Problem 24 - Master Resource Book in JEE Main Physics

Question

A glass tube \(80 \mathrm{~cm}\) long and open at both ends is half immersed in mercury. Then the top of the tube is closed and it is taken out of the mercury. A column of mercury \(20 \mathrm{~cm}\) long then remains in the tube. The atmospheric pressure (in \(\mathrm{cm}\) of \(\mathrm{Hg}\) ) is (a) 90 (b) 75 (c) 60 (d) 45

Step-by-Step Solution

Answer

The atmospheric pressure is 60 cm of Hg (option c).

Step 1: Analyze the Initial Condition

Initially, the glass tube is open at both ends and is half immersed in mercury. This means the top half of the tube is filled with air at atmospheric pressure, and the bottom half is filled with mercury.

Step 2: Understand the Situation After Closing the Tube

Once the tube is half-filled with mercury, the top is closed, and the tube is removed from the mercury bath. The bottom open end traps mercury inside the tube, forming a 20 cm column of mercury, while the rest is a vacuum.

Step 3: Apply the Concept of Pressure Equilibrium

When the tube is open at both ends under atmospheric pressure, pressure inside the tube equates to atmospheric pressure. However, when closed and a column is trapped, it balances the pressure difference. Use the principle that the pressure exerted by a column of fluid is proportional to its height, along with any atmospheric pressure exerted on an open column.

Step 4: Calculate the Atmospheric Pressure

Since 20 cm of mercury remains inside, the situation suggests the difference in pressure has balanced out. Using the concept of equilibrium: \( P_{atm} = h + P_{tube} \). Here, \( h = 20 \text{ cm} \text{ (the remaining mercury)}\), and \( P_{tube} = 0 \text{ because it has vacuum above it)} \). Therefore, atmospheric pressure \( P_{atm} = 80 - 20 = 60 \text{ cm Hg.}\)

Step 5: Conclusion

The atmospheric pressure, therefore, equates to a total column height that equals 60 cm of mercury. This means option (c) 60 is the correct answer.

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air above us in the atmosphere. It is an invisible force we often take for granted because it doesn't have an immediate impact on our daily actions. However, it is important in many scientific calculations and experiments. Atmospheric pressure can be measured in various units, but in this exercise, we use centimeters of mercury (cm Hg).

At sea level, standard atmospheric pressure is approximately 76 cm Hg. This means the weight of the column of air from the top of the atmosphere to sea level exerts a pressure that can sustain a column of mercury 76 cm high in a tube with a vacuum on top. Changes in altitude or weather conditions can affect atmospheric pressure. In this specific exercise, the calculated atmospheric pressure using the equilibrium principle resulted in a value of 60 cm Hg. This means that the external atmospheric pressure is capable of supporting a 60 cm column of mercury, given the conditions described.

Vacuum Pressure

Vacuum pressure refers to the condition where the pressure inside a container is less than the atmospheric pressure. Essentially, it is a partial vacuum. In scientific terms, this means the pressure is lower compared to the regular atmospheric pressure, contributing to the movement or inflow of air working to equalize pressure differences.

In the exercise provided, when the top of the tube is closed and the tube is removed from the mercury, a section above the mercury column becomes a vacuum. This is because no air is present between the closed top and the mercury surface, which exerts zero pressure in contrast to the atmospheric pressure outside the tube. This vacuum creates a balance where the column of mercury remains at a height of 20 cm — this represents a partial vacuum, showcasing how vacuum influences and stabilizes within the closed tube.

Manometer Principle

The manometer principle is vital in measuring pressure differences using columns of liquid. It works on the concept that the pressure difference between two points can be equated to the height of liquid columns. This is exactly what we observe in the exercise.

The principle uses a simple setup where liquid columns balance each other under different pressure conditions. If one side of the column is exposed to atmospheric pressure, while the other is under different pressure conditions, the height of the liquid in the column changes accordingly. This principle is effectively employed in the scenario where the tube is half immersed and then closed. The pressure at the closed end is zero (due to the vacuum), and the height of the mercury column provides the necessary data to calculate the external atmospheric pressure, resulting in a pressure equivalent to the weight of 60 cm of mercury. This clear demonstration of the manometer principle provides an intuitive approach to understanding how pressure difference measurement is done using liquid columns.