P v Surface Effects in Condensation If we compress a gas isothermally condensation is suppose to start at point O, and if we compress further the pressure is suppose to remain constant.. Actually the pressure will often follow the dotted line. Along this line the system is not in equilibrium, however, the slightest jar will abruptly reduce the pressure to the correct value. O’O’ O Similarly, if a liquid is expanded beyond the point O’ it will sometimes follow the dotted curve, but this would not be in equilibrium. These phenomena are respectively known as supersaturation and supercooling. supersaturation supercooling

Surface Effects in Condensation Vapor pressure as we have discussed this in class is the pressure at which a gas can co-exist in equilibrium with an infinitely large body of its own fluid. we denote this by the quantity, The pressure at which the gas can co-exist in equilibrium with a finite droplet, of radius r is not but a higher pressure, The difference between these pressures is due to the surface tension of the droplet. Suppose a droplet of liquid is placed in an external medium that exerts a pressure P on the droplet. Then the work done by the drop on expanding is and

Surface Effects in Condensation Integrating for a spherical drop of radius r, is the internal energy per unit volume of an infinite drop. The Gibbs potential for this can be written in the form For the droplet in equilibrium with its vapor at fixed pressure and temperature the Gibbs potential must be a minimum. This condition determines a relation between P and r for a given T.

Surface Effects in Condensation Let the mass of the droplet be m l and that of the vapor m v. Then the total Gibbs free energy of the system can be written as Suppose the drop undergoes an infinitesimal change in its radius owing to evaporation such that, Equilibrium is determined by the condition: Since:is the density of the liquid.

Surface Effects in Condensation Equilibrium is determined by the condition: This condition can only be met if the quantity in the bracket is zero Since Equilibrium is defined by Consider now a variation of this equilibrium with pressure. Recall:

Surface Effects in Condensation Also assume that the vapor is sufficiently dilute so that it behaves as an ideal gas, where m is the mass of a gas atom. Since the density of the liquid is much larger than that of the gas,

Surface Effects in Condensation Integrating both sides of this equation, we find P as a function of r at fixed temperature. Rearranging and solving for

Surface Effects in Condensation Recognizing that is the atomic volume of the liquid, we can rewrite this constant T This is known as the Kelvin Equation.

It is only “recently” that the Kelvin equation has been validated fluid/vapor equilibrium. cyclohexane T T’T’ T’T’ Surface Effects in Condensation Thermal shield