Question

A liquid with coefficient of volume expansion $\gamma $ is filled in a container of a material having coefficient of linear expansion $\alpha $. If the liquid overflows on heating, then:
$A)\text{ }\gamma =3\alpha $
$B)\text{ }\gamma >3\alpha $
$C)\text{ }\gamma <3\alpha $
$D)\text{ }\gamma ={{\alpha }^{3}}$

Answer 1

Hint: This problem can be solved by calculating the coefficient of volumetric expansion of the material of the container from the coefficient of linear expansion and comparing with the coefficient of volumetric expansion of the liquid keeping in mind that the liquid overflows from the container when the container is heated.
Formula used:
${{\gamma }_{solid}}=3\alpha $

Complete answer:
The coefficient of volumetric expansion of a substance is the measure of the expansion of the material when it is heated. Greater the coefficient of volumetric expansion, greater will be the expansion of the material when heat is given to it.
Now, the coefficient of volumetric expansion ${{\gamma }_{solid}}$ of a solid is related to the coefficient of its linear expansion $\alpha $ by
${{\gamma }_{solid}}=3\alpha $ --(1)
Now, let us analyze the question.
The coefficient of linear expansion of the material of the container is $\alpha $.
Let the coefficient of volumetric expansion of the material of the container be ${{\gamma }_{container}}$.
Therefore, using (1), we get
${{\gamma }_{container}}=3\alpha $ --(2)
Now, the coefficient of volumetric expansion of the liquid inside the container is given to be $\gamma $.
Now, since when the container and the liquid inside it is heated together, the liquid overflows, it means that the liquid expands by a larger amount than the container when the same amount of heat is given to it. This means that the coefficient of volumetric expansion of the liquid inside the container must be greater than the coefficient of volumetric expansion of the material of the container.
$\therefore \gamma >{{\gamma }_{container}}$ --(3)
Putting (3) in (2), we get
$\gamma >3\alpha $
Therefore, we have got the required relation.

Hence, the correct option is $B)\text{ }\gamma >3\alpha $.

Note:
Students must note that usually liquids have a larger coefficient of volumetric expansion than solids and gases have a further larger coefficient of volumetric expansion than liquids. This is the reason why gases are said to expand a lot upon heating followed by liquids and solids expand the least. Hence, more often than not, there is a possibility that the liquid in an already full container will overflow when heated than the liquid level going down with respect to the top level of the container.