Two moles of an ideal monoatomic gas is taken through a cycle ABCA as shown in the P-T diagram. During the process AB , pressure and temperature of the gas very such that `PT=Constant`. It `T_1=300K`, calculate

(a) the work done on the gas in the process AB and
(b) the heat absorbed or released by the gas in each of the processes. Give answer in terms of the gas constant R.

Explanation

Number of moles, n `= 2,T_(1)=300K`
During the process `ArarrB`
PT= constant
or `P^(2)v= constant = K(say)`
`Therefore, P=sqrt(K)/sqrt(V)`
`Therefore`,
`W_(ArarrB)=overset(V_B)underset(V_A)intP.dV=overset(V_B)underset(V_A)intsqrt(K)/sqrt(V)dV`
`=2sqrt(K)[sqrt(V_B)-sqrt(V_(A))]`
`=2[sqrt(KV_(B)-sqrt(KV_(A)]`
`=2[sqrt((P_B^2V_B))V_B-sqrt((P_A^2V_A))V_A](k=P^2V)`
`=2[P_(B)V_(B)-P_(A)V_(A)]`
`=2[nRT_(B)-nRT_(A)]`
`=2nRT(T_1-2T_1]=(2)(2)(R)[300-600]=-1200R`
`Therefore` work don on the gas in the process AB is 1200R Heat absorbed//released in different process.
Since the gas is monoatomic, therefore,
`C_(v)=(3)/(2)randC_P=(5)/(2)R and gamma=(5)/(3)`
Process A-B
`triangleU=nC_(v)triangleT=(2)((3)/(2)R)(T_B-T_A)`
`=(2)((3)/(2)R)(300-600)=-900R`
`Q_(A)rarrB=W_Ararr+triangleU=(-1200r)-(1900r)`
`Q_(ArarrB)=-2100R`(Heat released) Exactly 3 different flavours
`therefore` 2 alike and 2 different `rArr ^(6)C_(1)xx^(5)C_(2)=60`
`(^(6)C_(1)rarr`Select one variety out of 6 and take 2 of same kind)
(`^(5)C_(2)rarr` Out of remaining 5 different varieties select 2 and take one ice cream of each kind)
=60 ans. 2differenet flavours
2alike and 2 alike `^6C_2=15`
3alike and 1 different"`^6C_(1)xx^(5)C_(1)=(30)/(45)`
+3different flavous `=(60)/(105)`Ans