# Variation of Density with temperature

Variation of density of Solids and liquids with temperature

When a given mass of a solid or liquid is heated, its volume increases. Accordingly, the density of a solid or a liquid decreases on heating.

Let Vo and V be the volumes of a solid (or liquid) at Temperature T and T + ΔT respectively. If γ is its coefficient of cubical expansion, Then

V  =  Vo ( 1 + γ ΔT )   …….(i)

Let ρo and ρ be the densities of the solid or liquid at temperature T and T + ΔT . If the mass of the liquid or solid is M

Then , $\displaystyle \rho_0 = \frac{M}{V_0} ;\quad \rho = \frac{M}{V}$

Substituting for Vo and V in equation (i), we have

$\displaystyle \frac{M}{\rho} = \frac{M}{\rho_0}(1+\gamma \Delta T)$

$\displaystyle \rho = \frac{\rho_0}{1+\gamma \Delta T}$

$\displaystyle \rho = \rho_0 (1+\gamma \Delta T)^{-1}$

Expanding by binomial theorem and neglecting terms containing higher powers of γ ΔT , we have

$\displaystyle \rho = \rho_0 (1 -\gamma \Delta T)$

With increase in temperature, volume increases, so density decreases and vice-versa.

Note: (i) γ for liquids are in order of 10-3

(ii) For water, density increases from 0 to 4°C so γ is −ve (0 to 4°C) and for 4°C to higher temperature γ is +ve . At 4°C density is maximum.

Example : A sphere of diameter 7cm and mass 266.5 gm floats in a bath of liquid. As the temperature is raised, the sphere just begins to sink at a temperature of 35°C. If the density of the liquid at 0°C is , find the co-efficient of cubical expansion of the liquid. Neglect the expansion of the sphere.

Solution : The sphere will sink in the liquid at 35°C, when its density becomes equal to the density of liquid at 35°C.

The density of sphere ,$\displaystyle \rho_s = \rho_{35} = \frac{266.5}{\frac{4}{3}\times\frac{22}{7}\times(\frac{7}{2})^3}$

( density of sphere is constant)

ρ35= 1.483 gm/cm3

Now ,  ρ0 = ρ35[1 + γΔT]

1.527 =1.483[1 + γ × 35]

1.029 = 1 + γ × 35

γ = 0.00083 / °C