Question

Suppose that the pressure on a solid is increased quasistatically and isothermally from P to Pr. Show that the change in volume in an infinitesimal step in this process is given by

Answer 1

$PV=constant$

$PdV+VdP=0$

$B=\frac {dP}{\frac {dV}{V}}$ (Definition of the bulk modulus)

$dV=-\left (\frac {V}{B} \right )dP$

$W_{on}=-PdV=\frac {V}{B}P^2dP$

$W_{on}=\frac {V}{B}\int_{P_i}^{P_f}P^2dP$

$W_{on}=\left (\frac {V}{2B} \right )(P_f^2-P_i^2)$