An ideal gas having initial pressure P, volume V and temperature T is allowed to expands adiabatically until its volume becomes 5.66V while its temperature falls to T/2.
(i) How many degrees of freedom do the gas molecules have?
Obtain the work done by the gas during the expansion as a function of the initial pressure P and volume V.

Explanation

`underset(1)(P,V,T)overset("adiabetic Process")tounderset2(P',5.66V,(T//2)`
(a) `T_1V_1^(gamma-1)=T_2V_2^(gamma-1)`
`(T_1)/(T_2)=((V_2)/(V_1))^(gamma-1)`
`(T)/((T)/(2))=(5.66(V)/(V))^(gamma-1)`
`2=(5.66)^(gamma-1)`
Taking log on the base `e`
`ln2=(gamma-1)ln(5.66)`
`0.693=(gamma-1)(1.7334)`
`gamma-1=0.4impliesgamma=1.4=(7)/(5)`
`gamma=1+(2)/(f)`,`f:` degrees of freedom
`(7)/(5)=1+(2)/(f)impliesf=5`
(b) `DeltaW=(nR(T_1-T_2))/(gamma-1)=(nR((T-T)/(2)))/((7)/(5)-1)=((nRT)/(2))/((2)/(5))`
`=(5)/(4)nRT=(5)/(4)PV=1.25PV`