Let r₀ and r_T be densities of glycerin at 0°C and T°C respectively. Then,

${\rho _T} = {\rho _0}\left( {1 - \gamma \Delta T} \right)$

where $\gamma$ is the coefficient of volume expansion of glycerine and ${\Delta T}$ is rise in temperature.

${{{\rho _T}} \over {{\rho _0}}} = 1 - \gamma \Delta T \Rightarrow \gamma \Delta T = 1 - {{{\rho _T}} \over {{\rho _0}}}$

Thus, ${{{\rho _0} - {\rho _T}} \over {{\rho _0}}} = \gamma \Delta T$

Here, $\gamma = 5 \times {10^{ - 4}}{K^{ - 1}}$ and $\Delta T = {40^o}C = 40K$

$\therefore$ The fractional change in the density of glycerin

$= {{{\rho _0} - {\rho _T}} \over {{\rho _0}}} = \gamma \Delta T = \left( {5 \times {{10}^{ - 4}}{K^{ - 1}}} \right)\left( {40K} \right) = 0.020$