The coefficient of volume expansion of glycerin is 49 × 10^–5 K^–1. What is the fractional change in its density for a 30°C rise in temperature?

Given,

Coefficient of volume expansion of glycerin, α_V = 49 × 10^–5 K^–1

Rise in temperature, ΔT = 30°C = 30 K

Fractional change in its volume = ΔV/V

We know,

ΔV/V = α_VΔT

⇒ V_T2 - V_T1 = V_T1α_VΔT

⇒ (m/ρT₂)-(m/ρT₁) = (m/ρT₁)α_VΔT

⇒ [(1/ρT₂)-(1/ρT₁)]/(1/ρT₁) = α_VΔT

⇒ (ρ_T1 - ρ_T2)/ρ_T2 = α_VΔT

Where, m is the mass of glycerin

ρ_T1 is the initial density at T₁

ρ_T2 is the initial density at T₂

(ρ_T1 - ρ_T2)/ρ_T2 is the fractional change in density

So, fractional change in density of glycerine = 49 × 10^–5 K^–1 × 30 K

= 1.47 × 10^-2