## VAPOUR PRESSURE

**INTRODUCTION**: Liquids exhibit a free surface in the container whereas vapors and gases fill the full volume. Liquid molecules have higher cohesive forces and are bound to each other. In the gaseous state the binding forces are minimal. Molecules constantly escape out of a liquid surface and an equal number constantly enter the surface when there is no energy addition. The number of molecules escaping from the surface or re-entering will depend upon the temperature. Under equilibrium conditions these molecules above the free surface exert a certain pressure.

This pressure is known as vapor pressure corresponding to the temperature. As the temperature increases, more molecules will leave and re-enter the surface and so the vapor pressure increases with temperature. All liquids exhibit this phenomenon. Sublimating solids also exhibit this phenomenon. The vapor pressure is also known as saturation pressure corresponding to the temperature. The temperature corresponding to the pressure is known as saturation temperature. If liquid is in contact with vapor both will be at the same temperature and under this condition these phases will be in equilibrium unless energy transaction takes place. The vapor pressure data for water and refrigerants are available in tabular form. The vapor pressure increases with the temperature. For all liquids there exists a pressure above which there is no observable difference between the two phases. This pressure is known as critical pressure.

**Partial Pressure:** In a mixture of gases the total pressure P will equal the sum of pressures exerted by each of the components if that component alone occupies the full volume at that temperature. The pressure exerted by each component is known as its partial pressure.

**P = p1 p2 p3 ....**

Where p_{1 }= (m_{1}R_{1}T)/V; p_{2}= (m_{2}R_{2}T)/V in which T and V are the common temperature and volume.

For example air is a mixture of various gases as well as some water vapor. The atmospheric pressure is nothing but the sum of the pressures exerted by each of these components. Of special interest in this case is the partial pressure of water vapor. This topic is studied under Psychometric. The various properties like specific heat, gas constant etc. of the mixture can be determined from the composition.

**c _{m}**

**= Σ (**

**c**

_{i}**×**

**m**

_{i}**)/Σ**

**m**

_{i}
Where cm is the specific heat of the mixture and c_{i}and m_{i}are the specific heat and the mass of component i in the mixture.

**Problem**: A small thin plane surface is pulled through the liquid filled space between two large horizontal planes in the parallel direction. Show that the force required will be minimum if the plate is located midway between the planes.

Let the velocity of the small plane be u, and the distance between the large planes is h. Let the small plane be located at a distance of y from the bottom plane. Assume linear variation of velocity and unit area. Refer Fig.

Velocity gradient on the bottom surface = u/y

Velocity gradient on the top surface = u/ (h – y),

Considering unit area,

Force on the bottom surface = µ × (u/y), Force on the top surface = µ × u/ (h –y)

Total force to pull the plane = µ × u × {(1/y) [1/ (h – y)]}

To obtain the condition for minimization of the force the variation of force with respect to y should be zero, or dF/dy = 0, Differentiating the expression A,

dF/dy = µ × u {(–1/y^{2}) [1/(h – y)^{2}]}, Equating to zero, y^{2} = (h – y)^{2} or y = h/2

Or the plane should be located at the mid gap position for the force to be minimum the force required for different location of the plate is calculated using the following data and tabulated below.

**µ = 0.014 Ns/m ^{2,} u = 5 m/s, h = 0.1 m.**

Equation A is used in the calculation. Model calculation is given for y = 0.002 m.

**F = 0.014 × 5 × {(1/0.002) [1/0.01 – 0.002)]} = 43.75 N/m ^{2}**

**Note that the minimum occurs at mid position:**