Question

The value of coefficient of volume expansion of glycerin is $5\times {{10}^{-4}}{{K}^{-1}}$. The fractional change in the density of glycerin for a rise of ${{40}^{o}}C$ in its temperature is:
(A). $0.010$
(B). $0.020$
(C). $0.015$
(D). $0.025$

Answer 1

The density of a body is the mass per unit volume while volume is the three dimensional space occupied by an object. As volume changes with temperature, density also changes by an inverse relation. The change in volume for every unit change in temperature is given by coefficient of volume expansion. The energy of the system increases with increase in temperature.

Formulae used:
$\rho =\dfrac{M}{V}$
$\rho ={{\rho }_{0}}(1+\gamma \Delta T)$

Complete step-by-step solution:
Density of a body is the measure of its mass per unit volume. Its SI unit is $kg\,{{m}^{-2}}$. It is given as-
$\rho =\dfrac{M}{V}$
Here,
$\rho$ is the density
$M$is mass
$V$is volume
Let the initial volume of glycerin be ${{\rho }_{0}}$.

We know that, the new density on volume expansion is given as-
$\rho ={{\rho }_{0}}(1+\gamma \Delta T)$
Here,
$\gamma$ is the coefficient of volume expansion
$\Delta T$ is the change in temperature
Given temperature changes by 40 units, the value of $\gamma$ is $5\times {{10}^{-4}}{{K}^{-1}}$
We substitute the given values in the above equation to calculate new density
$\begin{aligned} & \rho ={{\rho }_{0}}(1+\gamma \Delta T) \\\ & \Rightarrow \rho ={{\rho }_{0}}(1+5\times {{10}^{-4}}\times 40) \\\ & \Rightarrow \rho ={{\rho }_{0}}(1+0.02) \\\ & \therefore \rho =1.02{{\rho }_{0}} \\\ \end{aligned}$
The fractional change in density is the change in density divided by initial density. Therefore,
$\begin{aligned} & \dfrac{\Delta \rho }{\rho }=\dfrac{1.02{{\rho }_{0}}-{{\rho }_{0}}}{{{\rho }_{0}}} \\\ & \therefore \dfrac{\Delta \rho }{\rho }=0.02 \\\ \end{aligned}$

Therefore, the fractional change in the density is $0.02$. So, the correct option is (B).

Note:
The coefficient of volume expansion is defined as the change in volume per unit change in temperature Density changes with volume but mass remains constant. Density is inversely proportional to the volume. As temperature increases, the volume of liquids also increases because as the molecules gain more energy they tend to move away from each other. The molecules in solids are very closely packed therefore; the change in volume in solids is negligible.