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Q.
The coefficient of volume expansion of liquid is $\gamma$. The fractional change in its density for $\Delta T$ rise in temperature is
Thermal Properties of Matter
Solution:
On thermal expansion,
Volumetric expansion is given by
$
V = V _{0}(1+\gamma \Delta T ) \ldots \ldots(1)
$
We know that, density, $d =\frac{\text { mass }}{\text { volume }}$ $d =\frac{ m }{ V }$
where, $m =$ constant $d \propto \frac{1}{ V }$
Density of the liquid varies as
$
\begin{array}{l}
d = d _{0}(1+\gamma \Delta T ) \\
d = d _{0}+ d _{0} \gamma \Delta T
\end{array}
$
Fractional change in density is
$
\begin{array}{l}
\frac{ d - d _{0}}{ d _{0}}=\gamma \Delta T \\
\frac{\Delta d }{ d _{0}}=\gamma \Delta T
\end{array}
$