Question

The coefficient of linear expansion of glass is $8 \times {10^{ - 6}}{/^ \circ }C$ and the coefficient of real expansion of mercury is $180 \times {10^{ - 6}}{/^ \circ }C$. The coefficient of apparent expansion of mercury in glass vessel is:
A: $204 \times {10^{ - 6}}{/^ \circ }C$
B: $7.5 \times {10^{ - 6}}{/^ \circ }C$
C: $18 \times {10^{ - 5}}{/^ \circ }C$
D: $156 \times {10^{\, - 6}}{/^ \circ }C$

Answer 1

When a body is exposed to a temperature change it undergoes some deformations. If the rise of temperature is positive i.e. the temperature is increasing the body expands, but if the temperature decreases i.e. the temperature change is negative then the body contracts.

Complete solution:
What is thermal expansion?
When a body is exposed to a rise in temperature the body expands. The expansion length is directly proportional to the product of its initial length and the rise in temperature. Thus mathematically it is given by;
$\Delta l \propto {l_0}\Delta T$
Thus, $\Delta l = \alpha {l_0}\Delta T$
Here, $\Delta l$ is the change in length and ${l_0}$ is the initial length, $\Delta T$ is the rise in temperature and $\alpha$ is the coefficient of linear expansion.
Now, if length can increase then so can area and volume.
Thus the surface area of the body increases when temperature increases, similarly, the volume increases as well.
But the relationship is not the same.
Because of dimensional issues the coefficient of linear expansion is not equal to coefficient of area expansion.
Similarly it is not equal to the coefficient of volumetric expansion.
However both the coefficients are related to the coefficient of linear expansion as given below;
$\beta = 2\alpha$ Here, $\beta$ is the coefficient of area expansion and $\alpha$ is the coefficient of linear expansion.
Also, $\gamma = 3\alpha$ Here, $\gamma$ is the coefficient of volumetric expansion and $\alpha$ is the coefficient of linear expansion.
Now let us consider a system of two bodies.
Let the first body be the vessel in which the second body is kept.
Then the equivalent coefficient of expansion of the system is called the coefficient of absolute expansion or the coefficient of real expansion of the second body.
If ${\gamma _{real}}$ is the coefficient of real expansion of the second body and ${\gamma _{apparent}}$ is the coefficient of apparent (normal volumetric) expansion. And ${\gamma _{vessel}}$ is the coefficient of apparent (normal volumetric) expansion of the first body then mathematically;
${\gamma _{real}} = {\gamma _{apparent}} + {\gamma _{vessel}}$
For instance in the question it is given that the mercury is stored in the glass vessel, so, the coefficient of real expansion of mercury is given as;
${\gamma _{real}} = 180 \times {10^{ - 6}}{/^ \circ }C$
Also the coefficient of linear expansion of glass is given;
${\alpha _{glass}} = 8 \times {10^{ - 6}}{/^ \circ }C$
Thus the coefficient of volumetric expansion of the glass vessel is given by;
${\gamma _{vessel}} = 3{\alpha _{glass}}$
$\Rightarrow {\gamma _{vessel}} = 3 \times 8 \times {10^{ - 6}} = 24 \times {10^{ - 6}}{/^ \circ }C$
Now since mercury is kept in the glass vessel the coefficient of real expansion of mercury is given by;
${\gamma _{real}} = {\gamma _{apparent}} + {\gamma _{vessel}}$
Substituting the values we get;
$\Rightarrow 180 \times {10^{ - 6}} = {\gamma _{apparent}} + 24 \times {10^{ - 6}}$
$\Rightarrow {\gamma _{apparent}} = (180 - 24) \times {10^{ - 6}}$
$\therefore {\gamma _{apparent}} = 156 \times {10^{ - 6}}{/^ \circ }C$

Therefore, option D is correct.

Note:
-The relations between all the coefficients must be remembered.
-The coefficients of real and apparent expansion are always the volumetric coefficients.
-Do not get confused with linear coefficients, you cannot store anything in a body with no volume.