Chapter 3, Problem 3.27NP

Interpretation Introduction

Interpretation:

For V as a state function:

  ${(\partial {(\partial V/\partial T)}_P/\partial P)}_T={(\partial {(\partial V/\partial P)}_T/\partial T)}_P.$

Using above relation, it is to be shown that the isothermal compressibility and isobaric expansion coefficient are related by ( $\partial \beta /\partial P{)}_T=-\left(\partial k/\partial T{)}_P\right)$ .

Concept Introduction :

The measure of change in relative volume of a state of matter in response to the change in pressure at constant temperature is known as isothermal compressibility factor. It is expressed by the symbol “ $\beta$ ” and its mathematical expression is:

  $\beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_P\mathrm{......}(1)$

The measure of change in relative volume of a state of matter in response to temperature change at constant pressure is known as isobaric volumetric thermal expansion coefficient. It is expressed by the symbol “k” and its mathematical expression is:

  $k=-\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_T\mathrm{......}(2)$

Here,

  $\beta$ = isobaric volumetric thermal expansion coefficient.

k = isothermal compressibility

  $\partial V$ = volume change

  $\partial P$ = pressure change

  $\partial T$ = temperature change