Question

The coefficient of volume expansion of mercury is 20 times the coefficient of linear expansion of glass. If the volume of mercury that must be poured into a glass vessel of volume V so that volume above mercury may remain constant at all temperature is $\dfrac{xV}{20}$ find x.

Answer 1

Hint: All materials expand when heated. The amount a material expands is dependent on its expansion coefficients. Here we are given the coefficient of volume expansion and coefficient of linear expansion. Coefficient of volume expansion is 3 times the coefficient of linear expansion for the same material.

Formula used:
\[\begin{align}
  & \gamma =3\alpha \\
 & \Delta V=\gamma V\Delta T \\
\end{align}\]

Complete step-by-step answer:
The coefficient of volume expansion of glass will be triple of the coefficient of linear expansion for the glass.
So, we have a coefficient on volume expansion of glass = $3\times \dfrac{1}{20}=\dfrac{3}{20}$times the coefficient of volume expansion of mercury.
Now the volume in the glass vessel above the mercury will remain constant if the amount of volume increases for both the mercury and the glass is the same. So, we will get the following expression
$\Delta {{V}_{mercury}}=\Delta {{V}_{glass}}$
The change in volume for both the objects must be the same. Even though the vessel is not completely made out of glass, we can still take the volume expansion as the linear expansion will take place in all three dimensions and that is similar to volume expansion.
$\begin{align}
  & {{\gamma }_{mercury}}\times \dfrac{xV}{20}\times \Delta T={{\gamma }_{glass}}\times V\times \Delta T \\
 & \Rightarrow {{\gamma }_{mercury}}\times \dfrac{x}{20}={{\gamma }_{glass}} \\
\end{align}$
We have calculated that
${{\gamma }_{glass}}=\dfrac{3}{20}\times {{\gamma }_{mercury}}$
When we compare both the expressions, we get the value of x as 3. Hence, the value of x will be 3.

Note: Students must take care that the formulas given are an approximation where we ignore the higher order terms due to them being very small in magnitude. So, if the change in temperature is very high, there may be different expressions. But for all practical purposes these expressions can be used.