Question

The volume coefficient of expansion of a liquid is 7 times the volume coefficient of expansion of the vessel. The ratio of absolute and apparent expansion of a liquid is:
A. $\dfrac{1}{7}$
B. $\dfrac{7}{6}$
C. $\dfrac{6}{7}$
D. $\dfrac{1}{6}$

Answer 1

Hint: Remember that expansion irrespective of whether it is absolute or apparent is proportional to its respective coefficient of expansion. Also, recall that the coefficient of absolute expansion is the sum of the coefficient of apparent expansion and coefficient of expansion of the vessel. Find the apparent expansion coefficient from this sum with help of the relation given in the question and then find the required ratio.

Formula used:
${{\alpha }_{abs}}={{\alpha }_{app}}+{{\alpha }_{ves}}$

Complete step-by-step answer:
Thermal expansion is a phenomenon that becomes handy in our day to day life. If we are to change the temperature of a body, then it results in change in its dimensions. So the resultant increase in dimensions of the body due to the increase in its temperature is what we call thermal expansion.
This expansion happens length wise, area wise and also volume wise.
Here are some of our day-to-day encounters with thermal expansion: tightly screwed bottles with metallic lids are put in hot water to get expanded, mercury in thermometer works based on this property, etc.
For a long rod with very small temperature change $\Delta T$ , the fractional change in length is proportional to change in temperature. That is,
$\dfrac{\Delta l}{l}$ ∝ ∆T
$\dfrac{\Delta l}{l}={{\alpha }_{l}}\Delta T$
Where ${{\alpha }_{l}}$ is coefficient of linear expansion. Similarly we have, for volume expansion, the fractional change in volume is proportional to change in temperature. That is,
$\dfrac{\Delta V}{V}$ ∝ ∆T
$\dfrac{\Delta V}{V}={{\alpha }_{V}}\Delta T$
Where, ${{\alpha }_{V}}$ is the coefficient of volume expansion.
${{\alpha }_{V}}=\left( \dfrac{\Delta V}{V} \right)\dfrac{1}{\Delta T}$
Though ${{\alpha }_{V}}$ is a characteristic of a substance, we should note that it isn’t always a constant. In general, ${{\alpha }_{V}}$ depends on temperature but becomes constant at higher temperatures.
The difference between absolute and apparent coefficients of volume expansion is nothing but, the dependency of apparent coefficient on the container in which it is placed, that is, the change is stated in relation to the volume of container which in turn changes with temperature.
We could represent coefficient of absolute expansion as the sum of coefficient of apparent expansion and coefficient of expansion of vessel. That is,
${{\alpha }_{abs}}={{\alpha }_{app}}+{{\alpha }_{ves}}$ ……………… (1)
Where, ${{\alpha }_{abs}}\to $coefficient of absolute expansion of liquid
${{\alpha }_{app}}\to $ coefficient of apparent expansion of liquid
${{\alpha }_{ves}}\to $ coefficient of volume expansion of vessel
We are given in the question that coefficient of volume expansion of liquid $({{\alpha }_{abs}})$ is 7 times the volume expansion of the vessel$\left( {{\alpha }_{ves}} \right)$, that is,
${{\alpha }_{abs}}=7\times {{\alpha }_{ves}}$ …………….. (2)
Substituting (2) in (1) we get the coefficient of apparent expansion as,
${{\alpha }_{app}}={{\alpha }_{abs}}-{{\alpha }_{ves}}=\left( 7\times {{\alpha }_{ves}} \right)-{{\alpha }_{ves}}$
Therefore,
${{\alpha }_{app}}=6\times {{\alpha }_{ves}}$ ………………. (3)
We could divide equation (2) by equation (3) to get the required ratio.
The ratio of absolute and apparent expansion of a liquid is,
$\dfrac{{{\alpha }_{abs}}}{{{\alpha }_{app}}}=\dfrac{7\times {{\alpha }_{ves}}}{6\times {{\alpha }_{ves}}}$
$\dfrac{{{\alpha }_{abs}}}{{{\alpha }_{app}}}=\dfrac{7}{6}$

So, the correct answer is “Option B”.

Note: We see that the ratio of the ‘coefficient’ of absolute and apparent expansions is not asked in the question. But we know that any expansion (fractional change in dimension) irrespective of whether linear, area or volume is directly proportional to their respective coefficients of expansion. So, we could substitute the ratio of these coefficients instead of the values of their expansion.