Coefficient of volume expansion of glycerin, ${\mathrm{\alpha }}_{\mathrm{V}}$ = 49 × 10${}^{-5}{\mathrm{K}}^{-1}$

Rise in temperature, ΔT = 30°C

The fractional change in its volume =$\frac{∆\mathrm{V}}{\mathrm{V}}$

So,

$$\begin{gathered} \frac{∆\mathrm{V}}{\mathrm{V}}={\mathrm{\alpha }}_{\mathrm{v}}∆\mathrm{T} \\ \Rightarrow {\mathrm{V}}_{{\mathrm{T}}_2}-{\mathrm{V}}_{{\mathrm{T}}_1}={\mathrm{V}}_{{\mathrm{T}}_1}\mathrm{\alpha }∆\mathrm{T} \\ \Rightarrow \frac{\mathrm{m}}{{\mathrm{\rho }}_{{\mathrm{T}}_2-}}-\frac{\mathrm{m}}{{\mathrm{\rho }}_{{\mathrm{T}}_1}}=\frac{\mathrm{m}}{\mathrm{\rho }}{\mathrm{\alpha }}_{\mathrm{V}}∆\mathrm{T} \end{gathered}$$

where m = mass of glycerine

 ${\mathrm{\rho }}_{{\mathrm{T}}_1}$ = Initial density at T${}_1$

 ${\mathrm{\rho }}_{{\mathrm{T}}_2}$=Final density at T${}_2$

$\Rightarrow \frac{{\mathrm{\rho }}_{{\mathrm{T}}_1}-{\mathrm{\rho }}_{{\mathrm{T}}_2}}{{\mathrm{\rho }}_{{\mathrm{T}}_2}}={\mathrm{\alpha }}_{\mathrm{v}}∆\mathrm{T}$

$\therefore$Fractional change in the density of glycerin = 49 ×10${}^{-5}$× 30 = 1.47 × 10${}^{-2}$