Question

Calculate the ratio of the heights to which water and mercury are raised by capillary action in the same glass tube.

Short answer

The water is gone rise up to \[{\rm{2}}{\rm{.78}}\] units of length from the surface of water level in the basin, level to the mercury which will fall \[{\rm{1}}\] unit.

Step-by-step solution

Conceptual Introduction

Fluid statics, often known as hydrostatics, is a branch of fluid mechanics that investigates the state of balance of a floating and submerged body, as well as the pressure in a fluid, or imposed by a fluid, on an immersed body.

Given data

The Cohesive forces are attractive forces between molecules of the same sort. Adhesive forces are the attractive forces that exist between molecules of various sorts. The surface of a liquid contracts to the lowest feasible surface area due to cohesive forces between molecules. This overall impact is referred to as surface tension is a term used to describe the force that exists.

The capillary action is the propensity of a fluid to rise or fall in a narrow tube, or capillary tube, according to the relative strength of the capillary’s adhesiveness and cohesion.

\[h = \frac{{2\sigma \cos \theta }}{{\rho gr}}\]

The angle of contact between glass and mercury is \[{\rm{140}}\] degrees.

The angle of contact between glass and water is \[{\rm{0}}\] degree.

Ratio of height of mercury and water

By putting all the value into the equation, we get:

Here,\[{\rho _w}\]is the density of water,\[{\rho _m}\]is the density of mercury,\[{\sigma _w}\]is the surface tension of water, and\[{\sigma _w}\]is the surface tension of the mercury.

Calculating further,

\[\begin{array}{c}\frac{{{h_w}}}{{{h_m}}} = \frac{{{\sigma _w}\cos {\theta _w}}}{{{\sigma _m}\cos {\theta _m}}} \times \frac{{{\rho _m}}}{{{\rho _w}}}\\\frac{{{h_w}}}{{{h_m}}} = \frac{{\left( {0.0728} \right)\cos \left( {0^\circ } \right)}}{{\left( {0.465} \right)\cos \left( {140^\circ } \right)}} \times \frac{{13600}}{{1000}}\\\frac{{{h_w}}}{{{h_m}}} = \frac{{2.78}}{{ - 1}}\end{array}\]

Therefore, water rises up to 2.78 units of length from the surface of water level in the basin, level to the mercury which falls one unit.