Given,
α=V1(dTdV)p
β=−V1(dPdV)T
Now, from the ideal gas equation,
PV=nRT
V=PnRT
Now, taking the differentiation with respect to the corresponding terms,
dTdV=PnR
if P is constant then,
(dTdV)P=PnR...(i)
now, taking the differentiation with respect to P,
(dPdV)T=P2nRT...(ii)
Now, taking the ratio of (i) and (ii)
(dPdV)T(dTdV)P=P2nRTPnR
(dT)P(dP)T=TP
(dP)T=(dT)P×TP
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