Question

A glass flask of volume $200c{{m}^{3}}$ is just filled with Mercury at ${{20}^{0}}C$. The amount of Mercury that will overflow when the temperature of the system is raised to ${{100}^{0}}C$ is:
$({{\gamma }_{glass}}=1.2\times {{10}^{-5}}/{{C}^{0}};{{\gamma }_{Mercury}}=1.8\times {{10}^{-4}}/{{C}^{0}})$
$\begin{aligned} & (a)2.15c{{m}^{3}} \\\ & (b)2.69c{{m}^{3}} \\\ & (c)2.52c{{m}^{3}} \\\ & (d)2.25c{{m}^{3}} \\\ \end{aligned}$

Answer 1

Upon heating the whole system, both the glass container and the Mercury inside it will undergo thermal expansion. Now, the amount of Mercury that will overflow will be equal to the change in volume of the glass container subtracted from the change in volume in Mercury present in the glass. Mercury will overflow because its coefficient of cubical expansion is more than that of the glass container.

Complete step-by-step answer:
Let the change in volume of the glass container be equal to $\vartriangle {{V}_{g}}$ and the change in volume of the Mercury present inside the container be equal to $\vartriangle {{V}_{Hg}}$.
Now, $\vartriangle V$ of any system under thermal expansion can be calculated using the formula:
$\Rightarrow \vartriangle V={{\gamma }_{system}}{{V}_{i,system}}(\vartriangle T)$
Where,
${{\gamma }_{system}}$ is the coefficient of cubical expansion of the system
${{V}_{i,system}}$ is the initial volume of the system
$\vartriangle T$ is the change in temperature
Now, we can calculate the change in volume for glass and Mercury.
Since, initial and final temperature for both the system is same, thus:
$\begin{aligned} & \Rightarrow \vartriangle T=(100-20) \\\ & \Rightarrow \vartriangle T={{80}^{0}}C \\\ \end{aligned}$
Also initial volume of both the systems are equal. And its value is $200c{{m}^{3}}$. Therefore, we have now:
For the glass flask:
$\begin{aligned} & \Rightarrow \vartriangle {{V}_{g}}={{\gamma }_{glass}}\times 200\times 80 \\\ & \Rightarrow \vartriangle {{V}_{g}}=1.2\times {{10}^{-5}}\times 200\times 80 \\\ & \Rightarrow \vartriangle {{V}_{g}}=0.192c{{m}^{3}} \\\ \end{aligned}$
Also, for Mercury:
$\begin{aligned} & \Rightarrow \vartriangle {{V}_{Hg}}={{\gamma }_{Mercury}}\times 200\times 80 \\\ & \Rightarrow \vartriangle {{V}_{Hg}}=1.8\times {{10}^{-4}}\times 200\times 80 \\\ & \Rightarrow \vartriangle {{V}_{Hg}}=2.88c{{m}^{3}} \\\ \end{aligned}$
Hence, the amount of Mercury that overflow from the glass flask can be calculated as:
$\begin{aligned} & =\vartriangle {{V}_{Hg}}-\vartriangle {{V}_{g}} \\\ & =(2.88-0.192)c{{m}^{3}} \\\ & =2.688c{{m}^{3}} \\\ & \approx 2.69c{{m}^{3}} \\\ \end{aligned}$
Hence, the amount of Mercury that will overflow from the glass container is equal to $2.69c{{m}^{3}}$.

So, the correct answer is “Option b”.

Note: Here, we saw the concept of relative expansion of two bodies. The body having greater coefficient of cubical expansion will expand more if all the other parameters are kept constant. This principle is used to tightly fit one object inside the other, that is when heated, the object with higher value of thermal expansion coefficient will fit tightly inside the other object having low thermal expansion coefficient.