Define the apparent and absolute coefficient of expansion of liquids. Find the relation between them.

Liquids cannot hold themselves alone but have to be contained in the vessel. Therefore when liquids are heated, naturally the vessel will also expand along with liquids. Thus when liquids are heated, they have also to fill the increased volume of the vessel. We cannot observe the intermediate state. We can only observe the initial and the final levels. This observed expansion of the liquid is known as the apparent expansion of the liquid and is less than its actual expansion.

The apparent coefficient of expansion of liquid is defined as the apparent increase in the volume of liquid in the vessel per unit original volume of liquid in the vessel per unit rise in temperature. It is represented by γ_a.

The absolute coefficient of expansion of liquid is defined as the actual increase in the volume of liquid (sum of the apparent increase in the volume of liquid in vessel and increase in the volume of the vessel) per unit original volume of liquid per unit rise in temperature. It is represented by γ_rRelation between γ_a and γ_r

Let us consider a vessel of volume V₀ at 0°C filled to brim with a liquid. Let γ_a and γ_r be the apparent and absolute coefficient of volume expansion of liquid respectively and γ_v is coefficient of cubical expansion of vessel.

Now heat the vessel to 0°C. Both vessel and liquid will expand. As the container is filled to the brim, therefore on heating, the liquid will overflow. Collect the overflow liquid. The liquid that overflows is the apparent increase in volume. Let it be ∆V_a. By definition, the apparent coefficient of expansion of liquid is,

               ${\mathrm{\gamma }}_{\mathrm{a}} = \frac{∆{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{o}}\mathrm{\theta }} \mathrm{or} ∆{\mathrm{V}}_{\mathrm{a}} = {\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{a}}\mathrm{\theta }$
Let ∆V_v be the increase in the volume of the vessel,

$∆{\mathrm{V}}_{\mathrm{v}} = {\mathrm{V}}_0{\mathrm{\gamma }}_{\mathrm{v}}\mathrm{\theta }$

Now the actual increase in the volume of liquid is,

$∆\mathrm{V}= ∆{\mathrm{V}}_{\mathrm{a}}+∆{\mathrm{V}}_{\mathrm{v}}={\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{a}}\mathrm{\theta }+{\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{v}}\mathrm{\theta }$                 ...(1)

Also, the actual increase in the volume of liquid is,

$∆\mathrm{V}={\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{r}}\mathrm{\theta }$                                                            ...(2)

From (1) and (2),

             ${\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{r}}\mathrm{\theta } = {\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{a}}\mathrm{\theta }+{\mathrm{V}}_{\mathrm{o}}{\mathrm{\gamma }}_{\mathrm{v}}\mathrm{\theta }$

or                ${\mathrm{\gamma }}_{\mathrm{r}} = {\mathrm{\gamma }}_{\mathrm{a}}+{\mathrm{\gamma }}_{\mathrm{v}}$

This is the required relation between ${\mathrm{\gamma }}_{\mathrm{a}} \mathrm{and} {\mathrm{\gamma }}_{\mathrm{r}}.$