Question

If the coefficient of absolute expansion of a liquid is 7 times the coefficient of cubical expansion of the vessel, the ratio of the coefficients of absolute and apparent expansions of the liquid:
(A) $ \dfrac{1}{7} $
(B) $ \dfrac{6}{7} $
(C) $ \dfrac{7}{6} $
(D) $ 7 $

Answer 1

Hint : The apparent expansion is equal to the difference between absolute expansion and volume expansion of the volume. We need to calculate for the apparent expansion relative to the volume expansion of the vessel.

Formula used: In this solution we will be using the following formula;
 $\Rightarrow {\gamma _{app}} = {\gamma _{abs}} - {\gamma _v} $ where $ {\gamma _{app}} $ is the apparent coefficient of expansion of the liquid, $ {\gamma _{abs}} $ is the absolute coefficient of expansion, and $ {\gamma _v} $ is the volume expansion of the vessel.

Complete step by step answer
Liquids in general have no constant shape, they simply take the shape of their volume. If expansion of the liquid is allowed by the application of the heat to the vessel, we can define two forms of expansion from this
Apparent expansion: This is the expansion of the liquid by measuring the initial and final volume without considering the expansion of the vessel.
Real expansion: This is the expansion of the liquid after considering the expansion of the vessel.
In the question, absolute expansion is 7 times the cubical expansions of the liquid
Hence, $ {\gamma _{abs}} = 7{\gamma _v} $ , then
The expression relating the absolute expansion, the apparent expansion and the volume expansion coefficient is given as
 $\Rightarrow {\gamma _{app}} = {\gamma _{abs}} - {\gamma _v} $ where $ {\gamma _{app}} $ is the apparent coefficient of expansion of the liquid, $ {\gamma _{abs}} $ is the absolute coefficient of expansion, and $ {\gamma _v} $ is the volume expansion of the vessel.
Then the apparent expansion coefficient can be given as
 $\Rightarrow {\gamma _{app}} = 7{\gamma _v} - {\gamma _v} = 6{\gamma _v} $ from substitution of $ {\gamma _{abs}} = 7{\gamma _v} $
Hence, the ratio of absolute to apparent is given as
 $\Rightarrow \dfrac{{{\gamma _{abs}}}}{{{\gamma _{app}}}} = \dfrac{{7{\gamma _v}}}{{6{\gamma _v}}} = \dfrac{7}{6} $
Hence, the correct answer is option C.

Note
For clarity, as seen, the absolute expansion is greater than the apparent expansion. This is because, when the container is heated, the volume increases and hence causes a drop in the level of the liquid and hence, allows expansion to appear smaller.