Question

You have $mkg$ of a certain liquid in a vessel, which is heated till there is a temperature change of $\Delta T$. If the coefficient of volume expansion is $\gamma$, what is the density of the liquid after heating, if the initial density was $\rho$?

• A. ${\rho }_2={\rho }_1(1+\gamma \Delta T)$
• B. ${\rho }_2=\frac{{\rho }_1}{1+\gamma \Delta T}$
• C. ${\rho }_2={\rho }_1(\gamma \Delta T-1)$
• D. ${\rho }_2=\frac{{\rho }_1}{1-\gamma \Delta T}$

Answer 1

The correct option is B ${\rho }_2=\frac{{\rho }_1}{1+\gamma \Delta T}$

The volumetric expansion can be expressed as,
$V_2=V_1(1+\gamma \Delta T)$
where, $V_1$ is the initial volume, $V_2$ is the final volume, $\gamma$ is the coefficient of linear expansion and $\Delta T$ is the change in temperature.
Let ${\rho }_1$ be the density of liquid when the volume is $V_1$ and ${\rho }_2$ be the density of the liquid when the volume is $V_2$.
We know $volume=\frac{mass}{density}$
Mass $m$ of the liquid remains constant.
$\Rightarrow \frac{m}{{\rho }_2}=\frac{m}{{\rho }_1}(1+\gamma \Delta T)$
or, ${\rho }_2=\frac{{\rho }_1}{1+\gamma \Delta T}$