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

Here, `V_(2)=5.66 V_(1), T_(2)= T_(1)//2`
In an adiabatic process,
`T_(2)V_(2)^(gamma-1)= T_(1)V_(1)^(gamma-1)`
`((V_(2))/(V_(1)))^(gamma-1)= (T_(1))/(T_(2))=2`
`(5.66)^(gamma-1)=2`
`(gamma-1)log5.66=log2`
`gamma=(log2)/(log 5.66)+1`
`=0.4+1= 1.4`
`1+2/n= gamma=1.4`
`2/n=1.4-1= 0.4= 2/5`
`n=(5xx2)/2=5`
Work done is adiabatic expansion
`W=(nRDeltaT)/(1-gamma)= (nR(T//2-T))/(1-1.4)`
`=1.25n RT= 1.25PV`