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Variation of Density with temperature

In this article we are going to discuss about Variation of Density with temperature of solids and liquids.

We know that on heating a substance, its volume increases but mass remains the same.

Hence we conclude that :-

Heating of bodies is accompanied by an increase in volume and decrease in density.

Let

M = mass of a substance

V₀ = initial volume, ρ₀ = initial density

ΔT = change in temperature

γ = coefficient of volume expansion of the substance

V = final volume, ρ = final density

Since mass remains the same ⇒ M = V₀ρ₀ = Vρ

Now final volume is given by V = V₀(1 + γΔT)

Dividing both sides by M, we get

$\displaystyle \frac{V}{M}=\frac{{{V}_{0}}}{M}(1+\gamma \Delta T)$

$\displaystyle \Rightarrow \frac{1}{\rho }=\frac{1}{{{\rho }_{0}}}(1+\gamma \Delta T)$

$\displaystyle \Rightarrow {{\rho }_{0}}=\rho (1+\gamma \Delta T)$

$\displaystyle \Rightarrow \rho =\frac{{{\rho }_{0}}}{1+\gamma \Delta T}$

$\displaystyle \Rightarrow \rho ={{\rho }_{0}}{{(1+\gamma \Delta T)}^{-1}}$

Now as γΔT<<<1, hence

$\displaystyle \Rightarrow \rho ={{\rho }_{0}}(1-\gamma \Delta T)$

This is the expression to find final density of a material after an increase in temperature of ΔT.

Example.

A sphere of diameter 7 cm and mass 266.5 gm floats in a bath of a liquid. As the temperature rises, the sphere begins to sink & at a temperature of 35ºC the sphere gets completely submerged. If the density of the liquid at $^{0º} C$ is 1.527 gm/cc, then neglecting the expansion of the sphere, find the coefficient of cubical(volume) expansion of the liquid.

Solution.

Volume of the sphere = $\displaystyle \frac{4}{3}\pi {{r}^{3}}=\frac{4}{3}\times \frac{22}{7}\times {{(7)}^{3}}=179.59c{{m}^{3}}$