b. Section 2.2.2 includes a discussion of the mean free path of a gas molecule and the number of collisions per second. i. Estimate the mean free path at STP. ii. Using your mean free path estimate, calculate the number of collisions per second with the average velocity given by equation 2-3b. 2.2.2 PRESSURE Momentum transfer from the gas molecules to the container walls gives rise to the forces that sustain the pressure in the system. Kinetic theory shows that the gas pressure, P, is related to the mean-square velocity of the molecules and thus, alternately to their kinetic energy or temperature. Thus, P=nMo/3NA = nRT/NA (2-4) where NA is Avogadro's number. From the definition of n, n/N, is the number of moles per unit volume and therefore, Eq. 2-4 is an expression of the perfect gas law. Pressure is the most widely quoted system variable in vacuum technology and this fact has generated a large number of units that have been used to define it under various circumstances. Basically, two broad types of pressure units have arisen in practice. In what we shall call the scientific system or coherent unit system (Ref. 4), pressure is defined as the rate of change of the normal component of momentum of impinging molecules per unit area of surface. Thus, the pressure is defined as a force per unit area, and examples of these units are dynes/cm (CGS) or newtons/meter? (N/m?) (MKS). Vacuum levels are now commonly reported in SI units or pascals; 1 pascal (Pa) = 1 N/m2. Historically, however, pressure was, and still is, measured by the height of a column of liquid, e.g., Hg or H,O. This has led to a set of what we shall call practical or noncoherent units, such as millimeters and microns of Hg, torr, and atmospheres, which are still widely employed by practitioners as well as by equipment manufacturers. Definitions of some units together with important conversions include 1 atm = 1.013 x 106 dynes/cm = 1.013 x 105 N/m = 1.013 x 105 Pa 1 torr = 1 mm Hg = 1.333 x 103 dynes/cm = 133.3 N/m? = 133.3 Pa 1 bar = 0.987 atm = 750 torr. The mean distance traveled by molecules between successive collisions, called the mean-free path, Amfp, is an important property of the gas that is dependent on the pressure. To calculate umfp we note that each molecule presents a target area redz to others where d, is its collision diameter. A binary collision occurs each time the center of one molecule approaches within a distance d. of the other. If we imagine the diameter of one molecule increased to 2d, while the other molecules are reduced to points, then in traveling a distance Imfp the former sweeps out a cylindrical volume teda nimfp One collision will occur under the condition that redz imfp n = 1. For air at room temperature and atmospheric pressure, amp. 500 , assuming N do ~ 5 . A molecule collides in a time given by Amfp/v, and under the above conditions, air molecules make about 101 collisions per second. This is why gases mix together rather slowly even though the individual molecules are moving at great speeds. The gas particles do not travel in uninterrupted linear trajectories. As a result of collisions, they are continually knocked to and fro, executing a zigzag motion and accomplishing little net movement. Since n is directly proportional to P, a simple relation for ambient air is amfp = 5 x 10-3/P, (2-5) with Amfp given in centimeters and P in torr. At pressures below 10-3 torr, impp is so large that molecules effectively collide only with the walls of the vacuum chamber. TA M) 1/2 directions can be defined; i.e., 1 dn, Muz flux) = exp - (2-2) ndy, 21RT 2RT and similarly for the y and z components. A number of important results emerge as a consequences of the foregoing equations. For example, the most probable (v.), average (T) and mean square (02) velocities are given, respectively, by 2RT (2-3) VM 8RT (2-3) TIM 5- - Surfwdo of(u)do ( s(vdu /S |* = = f*v*flopdo | S Floxo = 3RT/M; ( 3RT 2 ()dv (52)1/2 = (2-30) VM These velocities, which are noted in Fig. 2-1, simply depend on the molecular weight of the gas and the temperature. In air at 300 K, for example, the average molecular velocity is 4.6 x 104 cm/s, which is almost