An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0 . 1 mm is dipped vertically into the liquid through the airtight piston hole,

Question

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1 mm$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_{0}$ to $\frac{100}{101} V_{0}$ with the movable piston. Considering air as an ideal gas, the height $(h)$ of the liquid column in the capillary above the liquid level in $cm$ is _____.

[Given: Surface tension of the liquid is $0.075 N m^{- 1}$ , atmospheric pressure is $10^{5} N m^{- 2}$ , acceleration due to gravity $(g)$ is $10 m s^{- 2}$ , density of the liquid is $10^{3} kg m^{- 3}$ and contact angle of capillary surface with the liquid is zero]

TestHub · Accepted Answer

Let P f be the air pressure. As the process is isothermal, we can write P 0 V 0 = P f V f ⇒ P 0 V 0 = P f 100 101 V 0 ⇒ P f = 101 × 10 3 Pa ∵ P 0 = 10 5 N m - 2 Now, consider the 4 points shown in diagram P d - P c = 2 T R ∵ P d = P 0 ∴ P c = P 0 - 2 T R Now, P a = P b (also, P a = P f ) ⇒ P f = ρ g h + P c ⇒ 101 × 10 3 = 10 3 × 10 × h + 10 5 - 2 × 0 . 075 0 . 1 × 10 - 3 ⇒ h = 1 4 m = 25 cm

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