Show that the coefficient of surface tension T is given by T=rhPg/2 where P, h and r are density, height and radius of the system respectively.?
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To prove that the coefficient of surface tension T is given by T = rhPg/2, we can start by understanding the concept of surface tension and its relation to the given variables.
Surface tension is a property of liquids that arises due to the cohesive forces between molecules at the surface. It can be defined as the force per unit length acting perpendicular to an imaginary line drawn on the surface of a liquid.
Consider a liquid in a container or a tube, such as a capillary tube. Let's assume the liquid forms a meniscus, which is the curved surface at the liquid-air interface. The forces acting on this liquid-air interface are the weight of the liquid column above the meniscus and the surface tension acting along the circumference of the meniscus.
Now, let's break down the equation T = rhPg/2 and understand the variables involved:
- T represents the coefficient of surface tension.
- r is the radius of the system, which can be taken as the radius of the capillary tube or any other system in which surface tension is observed.
- h is the height or the vertical distance from the liquid-air interface to the center of the system. In the case of a capillary tube, h represents the height of the liquid column above the meniscus.
- P is the density of the liquid.
- g is the acceleration due to gravity.
To derive the equation T = rhPg/2, we need to consider the equilibrium of forces acting on the curved meniscus. We assume that the meniscus is in a state of equilibrium, meaning the forces acting on it are balanced.
The weight of the liquid column above the meniscus is given by the formula W = Pghπr^2, where W is the weight, P is the density, g is the acceleration due to gravity, h is the height, and r is the radius.
Since the liquid-air interface is curved, the surface tension acts along the circumference of the meniscus in both directions. As a result, the force exerted by surface tension is given by F = 2Tπr, where F is the force and T is the surface tension coefficient. The factor of 2 arises because there are forces acting in both directions along the circumference.
To balance the forces, we equate the force due to surface tension (F) and the weight of the liquid column (W). Therefore, we have:
2Tπr = Pghπr^2
Simplifying the equation by dividing both sides by 2πr and canceling out the common terms πr gives us:
T = Pghr/2
And rearranging the terms, we get:
T = rhPg/2
Thus, we have derived the equation T = rhPg/2, showing that the coefficient of surface tension is given by this formula in terms of the given variables.