The coefficient of volume expansion of glycerine is 49 × 10^-5 K^-1. Find the fractional change in its density for a 20° C rise in temperature?

Question

Question

This question was previously asked in

AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 3)

Answer 1

CONCEPT

According to the formula for the increase in volume due to temperature change

\(V_f = V_0(1+γΔ T)\)

Where γ = coefficient of volume expansion

ΔT is the change in volume

V₀ and V_f are the initial and final volumes respectively

CALCULATION

Given: γ = 49 × 10^-5K^-1, ΔT = 20^oC

Now

V_f = V₀(1+ 49 × 10^-5× 20)

\(\frac{V_f}{V_0} = 1.0098\)

For the change in density

\(\rho = \frac{m}{V}\)

the fractional change in density

\(\frac{\rho_f - \rho_0}{\rho_0} = \frac{\frac{m}{V_f} - \frac{m}{V_0}}{\frac{m}{V_0}}\)

\(⇒ \frac{\Delta \rho}{\rho_0}= \frac{V_0}{V_f} -1\)

\(⇒ \frac{\Delta \rho}{\rho_0} = \frac{1}{1.0098} -1\)

\(⇒ \frac{\Delta\rho}{\rho_0} = -9.8 × 10^{-3}\)

\\(⇒ \frac{\Delta\rho}{\rho_0} = -0.98× 10^{-2}\)

The negative sign represents there is a decrease in the density of the glycerine.

Hence the correct answer is 0.98 × 10^-2