2.2 Pressure, temperature, and rms speed  (Page 7/18)

Another important application of partial pressure is vapor pressure    , which is the partial pressure of a vapor at which it is in equilibrium with the liquid (or solid, in the case of sublimation) phase of the same substance. At any temperature, the partial pressure of the water in the air cannot exceed the vapor pressure of the water at that temperature, because whenever the partial pressure reaches the vapor pressure, water condenses out of the air. Dew is an example of this condensation. The temperature at which condensation occurs for a sample of air is called the dew point . It is easily measured by slowly cooling a metal ball; the dew point is the temperature at which condensation first appears on the ball.

The vapor pressures of water at some temperatures of interest for meteorology are given in [link] .

The relative humidity (R.H.) at a temperature T is defined by

A relative humidity of $100\%$ means that the partial pressure of water is equal to the vapor pressure; in other words, the air is saturated with water.

Mean free path and mean free time

We now consider collisions explicitly. The usual first step (which is all we’ll take) is to calculate the mean free path    , $\lambda ,$ the average distance a molecule travels between collisions with other molecules, and the mean free time $\tau$ , the average time between the collisions of a molecule. If we assume all the molecules are spheres with a radius r , then a molecule will collide with another if their centers are within a distance 2 r of each other. For a given particle, we say that the area of a circle with that radius, $4\pi r^2$ , is the “cross-section” for collisions. As the particle moves, it traces a cylinder with that cross-sectional area. The mean free path is the length $\lambda$ such that the expected number of other molecules in a cylinder of length $\lambda$ and cross-section $4\pi r^2$ is 1. If we temporarily ignore the motion of the molecules other than the one we’re looking at, the expected number is the number density of molecules, N / V , times the volume, and the volume is $4\pi r^2\lambda$ , so we have $(N\text{/}V)4\pi r^2\lambda =1,$ or

Taking the motion of all the molecules into account makes the calculation much harder, but the only change is a factor of $\sqrt{2}.$ The result is