The Problem of Counting the Number of Molecules and Calculating Thermodynamic Properties An Experimental Approach Luis Alfonso Torres, Imelda Hernhdez-Contreras, and Juan Antonio Guardado Laboratorio de Terrnoquirnica, Depto. de Quimica, Centro de lnvestigacion y de Estudios Avanzados dei I.P.N. Apdo, Postal 14-740.07000 Mexico, D.F. Mexico Many physical chemistry and thermodynamic textbooks state that the thermodynamic properties of a system can be considered as the average of mechanical variables. However, not many examples of an experimental approach to illustrate this statement have been given. The problem begins with the question of the meaning of average. In reality, there are several averaging procedures that can be used, as described in ref I. As an example let us consider the pressure of a gas. This quantity is related to the momentum transported by the molecules of the gas through a unit area per unit time. For practical purposes let us consider a vessel containing a gas constituted by only one type of particles. Imagine that within the vessel a small window of unit area is hung from the top of the container. The average of the momentum can be calculated if a microscopic observer seated a t the window's comer (Fig. l a ) is able to carry out the extremely boring task of counting the number of molecules crossing the window and determining their velocities. The total momentum is computed and divided by the observation time.
above procedures, it is necessary a t least to imagine the possibility of counting the molecules and determining their velocities. Unfortunately, there is not a microscopic observer working like a Maxwell demon and carrying out this formidable task. The Knudsen Effusion Method To Count the Number of Molecules Fortunately, Martin Knudsen (2, 3), a t the begining of this century, while developing the kinetic theory of gases, devised a method using macroscopic procedures to determine the pressure of a gas or to measure the average of the momentum transported in an indirect manner. Knudsen considered that pressure arises from forces exerted when particles strike the unit area of the wall of the gas container per unit time. The gas particles are moving through all the available volume. At equilibrium, the molecular velocities are described by the Maxwell-Boltzman distribution function. If the vessel is completely closed, the gas molecules collide against the wall. The transfer of momentum during these collisions is revealed as a force that, when expressed per unit area is normally known as the pressure. Therefore, the problem of determining the pressure reduces to knowing the number of molecules that can reach the unit area of the wall in a second (Fig. 2a). Knudsen showed that this number, or the pressure, can be calculated if an orificeof area A is pierced a t the top of the vessel
Figure 1. Schematic representation of averaging procedure over the molecules (la)and over the time (1b). A second way of averaging is really funny: Have an observer ride on a molecule, and take a very long trip during which the molecule and the observer cross the window of unit area many times (Fig. lb). The average of the momentum transported can be calculated by counting the number of times that the molecule (and rider) crosses the imaginary window. Determine the correspondingvelocities, and then divide by the traveling time. A third way is not so funny because it involves some mathematical work that consists of calculating the momentum transported by the molecules crossing the unit area per unit time. The average is obtained by multiplying the quantity (mv) by the distribution function, and then integrating over the velocities and the volume. In the
Figure 2. Molecules inside the cylinder can reach the container's wall (2a);schematic representation of the Knudsen effusion process (2b). Volume 72 Number 1 January 1995
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(Fig. 2b) maintained a t temperature T,and the rate of loss of mass (dmldt) is observed. Such an experimental device is called a Knudsen effision cell. The pressure is given by the Knudsen equation (eq 1).
whereR is the gas constant, and M is the molecular weight of the gas. In order to ensure the distribution fundion and the equilibrium in the svstem. several exoerimental conditions are assumes that the gas has a n imposed. The Ea~cu~ition ideal behavior. Therefore, the pressure inside the container must be kept as lowas po&.ible. The effusion orifice must be smaller than the mean free pathway between successive collisions, and the thickness-of the orifice must be negligible in order to guarantee that a molecule reaching the orifice always leaves the container. This last condition cannot be experimentally satisfied, and a Clausing probability factor (4) must be applied. The Vapor Pressure as an Example The above restrictions seem to have been exactly designed for the measurement of the vapor pressure of an organic or organometallic solid compound, ranging typically from 0.1 to 10 Pa. Many laboratories worldwide carry out measurements of vapor pressure as a function of the temoerature usine this method. This ~rincioleis also aoplied for vapor pressure measurements of metals at higher temoeratures. Some technoloeies for the preparation of deiosiiion of thin somk semiconductor rnaterials~e~uire films from molecular flow of vapors escaping from Knudsen cells. In addition, other thermodynamic properties can be derived from the vapor pressure as a function of temperature. In particular the enthalpy of sublimation can be obtained from the well-known Clausius-Clapeyron equation.
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I
To the Vacuum System
Figure 3. Schematic representation of the sublimation process.
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Substituting eq 1into eq 2 followed by integration, assuming that the enthalpy change of the phase transition is independent of the temperature, we get eq 3.
where in B includes the term (l/A)(2rrRIM),which remains constant durine an emeriment. From this eauation the enthalpy of sublimationcan be easily obtained'from the slope of the straight line resulting from plotting the left-hand term of eq 3 versus 1IT. The sublimation process can be schematically represented in Figure 3. The solid sample is placed a t the boG tom of the Knudsen cell and placed into a holder connected to a high vacuum system. When the pump is started the residual air is pumped out. and the sublimation process begins. If the tekpe;ature is fixed, when the equilibrium is reached the pressure of the vapor phase becomes fixed because the phase rule must be satisfied. (The system is one component with two phases in equilibrium, so there is one degree of freedom.) The geometrical characteristics of the cell must be carefully established because the equilibrium must not be perturbed by the presence ofthe orifice. In this case, even when the vapor molecules are continuously esc a ~ i n ethrough the orifice. the molecules are beine continuo& repraced from thesubliming solid phase, a"d the distribution function throueh - the -eas is re-established. Typically, the effusion rate of the vapor in equilibrium with its solid is around g s-', using orifices ranging from 0.1 to 1mm at temperatures between 0 and 140 "C. Traditional vacuum electrobalances are not the best instruments for measuring such a small mass flow because their sensitivity may be just 0.1 mg, and a reliable determination of dmldt requires a long time. Actually, the enthalpy of sublimation can be obtained from eq 3 even if the rate of loss of mass is not exactly known. Instead only a proportional quantity is experimentally available. The Piezoelectric Effect At this point it is important to take a look a t the really attractive piezolectric effect shown by many materials. Among them, the most popular is the quartz crystal. Quartz crystals are made of silicon and oxygen atoms in a periodical three-dimensional arrangement. Therefore, several cuts (e.e.. - . AT.. BT.. CT.X. Yl can be made a t different angles relative to a major &*(Kg. 4,. These pieces of crystal oresent the oiezoelectnc effect (fi.om the Greek oiezin. meaning to pre& discovered a t the end of the 19b cen;
Figure 4. Ctystai structure of quartz and cuts of piezoelectric pieces.
Mechanical
Figure 5. Schematic representation of the piezoelectric effect tury. It can be defined as an electric polarization produced by mechanical strain. In other words, if pressure is applied over a piece of quartz crystal, a charge separation occurs, and a source of electromotive force is created. Alternatively, when a voltage is applied across the crystal, mechanical movement is produced. In addition when a voltage is supplied as pulses a t the proper frequency, the crystal vibrates and produces a steady signal. For each cut of the nystal, the mode of vibration (bending, shear, torsion, etc.) and frequency resonance are characteristic. These phenomena are represented in Figure 5. From this physical principle, a quartz crystal can be used as a weighing device to count the molecules if we are able to keep these molecules over its surface. It is evident that when molecules are being condensed on the surface of the crystal, the thickness increases continuously, so more and more energy is necessary to make it vibrate. This fact is the basic principle for Knudsen effision measurements. For our purpose, the cut AT with thickness shear mode of vibration is adequate. The molecules escaping through the orifice of the vessel can be trapped on the liquid-nitrogen-cooled surface of the crystal. The mechanical proper-
Figure 6. Schematic diagram of the experimental setup as described in the text.
Figure 7. Details of the Knudsen effusion cell. ties are affected by the thickness change, producing a frequency resonance variation (dvldt)that displays a number proportional to the rate of loss of mass of the cell (dmldt) and to the vapor pressure. The sensitivity of such a microbalance is about 104g s-'(or a variation of 1pg in 100 s), allowing a few tie-consuming experiments. A series of 10 to 12 measurements a t different temperatures can be made in a day's work. Also, the microbalance is not sensitive to mechanical vibrations in the environment. Microbalances working on this principle have also been described by Glukhova (5)and Burkinshaw (6). Details of the physics and instrumentation have been extensively described by Lu and Czanderna (7). Experimental Setup
An experimental setup is schematically displayed in Figure 6. The geometrical design and materials of the Knudsen effision cell A are detailed in Figure 7. The cell holder B is made of aluminum and works as a thermal stabilizer. It also houses the temperature sensors C (thermistors or platinum resistances). The holder has a flange for quick raccordii - to other vacuum-line elements. During- an experiment the holder is immersed in a temperature-controlled water bath D. Acrossoiece was used t o introduce a cold finger E, supporting the quartz crystal F, just in front of the effision orifice. The crystal is electrically connected to the reference oscillator H, using a feedthrough G. Com-
Figure 8. Graphical representation of experimental results to determine the enthalpy of sublimation of naphthalene. Volume 72 Number 1 January 1995
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parison of the frequencies from the reference oscillator and the crystal is detected at a frequency meter I. A Sample of Experimental Results An experimental run was camied out with a sample of a pure compound, in this case naphthalene. Measurements of resonance frequency and time were taken at a given temperature while the vapor molecules were being condensed. These values lead to a straight line from which the slope (dvldt)~was calculated. The procedure was repeated at different temperatures, and the results were analyzed using eq 3 to obtain the enthalpy of sublimation. A typical experimental data set is summarized in the table while the graphical representation of eq 3 is shown in Figure 8. The value AHsub = (73.19 0.52) kJ mol-' can be assigned to the average temperature (291.16 K) because the experiments were made between 286.19 and 296.13 K. Comparison with the reported values (8)of enthalpy of sublimation was possible taking into account that
Experimental Sample Data for a Knudsen Effusion Experiment of Naphthalene Using a Quartz Crystal Microbalance
296.127 295.240 294.052 293.079 290.872 289.943 288.720 287.438 286.195
32.2416 29.4833 25.8850 23.5800 18.8516 17.0016 15.0850 13.1983 11.7500
3.3769 3.3871 3.4008 3.4120 3.4379 3.4490 3.4636 3.4790 3.4941
6.3187 6.2277 6.0955 6.0006 5.7730 5.6681 5.5464 5.4106 5.2922
Orifice diameter is 150 pm. Leasf-squares analysis leads loA&a = 73.19 *0.52 W mol-'.
reported by De Kruif (9). This allows us to calculate the corrected value a t 298.15 K as
for this reference material. Acknowledgment We acknowledge financial support from the National Council of Science and Technology (CONACYT-MEXICO) for purchase of the equipment. One of us (J. A. Guardado) wishes to thank CONACYT for his scholarship. Literature Cited 1. Berry,R. S.;Riee. S.A;RoaaJ.PhysiealChemiatw,PPort b, M~otferinEquilB~ium: Statistied M~choniesand T b m a l v m i c s : John Wile" and sona: New Ymk.
Assuming that ACp is constant over the integration temperature range, we get
The latter value is in good agreement with the established value (10)of
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1980: Chapter 15. 2. Knudsen,M.Ann.Phwik 1809,28,75. 3. Knudsen, M. Kinefie Theow D ~ G C I SSome ~ S , Modern Asp~cts,2nd ed.; Mothupn' Monogmphs ofPhysicol Subjects;Methuen:London. 1946. 4. Clausing P h . Physik 1932,IZ,961. 5. Glukhova,O.T.;Arkhangelova,N.M.;Qpll~ky,AB.;S&odud,L.E;Yaoson,I.K T h o c h i m i c a Acfa 1985.95.133, 6. Burkinshay P M.;Mortimer,C.T.J Chem. Soc Dollon k m .1%. 75. 7. Lu,C.;Czanderna,A.W A p p l i m f i o n s o f P i e ~ ~ I ~ l rCrysfolMiembolonws, ~Quo~ Val. 7,MelhDdF ond Ph~nomeno.Thair Applications in S & m and 'lkhnolagy; E1seuier:Amnterdan. 1984. 8. Torreffidrnez. L. A.; BarreireRadriguez, G.: Galarea-Mondrag6n. A. ehirniecl Aefa 1988,224,229-233. 9. De M. C. G. J. Ciwm. Thermodyn. l981,23,1081. 10. Head.A. J.; Sabbah, R. Remmmendei Refinnc. Mderkzls for the Rdizaflon of Phvs~mhernimlPmmrties Internationel Union o f Pum and Aoolied Chembtm: ~ & h ,K N., Ed.;~iackwell:Oxford, 198; ~hapte;7, p 272.
