An incompressible liquid is filled in a container. There is air above the liquid and a massless movable piston with a hole is also placed on the air. A thin capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the hole on the piston as shown in the figure. The air in the container is isothermally compressed from the original volume V0 to \(\frac{100}{101}\)V0 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 Twater =0.075N/m, P0=105N/m2 , g=10m/sec2, \(\rho l\)=103 kg/m3, \(\theta=0\))
(given Twater =0.075N/m, P0=105N/m2 , g=10m/sec2, \(\rho l\)=103 kg/m3, \(\theta=0\))
Correct Answer: 25
Solution and Explanation
Pressure-Volume Relation and Height Calculation
Given that \( T \) is constant: \[ PV = \text{constant} \]
From this, we get the relation between pressure and volume: \[ P \propto \frac{1}{V} \]
If the volume changes by a factor of \( \frac{100}{101} \), the pressure changes by a factor of \( \frac{101}{100} \). Thus, the pressure becomes: \[ P_f = 1.01 \times P_0 \]
The equation for the pressure difference is: \[ P_0 - \frac{2T}{R} = 1.01 P_0 - \rho g h \]
Rearranging this to solve for \( \rho g h \): \[ \rho g h = 0.01 P_0 + \frac{2T}{R} \]
Now, substituting the given values: \[ (10^3)(10)h = (0.01)(10^5) + \frac{2 \times 0.075}{0.1 \times 10^{-3}} \]
Solving for \( h \): \[ h = \frac{2500}{10^4} \, \text{m} = 25 \, \text{cm} \]
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Concepts Used:
Mechanical Properties of Fluid
The science of the mechanical properties of fluids is called Hydrostatics. A fluid is a substance that relents to the slightest pressure. Fluids are categorized into two classes famed by the names of liquids, and elastic fluids or gases, which later comprehend the air of the atmosphere and all the different kinds of air with which chemistry makes us acquainted.
Streamline Flow:
A streamline is a curve the tangent to which at any point provides the direction of the fluid velocity at that point. It is comparable to a line of force in an electric or magnetic field. In steady flow, the pattern of the streamline is motionless or static with time, and therefore, a streamline provides the actual path of a fluid particle.
Tube of Flow:
A tubular region of fluid enclosed by a boundary comprises streamlines is called a tube of flow. Fluid can never cross the boundaries of a tube of flow and therefore, a tube of flow acts as a pipe of the same shape.
Surface Tension and Viscosity:
The surface tension of a liquid is all the time a function of the solid or fluid with which the liquid is in contact. If a value for surface tension is provided in a table for oil, water, mercury, or whatever, and the contacting fluid is unspecified, it is safe to consider that the contacting fluid is air.