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The coefficient of volume expansion of glycerin is $49 \times 10^{-5} \;K ^{-1} .$ What is the fractional change in its density for a $30\,^{\circ} C$ rise in temperature?
$3.64 \times 10^{-1}$
$8.48 \times 10^{-2}$
$1.47 \times 10^{-2}$
$4.75 \times 10^{-3}$
Solution
Coefficient of volume expansion of glycerin, $\alpha_V=49 \times 10^{-5} K ^{-1}$
Rise in temperature, $\Delta T=30^{\circ} C$
Fractional change in its volume $=\frac{\Delta V}{V}$ This change is related with the change in temperature as:
$\frac{\Delta V}{V}=\alpha_{V} \Delta T$
$V_{T_{2}}-V_{T_{1}}=V_{T_{1}} \alpha_{V} \Delta T$
$\frac{m}{\rho_{T_{2}}}-\frac{m}{\rho_{T_{1}}}=\frac{m}{\rho_{T_{1}}} \alpha_{V} \Delta T$
Where, $m=$ Mass of glycerine
$\rho_{T_{1}}=$ Initial density at $T_{1}$
$\rho_{r_{1}}=$ Final density at $T_{2}$
$\frac{\rho_{T_{1}}-\rho_{T_{2}}}{\rho_{T_{2}}}=\alpha_{V} \Delta T$
Where,
$\frac{\rho_{T_{1}}-\rho_{T_{2}}}{\rho_{T_{2}}}=$ Fractional change in density
$\therefore$ Fractional change in the density of glycerin $=49 \times 10^{-5} \times 30=1.47 \times 10^{-2}$
