Diffusion of Water Vapor A Physical Chemistry Laboratory Experiment Robert N. Nelson Georgia Southern University, Landrum Box 8064, Statesboro, GA30460-8064
Gas diffusion is usually one of the main topics of the study of transport properties i n physical chemistry courses (see, for example, Atkins ( 1 ) or Levine (2)).There are, however, few simple experiments to determine the values of diffusion coefficients i n gases. Shoemaker e t al. (3) describe a n experiment using the time-dependent "Loschmidt" technique, but this requires extreme care in creating a leak-tight system and in analyzing the contents of the two portions of the diffusion cell. This new experiment makes use of a simple apparatus similar to the Palmes passive diffusion samplers, which are used for environmental monitoring (4). The apparatus is easily constructed and gives very good results with respect to reproducibility and agreement with theory. The original impetus for this experiment was a n observation made i n 1977 by a group of freshman chemistry students a t Colgate University who were trying to calibrate glassware by weighing water i n open beakers: They had to chase the weight on the balance a s i t continually dropped due to evaporation. A rough calculation verified that the main contribution was simply diffusion of water vapor.
Support Hook
Filter Paper Pad
Flow Straightener Figure 1 . Cutaway diagram of the diffusion cell. of water vapor (desiccant tray) is placed a t the open end of the cell. Once a steady state i s established, the concentration of water molecules varies linearly with distance from the end of the cell. so the value of dCIdz is a constant riven
Theory Methods for the Determination of Diffusion Coefficients
Methods for the determination of diffusion coefficients of gases can be divided into two broad types: time-dependent and time-independent or steady-state. The time-dependent methods, as typified by the Loschmidt technique, start with a sharp gradient in concentration and follow the concentration changes a s a function of time. Although they have the advantage of being absolute methods, which are substantially independent of apparatus geometry, they have the disadvantage of requiring determination of gas compositions, either a t a single time or as a function of time. They also require a method of generating the sharp initial boundary and preserving this boundary when the two halves of the cell are brought into contact. Time-independent methods, on the other hand, depend on establishment of a fixed concentration gradient, independent of time, so that a determination of the flux of one species is sufficient to determine the diffusion coefficient. One disadvantage is that the measured flux is directly dependent on apparatus geometry, so this geometry must be well-known or calibrated out. Adetailed review of the various c!xpcrimmt:il techniques and their edvnnt:ip~.innd disadvantages can be (bund in Dunlop 151and Shuoter r 6 i . Determining the Rate of Mass Loss Fick's first law governs the steady-state situation. dC J = -Ddz
(1)
where J ( c m ? s-') is the flux of molecules; D (cmZs-'1 is the diffusion coefficient; and dC/dz (cm4) is the gradient in number density of molecules. In this apparatus a source of water vapor (wet filter-paper pad) is placed a t the closed end of the cell, and a sink
where C,,,,,, i s the number density of water molecules a t the source (wet filter paper); CSi* = 0 is the number density of water molecules a t the top of the desiccant tray; PH,O tl'a is the v;ipor pres,ure of water; /,I: 15 the distance bet u w n thc snurce and the sink; and kT is the product ufthe Boltzmann constant and the Kelvin temperahre. If the temperature remains constant, the gradient is determined only by cell length. Computer modeling shows that within 3 min after the water is placed on the filter-pap w pad and thc cell is aiscmblcd, the gradient is constnni I'i in thisexperimrnr, in runtrmr to to u , ~ t h berterthan ~n the Palmes diffusion sampler, the source, rather than the sink, i s a t the closed end of the cell. As water diffuses out of the cell, the mass of water vapor diffusing from the cell can be determined as a function of time to evaluate the diffusion coefficient according to
where mCell(g) is the mass of the diffusion cell; mHzo (g) is the mass of a single water molecule; and A,,a (cm2)is the cross-sectional area of the cell. Other terms are defined in equation 2. Thus, a measurement of the rate of mass loss, coupled with knowledge of the temperature and cell dimensions, suffices to determine the diffusion constant. Procedure Apparatus
Figure 1shows a cutaway view of the diffusion cell used in this experiment. It consists of a 2-in. by 3-in. pill vial (available from most pharmacies), a filter-paper pad cut to Volume 72 Number 6 June 1995
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fit the base of the vial, and a flow-straightener, which consists of a piece of aluminum honeycomb (Hexcel, Norcross, GA or Goodfellow Metals) in this cell. The filter paper was glass fiber because normal paper will wrinkle. The support loop was bent from wire and taped to the closed end of the plastic vial. The flow-straightener was needed to prevent drafts and convection a t the open end of the cell. Without it, the measured diffusion coefficients are several times too large and the rate of mass loss i s not stable. You could also cut plastic drinking straws into 1-cm lengths and place a close-packed array of them in the open end of the cell.
Diffusion of Water Vapor Observed and Fitted Data
,p
MD data offset +I00 seo
Measurements
The filter-paper pad (glass fiber is less prone to wrinkling and disintegration) is moistened with about 2 mL of distilled water. and the cell is hune from the Dan s u ~ ~ o r t of a balance. Either a n analyticz balance with 0:i-mg readabilitv or a n electronic balance with 1.0-me readability can b e i s e d , although with the electronic barance more time is required to acquire data. A petri dish containing a layer of desiccant, such as Drierite, is placed just below the open end of the cell, and a 5-min period is allowed for the gradient and temperature to stabilize. The mass is then read a t intervals to determine the mass loss as a function of time. For the analytical balance, i t i s more convenient to record the time when the optical scale reads a whole number of millierams of mass lost rather than to trv follow in^ " the changing mass'on the optical scale. Figure 2. Plot of experimental results. The data points are the experiFor the cell used in these experiments, the rate of mass mental values, and the lines are a linear least-squares fit to the data. loss h l A t i s about 10 pgls, so 1mg requires about 100 s. The data forthe run by M. Davis are offset 100 s to avoid overlap with those of E. Carleton. Symbol sizes are comparable to experimental The mass is followed for a total mass change of perhaps errors in time and mass. 30-40 mg depending on the rate of loss. When the electronic balance is used with 1-mg readability, mass readings every 3-5 rnin (2-4 mg change) for about 45-60 min University carried out this experiment. The results for the will give sufficient data points, whereas with the analyti1993 class are shown in Figure 2 and summarized in the cal balance (0.1-mg readability) the readings will be a t table. Estimated errors in mass loss are f0.2 mg (due to 1-mg intervals corresponding to 100 s. The temperature reading the optical scale) and i 1 2 s (the corresponding near the cell is recorded periodically with a thermometer time uncertainty). The lines are linear-regression fits to inside the balance case. The barometric pressure i s also the data points. The literature value (7) for D (corrected to recorded. experimental temperature of the runs) is 0.026 cm2 s-', Analysis which is within the experimental uncertainties of the data. Aplot of mass loss vs. time should be a straight line. ExamiAcknowledgment nation of the residuals from a linear least-squares fit will permit exclusion of nonlinearity due to temperature change durI would like to acknowledge the work of Travis Paul and ine the run or outliers due to data recordine errors or Tatsuya Naganawa, Georgia Southern '92, and of Ernest h6teresis in the balance. This last problem seems to plague Carleton and Michele Davis, GSU '93, students in CHE583 manv electronic balances for several minutes after the tare or who carried out the experimental data collection. I would also like to acknowledge with great pleasure my conversations with Edward D. ~ a l m e as n d k ~ u n n i s o of n the Institemperature based on the hard-sphere approximation from tute of Environmental Medicine. New York Universitv. ",for kinetic molecular theory. discussions during the development of the passive-diffusion samplers in the mid-1970's.
-
where h i s the mean free path; p is the reduced mass: o is the collision cross section; a n d P is the barometric pressure. This expression implies a TU dependence for D and may be used to estimate the literature value of D a t the experimental temperature. Results
Results of 1993 Runs
Parameter
E. Carleton's Data Set
no, of data points AmiAt (pgls) T M) D (cm2/s)
28 9.61 + 0.04 29? 31 0.236 k 0.019 -9.2% -1.26
Error vs. Literature t-value
M.
Davis's Data Set 35 9.95 + 0.04 29k31 0.245 i 0.019 -5.8% -0.8
Students in the lgg2 and lgg3 The uncertainty in D is due primarily to the uncefiainty in the vapor pressure of water because a +I K temperature spring-quarter physical chemis- uncertainty translates into ai6% uncertainty in the vapor pressure. Additional contributions are due to the uncertainties try classes a t Georgia Southern in the length and area of the cell (each contributes about 2.5%). Thus. the relative uncertainties of 8% listed for D
568
Journal of Chemical Education
Literature Cited 1. Atkins, P W Phwico! Chemistry. 4th ed.: Oxford University: New Y d . 1989; pp 735-738. 2. Leuine. I. N. PhyrvalChmislry. 3rded.; McGraw-Hill: New York, 1988:PP485-486. 3. Shoemaker, D. P,Garland,C. W.; Steinfeld, J. 1;Nibler. J. WExprimw~fsm Physim! Chamintry, 4th ed.: Mffirau-Hili: New York. 1982; pp 107-119. 4. Palmes. E. D.; Gunniron, A. F: DiMattio, J.: Tomczy, C.Am. Ind Hyg Assm. J. 1976, 37.570-177.
5 . Dunlop, P J.; Harrla, K.R.;Young. D. J. In Physrcoi Methods ofCh:hpmistry,Delrrmi~ nation ofThermody~micProprrlies,2nd ed.: Rorritar, B. W.:Baetrold, R. C.. Eds; Wiley: New York. 1992: Vol.6, p 183.
6. Shooter, D. J. Chem E d n c 1SSS.70,A133-A140. 7. Reid, R. C.:Prsurniu. J. M.: Sherwood.T K. T h e h p r t k e ofGosrsondLiquuis, 3rd od.; McCraw-Hill: New Yoik, 1977; p 557.
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