Question

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

Answer 1

Hint: The relation between ${\rm{\alpha and \ gamma }}$ is $\dfrac{{\rm{\alpha }}}{{\rm{1}}}{\rm{ = }}\dfrac{{\rm{\gamma }}}{{\rm{3}}}$or ${\rm{\gamma }}\,{\rm{ = }}\,{\rm{3\alpha }}$. We need to consider the knowledge of linear thermal expansion and volumetric coefficient of expansion.

Complete step by step solution:
The coefficient of volume expansion is three-time of the coefficient of linear expansion. The linear expansion coefficient is the change in length of one unit long of a specimen when its temperature changes by one degree. Different materials expand by varying quantities.
Given,
Linear thermal expansion is given by ${\rm{\Delta L = }}\,{\rm{\alpha L\Delta T}}$, here ${\rm{\Delta L}}$ is the change in length${\rm\;{L}}$, ${\rm{\Delta T}}$ is the change in temperature, and ${\rm{\alpha }}$ is the coefficient of linear expansion, which varies slightly with temperature. The change in area due to thermal expansion is ${\rm{\Delta A}}\,{\rm{ = }}\,{\rm{2\alpha A\Delta T}}$, here ${\rm{\Delta A}}$ is the change in the area.

The volumetric coefficient of expansion of the material is ${\rm{3\alpha }}$.
As the system is heated, the liquid starts flowing out of the box. It means the volumetric coefficient of expansion of liquid is greater than of the container.
So, ${\rm{\gamma }}\, > 3{\rm{\alpha }}$

Hence, the correct option is (B).

The Thermal Expansion Coefficient describes how an object's size changes with a temperature change. In particular, at constant pressure, it measures the fractional change in size per degree in temperature. It has developed several types of coefficients: volumetric, area, and linear.

Note: The linear thermal expansion defines the fractional length increase, while the volumetric expansion defines the fractional volume increase per unit temperature rise. The ratio increase in length from the original length for 1-degree rise in temperature is called the coefficient of linear expansion. The coefficient of linear expansion is three times the
coefficient of volume expansion.