Thermal Expansion

VARIATION OF DENSITY OF A SUBSTANCE WITH TEMPERATURE

The volume of a solid increases on heating while its mass remains constant, hence the density of the substance decreases. Let a solid of mass m has volumes v₁ and v₂  and densities d₁  and d₂ at temperatures t₁°C and t₂°C respectively, hence we 
have m = v₁d₁  = v₂d₂
v₁d₁  = v₂d₂                                                  .........(1)
We know that v₂ =v₁  [1+ \(\gamma \) (t₂ - t₁)]       .........(2)
Where  is coefficient of volume expansion of solid. Substituting equation (2) in (1), then
v₁d₁  =  v₁ [1+ \(\gamma\) (t₂ - t₁)]d₂
d₁ = d₂ [1+ \(\gamma\) (t₂ - t₁)]                          ...........(3)
d₁ =   d₂ + d₂ \(\gamma\)  (t₂ - t₁)
\(\gamma = \frac{{{d_2} - {d_1}}}{{{d_2}\left( {{t_2} - {t_1}} \right)}}\)                                          ..........(4)
 If given solid is heated from 0°C to t°C then its density changes from d₀ to d_t
Substituting these in equation (4) 
\(\gamma = \frac{{{d_0} - {d_t}}}{{{d_t}\left( t \right)}}\)                                               ..........(5)
and \({d_0} = {d_t}[1 + \gamma t]\)
(or) \({d_t} = \frac{{{d_0}}}{{1 + \gamma t}} \simeq {d_0}\left( {1 - \gamma t} \right)\)