Question

At the middle of the mercury barometer tube, there is a little column of air with the length $l_0$ and there is vacuum at the top as shown. Under the normal atmospheric pressure and the temperature of $300\text{K}$, $l_0=10\text{cm}$. What will be the length of the air column $($in $\text{cm})$ if the temperature rises to $330\text{K}$ ?

$($Neglect expansion of the tube$)$

Answer 1

Pressure in the air inside the column is equal to the weight of mercury over the air divided by the internal cross-sectional area of the tube.

When the temperature increases, the weight of the upper part of the mercury column does not change. That is why the pressure in the air is also constant.

For the isobaric process, the change in volume is proportional to the change in temperature $i.eA\Delta l\propto \Delta T$, where $A$ is the cross-section of the tube and $l$ is the length of the air column.

So, for same $A,\Delta l\propto \Delta T$

$\therefore \frac{l}{l_0}=\frac{T}{T_0}\Rightarrow l=l_0\left(\frac{T}{T_0}\right)=10\times \frac{330}{300}=11\text{cm}$