Question

The coefficient of volume expansion of a liquid is $5\times {10}^{-4}{{}^{\circ }\text{C}}^{-1}$. If its temperature is increased by ${30}^{\circ }\text{C}$, the percentage change in its density is

Answer 1

Given:
Coefficient of volume expansion of liquid, $\gamma =5\times {10}^{-4}{{}^{\circ }\text{C}}^{-1}$
Change in temperature, $\Delta T={30}^{\circ }\text{C}$
We know that, due to thermal expansion, density as a function of temperature is given by, $\rho =\frac{{\rho }_0}{(1+\gamma \Delta T)}$
$\Rightarrow \rho (1+\gamma \Delta T)={\rho }_0$
$\Rightarrow \rho -{\rho }_0=-\rho \gamma \Delta T$
$\Rightarrow \frac{\rho -{\rho }_0}{\rho }\times 100=-\gamma \Delta T\times 100$
$=-5\times {10}^{-4}\times 30\times 100=-1.5\%$
The negative sign indicates that the density decreases with increase in temperature.