Answer to Question #169514 in Molecular Physics | Thermodynamics for Khushi

Question #169514

For a thermodynamic system, isobaric coefficient of volume expansion (alpa) and isothermal compressibility (bita) are defined as

ailpa=1/V(dV/dT)p

bita=-1/V(dV/dT)T

Show that for an isochoric change, dp = dT.



1
Expert's answer
2021-03-09T15:29:47-0500

Given,

α=1V(dVdT)p\alpha = \frac{1}{V}(\frac{dV}{dT})_p


β=1V(dVdP)T\beta = - \frac{1}{V}(\frac{dV}{dP})_T


Now, from the ideal gas equation,


PV=nRTPV=nRT


V=nRTPV=\frac{nRT}{P}


Now, taking the differentiation with respect to the corresponding terms,


dVdT=nRP\frac{dV}{dT}=\frac{nR}{P}


if P is constant then,


(dVdT)P=nRP...(i)(\frac{dV}{dT})_P=\frac{nR}{P}...(i)


now, taking the differentiation with respect to P,

(dVdP)T=nRTP2...(ii)(\frac{dV}{dP})_T=\frac{nRT}{P^2}...(ii)

Now, taking the ratio of (i)(i) and (ii)(ii)


(dVdT)P(dVdP)T=nRPnRTP2\frac{(\frac{dV}{dT})_P}{(\frac{dV}{dP})_T}=\frac{\frac{nR}{P}}{\frac{nRT}{P^2}}


(dP)T(dT)P=PT\frac{(dP)_T}{(dT)_P}=\frac{P}{T}


(dP)T=(dT)P×PT(dP)_T=(dT)_P\times \frac{P}{T}


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