Question

The coefficient of volume expansion of glycerine is $49\times {10}^{-5}{K}^{-1}.$ What is the fractionl change in its density for a $30{}^{\circ }C$ rise in temperature?

Answer 1

Given, coffcient of volume expansion of glycerine,${\alpha }_V=49\times {10}^{-5}{K}^{-1}$ Rise in temperature, $\Delta T=30{}^{\circ }C$
Let the volumes of glycerine at $T_2\text{and}{T}_2\text{be}{V}_1\text{and}{V}_2,$ then $V_{T_2}=V_{T_1}(1+{\alpha }_V\Delta T)$

As we know volume = mass/density
So, $\frac{m}{{\rho }_{T_2}}=\frac{m}{{\rho }_{T_1}}(1+{\alpha }_V\Delta T)$

Here, $m=\text{mass of glycerine}$
${\rho }_{T_1}=\text{Initial density at}{T}_1$
${\rho }_{T_2}=\text{final density at}{T}_2$

Therefore,
$\frac{m}{{\rho }_{T_2}}-\frac{m}{{\rho }_{T_1}}=\frac{m}{{\rho }_{T_1}}={\alpha }_V\Delta T$

$\frac{{\rho }_{T_2}-{\rho }_{T_2}}{{\rho }_{T_2}}={\alpha }_V\Delta T$

$\frac{{\rho }_{T_1}-{\rho }_{T_2}}{{\rho }_{T_2}}=49\times {10}^{-5}\times 30$

$\frac{{\rho }_{T_1}-{\rho }_{T_2}}{{\rho }_{T_2}}=1.47\times {10}^{-2}$

Fractionl change in density of glycerine

$\approx \frac{{\rho }_{T_1}-{\rho }_{T_2}}{{\rho }_{T_1}}$

$1.47\times {10}^{-2}$