Question

The coefficient of volume expansion of glycerin is $49\times {10}^{-5}{K}^{-1}$. What is the fractional change in its density for a ${30}^{o}C$ rise in temperature?

Answer 1

Coefficient of volume expansion of glycerin, $\gamma =49\times {10}^{-5}{K}^{-1}$
Rise in temperature, $T={30}^{o}C$
Fractional change in its volume $=\Delta V/V$
This change is related with the change in temperature as:
$\frac{\Delta V}{V}=\gamma \Delta T$
$\frac{\Delta V}{V}=49\times {10}^{-5}\times 30=0.0147$

Let $V:$ volume, $m:$ mass, $\rho :$ density

$m$ is constant by law of conservation of mass.

$m=\rho V$

$0=\rho dV+Vd\rho$

$\frac{d\rho }{\rho }=-\frac{dV}{V}=-0.0147$