((P_(C ))/(T_(C )))` or `((P_(B))/(P_(C ))) = ((T_(B))/(T_(C )))`
For path CD
`((P_(D))/(T_(B)))^(gamma - 1) = ((T_(D))/(T_(C )))^(gamma)`
Fro path AD
`((P_(A))/(P_(D))) = ((T_(A))/(T_(C )))`
Dividing Eq. (viii) by Eq. (vii), we get
`((P_(A))/(P_(B)) xx (P_(C ))/(P_(D)))^(gamma -1)` = `((T_(A))/(T_(B)) xx (T_(C ))/(T_(D)))^(gamma)`
Dividing Eq. (viii) by Eq. (iv) we get
`((P_(A))/(P_(B)) xx (P_(C ))/(P_(D)))` = `((T_(A))/(T_(B)) xx (T_(C ))/(T_(D)))` or `((P_(A))/(P_(B)) xx (P_(C ))/(P_(D)))^(gamma -1) = ((T_(A))/(T_(B)) xx (T_(C ))/(T_(D)))^(gamma - 1)`
`((T_(A))/(T_(B)) xx (T_(C ))/(T_(D)))^(gamma) = ((T_(A))/(T_(B)) xx (T_(C ))/(T_(D)))^(gamma - 1)`
`T_(A) T_(C ) = T_(D) T_(B)`
`1000 xx 425 = T_(D) xx 850`
`T_(D) = 500 K`