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

Question

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.

Answer 1

Given,

$\alpha = \frac{1}{V}(\frac{dV}{dT})_p$

$\beta = - \frac{1}{V}(\frac{dV}{dP})_T$

Now, from the ideal gas equation,

$PV=nRT$

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

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

$\frac{dV}{dT}=\frac{nR}{P}$

if P is constant then,

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

now, taking the differentiation with respect to P,

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

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

$\frac{(\frac{dV}{dT})_P}{(\frac{dV}{dP})_T}=\frac{\frac{nR}{P}}{\frac{nRT}{P^2}}$

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

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