Question

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

a=(1/v) (dv/dT)_p    b= -(1/v) (dv/dp)_T

Show that for an isochoric change, b dp = a dT.

Answer 1

$$\frac{a}{b}=-\frac{(\frac{\partial V}{\partial T})_p}{(\frac{\partial V}{\partial p})_T}\\\frac{a}{b}=-\left(\frac{\partial V}{\partial T}\right)_p\left(\frac{\partial p}{\partial V}\right)_T\\\frac{a}{b}=\left(\frac{\partial p}{\partial T}\right)_V$$