A gas has molar heat capacity `C = 37.55 J "mole"^(-1)K^(-1)`, in the process PT = constant, find the number of degree of freedom of the molecules of the gas.

Explanation

Given that PT=K(constant)....(i)
But `PV=RTrArrP=(RT)/(V)`
`therefore (RT)/(V)xxT=K`
`rArrV=(RT^(2))/(K):(dV)/(dT)=(2RT)/(K)`
`(dV)/(dT)=(2R)/(P) (because PT=K)` ....(ii)
From first law of thermodynamics
`dQ=dU+dW , CdT=C,dT+PdV`
`C=C_(v)+P((dV)/(dT))` as , `C=C_(v)+P(dV)/(dT)`
` C=C_(v)+P((2R)/(P))rArrC-C_(v)=2R`
`rArr C_(v)=C-2R` .....(iii)
But `C _(v)=(fR)/(2)` ..... (iv) from (iii) and (iv)
`(fR)/(2)=C-2RrArrf=(2(C-2R))/(R)`
`=(2(37.55-2xx8.3))/(8.3)=5`