In a capillary with diameter d=0.5 mm water will rise to a height (2 α)/(ρg r)=(4 α)/(ρg d)

step 1 Hello students in this question we have a vertical capillary of insert diameter D equals to 0 .5

In a capillary with diameter d=0.5 mm water will rise to a...

In a capillary with diameter $d=0.5 \mathrm{~mm}$ water will rise to a height $$ \begin{gathered} \frac{2 \alpha}{\rho g r}=\frac{4 \alpha}{\rho g d} \\ =\frac{4 \times 73 \times 10^{-3}}{10^{3} \times 9.8 \times 0.5 \times 10^{-3}}=59.6 \mathrm{~mm} \end{gathered} $$ Since this is greater than the height $(=25 \mathrm{~mm})$ of the tube, a meniscus of radius $R$ will be formed at the top of the tube, where $R=\frac{2 \alpha}{\rho g h}=\frac{2 \times 73 \times 10^{-3}}{10^{3} \times 9.8 \times 25 \times 10^{-3}} \sim 0.6 \mathrm{~mm}$

In a capillary with diameter $d=0.5 \mathrm{~mm}$ water will rise to a height
$$
\begin{gathered}
\frac{2 \alpha}{\rho g r}=\frac{4 \alpha}{\rho g d} \\
=\frac{4 \times 73 \times 10^{-3}}{10^{3} \times 9.8 \times 0.5 \times 10^{-3}}=59.6 \mathrm{~mm}
\end{gathered}
$$
Since this is greater than the height $(=25 \mathrm{~mm})$ of the tube, a meniscus of radius $R$ will be formed at the top of the tube, where $R=\frac{2 \alpha}{\rho g h}=\frac{2 \times 73 \times 10^{-3}}{10^{3} \times 9.8 \times 25 \times 10^{-3}} \sim 0.6 \mathrm{~mm}$

Key Concepts

Capillary Action

Capillary action refers to the phenomenon where a liquid spontaneously rises or falls in a narrow tube due to the balance between adhesive forces between the liquid and the tube’s wall and cohesive forces within the liquid. This behavior results from the interactions at the liquid-solid interface, and it is most noticeable in small diameter tubes or porous media.

Surface Tension

Surface tension is the energy, or force per unit length, present at the interface between a liquid and another phase (solid or gas). It arises from cohesive forces between liquid molecules and plays a crucial role in capillary phenomena by enabling liquids to form curved surfaces such as menisci, ultimately influencing the height to which the liquid will rise in a capillary tube.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational force acting on it. In the context of capillary action, the rising liquid column develops hydrostatic pressure, which opposes the upward force generated by surface tension. Equilibrium is reached when the upward capillary force balances the downward hydrostatic pressure.

Meniscus Formation and Curvature

Meniscus formation occurs when the liquid interface curves near the wall of a container or tube, as a result of the competing adhesive forces between the liquid and the container material and the cohesive forces within the liquid. The curvature of the meniscus, which can be described by the radius of curvature, determines the pressure difference across the liquid surface (known as the Laplace pressure) and is critical in determining the capillary rise limit when physical constraints are imposed.

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