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1 EPS/Chem 182, ver. 4/4/2022 Knudsen Effusion Method for Vapor Pressure Determination Knudsen effusion can be used as a dynamic method of vapor pressure determination provided that it is restricted to the measurement of very low pressures. The maximum pressure that can be measured by this method is between 10 − 2 and 10 − 3 Torr (mmHg). The technique is usually used for the determination of vapor pressures at high temperatures. However, high temperature is not an essential feature of the method. The only requirement is that the pressure be low. The material whose vapor pressure is to be determined is placed in a container with a small hole. The vapor pressure is related to the amount of material that escapes through this hole in a given length of time. It is assumed that the temperature is maintained constant at some known value. Since the pressures are so low, no error is introduced by the use of the gas kinetic theory and the assumption that the behavior of the gas is ideal. The theoretical problem is to evaluate the number of molecules that pass through this hole in a given period of time. Kinetic theory can rigidly solve this effusion problem for certain given conditions, hole diameter, hole length, and pressure. For the present problem, consider a container divided into two parts, A and B, by a partition. Suppose that the sample whose vapor pressure is to be determined is in part A, and that the pressure in part B is maintained at essentially zero by some appropriate method, such as using a high vacuum. The purpose of this setup is that we then need to consider motion of the molecules in one direction only (out through the hole!). Kinetic theory can be used to predict the number of molecules that hit a given area in a given period of time. The problem is then to relate the number of molecules which pass through the opening with those that enter the opening. If the mean free path of the molecules is large compared to the diameter and depth of the opening, then all the molecules which hit the orifice will pass through to the other side. If, on the other hand, this condition is not fulfilled, a number of molecules, due to collisions, will be deflected back to the part of the container from which they originated. It is not always possible to make a correction for these deflected molecules. Thus, a necessary condition for using the equations from gas kinetic theory that appear below is that the mean free path must be large compared to the dimensions of the hole . It is this condition that restricts the method to low vapor pressures. Detailed treatment of the effusion of molecules may be found in texts on kinetic theory (including the Atkins Chapter 8 under “Multi-Lab Resources” on bcourses). Two other conditions must also be satisfied if this Knudsen Effusion method is to be used to calculate vapor pressures of substances: (1) the rate of effusion through the hole must be slow compared to the rate of vaporization, and (2) the result of the effusion in the vicinity of the hole must be such that the velocity distribution of the molecules is essentially unchanged. These two requirements also make it necessary to have a small hole. In the following, we derive or collect the various equations and relationships from gas kinetic theory needed to calculate the vapor pressure of a substance in a Knudsen Effusion cell. First, it is relatively simple to determine the number of molecules striking a given area per unit time. Designating “ N ” as the number of molecules that collide with a unit area in one second, “N” is given by: ⑴ 𝑁𝑁 = 𝑛𝑛𝑆𝑆 𝑥𝑥 �� 2 = 𝑛𝑛𝑆𝑆 ̅ 4

2 where n = number of molecules/m 3 = the average molecular speed (in all directions) = the average molecular speed in the x-direction. Equation 1 can be derived from basic principles of molecular velocity distributions, and their averages can be found in fundamental texts on the kinetic model of gases (e.g., Atkins 0F 1 which is available on bcourses). From kinetic theory (see the Atkins reference on bcourses), we know that the average speed of molecules with molar mass, M, at temperature, T, is given by Equation 2: ⑵ 𝑆𝑆 ̅ = � 8𝑅𝑅𝑅𝑅 𝜋𝜋𝜋𝜋 � 1 2 where M = molecular mass of compound (kg/mol) R = gas constant (8.3145 J K − 1 mol − 1 ) (Remember that a joule, J, is kg m 2 s − 2 ). Note how this equation is derived from kinetic theory considerations in the Atkins reference. The mass of material escaping from the hole per second is is given by Equation 3, ⑶ where m = mass of one molecule (kg) A = area of hole (m 2 ). Note that the product mn has units of kg/m 3 so that ⑷ 𝑚𝑚𝑛𝑛 = 𝜋𝜋 𝑉𝑉 where V = molar volume of the gas (m 3 /mol). Therefore, can be expressed as: ⑸ 𝐺𝐺 ̇ = 𝜋𝜋𝑆𝑆 ̅ 𝐴𝐴 4𝑉𝑉 Now we can employ the ideal gas law (Equation 6) 1 Atkins, P.; de Paula, J. Physical Chemistry: Thermodynamics and Kinetics ; W. H. Freeman: Great Britain, 2006; vol. 1, pp 241-251. kg sec

3 ⑹ 𝑃𝑃𝑃𝑃 = 𝑅𝑅𝑅𝑅 where P = pressure (Note that pressure is a measure of force per unit area, which in SI units is 1 Pascal (Pa) = 1 N/m 2 = 1 (kg m s -2 ) / m 2 = 1 kg m -1 s -2 ). Solving for V in Equation 6 and substituting it into Equation 5 gives: ⑺ 𝐺𝐺 ̇ = 𝑃𝑃𝜋𝜋𝑆𝑆 ̅ 𝐴𝐴 4𝑅𝑅𝑅𝑅 Next, substituting Equation 2 for the average molecular speed, , into Eq. 7 yields Eq. 8: ⑻ 𝐺𝐺 ̇ = 𝑃𝑃𝑃𝑃 � 𝜋𝜋 2𝜋𝜋𝑅𝑅𝑅𝑅 � 1 2 which can then be rearranged into expressions for pressure, shown in Equations 9a and 9b: (9a) 𝑃𝑃 = 𝐺𝐺 ̇ 𝐴𝐴 × � 𝑀𝑀 2𝜋𝜋𝜋𝜋𝜋𝜋 � 1 2 , 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 (9b) 𝑃𝑃 = 𝐺𝐺 ̇ × � 2𝜋𝜋𝜋𝜋𝜋𝜋 𝑀𝑀 � 1 2 𝐴𝐴 ∝ 𝑘𝑘𝑘𝑘 𝑆𝑆 × 𝑚𝑚 𝑆𝑆 𝑚𝑚 2 Equation 9b shows that the pressure in the Knudsen Effusion cell is equal to the product of the rate of mass loss through the hole, (in kg s − 1 ), and the average speed of the molecules in the x-direction, (in m s − 1 ) – i.e., their speed out the hole – divided by the area of the hole, A (in m 2 ). Thus, by measuring and A and knowing from the kinetic theory of gases, the pressure in the cell can be determined. In this experiment, this pressure corresponds to the vapor pressure of the substance in the cell at the temperature of the cell . In addition to determining the vapor pressure of a substance, the data collected in such an experiment can yield the thermodynamic properties of the substance as well. Trouton’s Rule states that the heat of vaporization of a substance divided by the absolute temperature of the boiling point is typically ~88 J mol − 1 K − 1 : ⑽ ∆𝑆𝑆 𝑣𝑣𝑣𝑣𝑣𝑣 = ∆𝐻𝐻 𝑣𝑣𝑣𝑣𝑣𝑣 𝑅𝑅 𝐵𝐵𝐵𝐵 ≈ 88 𝐽𝐽 𝑚𝑚𝑚𝑚𝑃𝑃 −1 𝐾𝐾 −1 Although Trouton’s Rule is only approximate, it is often reasonably accurate enough (given that the entropy change on going from a liquid to a gas is relatively independent of the identity of the substance), so that an estimate of ΔH vap can be obtained.

4 The differential form of the Clausius-Clapeyron Equation is given in Equation 11 and allows calculation of the equilibrium vapor pressure of a substance at different temperatures if ∆ H vap is known: ⑾ � 𝑑𝑑ln𝑃𝑃 𝑑𝑑 1 𝜋𝜋 � = − ∆𝐻𝐻 𝑣𝑣𝑣𝑣𝑣𝑣 𝑅𝑅 where R is the gas constant (8.314 J mol − 1 K − 1 ). (Note that this equation is used a lot in both chemistry and earth science!). The integral form of Equation 11 is ⑿ ln 𝑃𝑃 = 𝑃𝑃𝑚𝑚𝑛𝑛𝑃𝑃𝑐𝑐 − � ∆𝐻𝐻 𝑣𝑣𝑣𝑣𝑣𝑣 𝑅𝑅 � 1 𝑅𝑅 And between 2 specific limits in Equation 13: ⒀ ln � 𝑃𝑃 1 𝑃𝑃 2 � = − � ∆𝐻𝐻 𝑣𝑣𝑣𝑣𝑣𝑣 𝑅𝑅 � � 1 𝑅𝑅 1 − 1 𝑅𝑅 2 � In the questions below, Equations 10 and 13 together will be used below to estimate the time necessary to wait for naphthalene to effuse out the orifice in order to obtain a measureable decrease in initial starting weight at two other temperatures beyond the one measured in this laboratory exercise. Finally, it is useful for considerations in this experiment to be able to calculate the mean free path of molecules as a function of temperature and pressure and their size. The equation from gas kinetic theory for the mean free path of an ensemble of gas molecules is ⒁ 𝐿𝐿 = 𝑘𝑘𝑅𝑅 2 1 2 𝜋𝜋𝜎𝜎 2 𝑃𝑃 where k = Boltzmann’s constant (1.381x10 -23 J K − 1 ), T= temperature in K P = pressure in Pa, and σ = collision diameter for the gas molecule. (Note that a C-C bond in an aromatic ring is typically around 1.40x10 − 10 m). Experimental Procedure Important : Be sure to make estimates of the errors associated with each measurement you make (e.g. pressure, temperature, length, etc.). A sophisticated error analysis will be an important part of the grading for this lab. Also, you’ll need to know the details of how the mechanical and diffusion pumps work, and how to make pressure measurements. Recall that a very good reference book,

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