Question

A glass vessel of volume ${{V}_{0}}$ is completely filled with a liquid and its temperature is raised by $\Delta T$. What volume of the liquid will overflow? Coefficient of linear expansion of glass = ${{\alpha }_{g}}$ and coefficient of volume expansion of the liquid =$\gamma$.

Answer 1

Recall the expression for volume expansion. We know that the difference in thermal expansion of the glass vessel and the liquid causes the overflow of liquid. Using the above expression find this difference and hence the answer. Also, remember to convert the given coefficient of linear expansion of glass to that of volume expansion.

Formula used:
Expression for coefficient of volume expansion,
$\dfrac{\Delta V}{V}=\gamma \Delta T$

Complete step by step answer:
We are given a glass vessel that is filled up to its brim with a liquid of coefficient of volume expansion$\gamma$. We are raising the temperature by $\Delta T$ and we are asked to find the volume of the liquid that will overflow.
We know that when an object is subjected to change in temperature, the fractional change in length if proportional to $\Delta T$
$\dfrac{\Delta l}{l}\propto \Delta T$
$\Rightarrow \dfrac{\Delta l}{l}=\alpha \Delta T$ …………………………….. (1)
Where, ‘$\alpha$’ is the coefficient of linear expansion. Similarly, the fractional change in volume of a substance subjected to $\Delta T$ temperature change is given by,
$\dfrac{\Delta V}{V}=\gamma \Delta T$ ……………………………… (2)
Where, ‘$\gamma$’ is a coefficient of volume expansion.
We know from our prior knowledge that, the coefficient of volume expansion is three times the coefficient of linear expansion, that is,
$\gamma =3\alpha$
From the given coefficient of linear expansion of glass, we get the coefficient of volume expansion of glass as,
${{\gamma }_{g}}=3{{\alpha }_{g}}$ ……………………………………. (3)
We know that the volume of liquid overflowed will be the difference in volume expansions of the glass and the liquid. So, let the volume of liquid overflowed be V, then,
$V={{\left( \Delta V \right)}_{l}}-{{\left( \Delta V \right)}_{g}}$
Where, ${{\left( \Delta V \right)}_{g}}$and ${{\left( \Delta V \right)}_{l}}$ change in volumes in glass and liquid respectively. Also, the initial volume and the temperature change undergone are same for glass as well as liquid.
Now from (1) and (2),
$V=\left( \gamma {{V}_{0}}\Delta T \right)-\left( {{\gamma }_{g}}{{V}_{0}}\Delta T \right)$
$\Rightarrow V={{V}_{0}}\Delta T\left( \gamma -{{\gamma }_{g}} \right)$
From (3),
$V={{V}_{0}}\Delta T\left( \gamma -3{{\alpha }_{g}} \right)$
Therefore, the volume of the liquid that will overflow is ${{V}_{0}}\Delta T\left( \gamma -3{{\alpha }_{g}} \right)$

Note:
We can tell the units of resistivity in terms of Siemen because resistivity will be the inverse of conductivity and we can express the units of resistivity in terms of ohm too. If we convert all these units in the form of fundamental quantity units, we can get the dimensional formula too.