In the Laboratory
Carbon Dioxide Dissolution as a Relaxation Process A Kinetics Experiment for Physical Chemistry Peter G. Bowers,* Mordecai B. Rubin,† Richard M. Noyes,‡ and Dagmar Andueza Department of Chemistry, Simmons College, Boston, MA 02115 In chemistry, the term “relaxation” is used in general to describe events that occur as a system returns to equilibrium after undergoing a (usually small) perturbation. In spectroscopy the term refers to the restoration of thermal equilibrium in the population of specific energy levels, such as vibrational relaxation following photochemical excitation or spin relaxation in NMR. In chemical kinetics, with which this article is concerned, the word is used to describe firstorder return to chemical equilibrium after a system has been given a small perturbation, by a sudden change in temperature (T-jump), pressure (P-jump), or concentration (C-jump). The perturbation may be produced, for example, by an electrical impulse, by a shock wave, or by ultrasound. Physical chemistry texts (1–4) give a kinetic analysis for equilibria of varying complexity. The simplest is A
B
(1)
If k1 and k2 are the forward and reverse rate constants for the elementary steps and a system containing both A and B in some way reaches a position close to equilibrium at t = o, with [A] = [A]o, and then relaxes to equilibrium where [A] = [A]eq, then detailed treatment gives ∆[A] t = ∆[A]o exp { t/τ
(2)
where ∆[A]t and ∆[A]o are positive values of displacement parameters, defined as |[A]t – [A] eq| and |[A]o – [A]eq|, respectively, and τ is the relaxation time: τ = 1/(k1 + k2)
(3)
Thus the displacement parameter changes in a first-order manner during relaxation. Also, since reactions 1 and 2 are elementary steps, the rate constants are related to the equilibrium constant by eq 4. Keq = k1/k2
(4)
Evaluation of τ from a suitable kinetic plot, together with a knowledge of the equilibrium constant Keq, allows determination of both k1 and k2. Most relaxation times of interest are much shorter than mixing times, and the jump techniques were developed to overcome this by starting with systems already mixed and at equilibrium. The most frequently cited relaxation processes are the fast reactions first investigated by Eigen using the T-jump method. An example is the dissociation of water H3O+ + OH{, τ ~ 10{6 s). A common undergraduate (2H2O laboratory experiment is the study of relatively slow relaxation in the chromate/dichromate system in aqueous solution (Cr2O 72{ + H 2O 2CrO42{ + 2H+, τ ~ 100 s). This relaxation can be studied spectrophotometrically using an acid/base in*Corresponding author. † Permanent address: Department of Chemistry, Technion, Haifa, Israel. ‡Permanent address: Department of Chemistry, University of Oregon, Eugene, OR 97403.
dicator to follow the change in pH after mixing the reagents to a point close to equilibrium (5). Transfer of Carbon Dioxide into Water The air–sea exchange rate for CO2 is important in describing the global CO 2 balance and extensive laboratory and field studies have been made of the kinetics (6–10). Carbon dioxide dissolution is also important industrially, where it is usually followed by rapid reaction, as in the removal of CO2 from natural gas by absorbing it into basic solution. Even for nonreactive CO2 transfer, such as air–sea exchange, interpretation of the results can be complicated by such factors as agitation or motion of either phase, the presence of other gases, and thermal effects. However, essentially all studies find that gas transfer can be described by an experimental equation of the form J = {ktr(cs – c seq)
(5) cm{2 s{1 ),
The flux of gas crossing the surface, J (in mol is taken to be positive when gas enters the liquid, and ktr (in cm s{1) is a mass transfer coefficient or exchange coefficient. The quantity (cs – cseq) is the difference between the dissolved gas concentration at any time and the equilibrium value at the existing partial pressure of the gas at that time. Values of ktr for CO2 /water range over two orders of magnitude, from 10{4 to 10{2 cm s{1, depending on the method and the relative importance of the complicating factors mentioned above. Alternative mechanisms have been developed to describe interphase gas transfer. Most of these invoke some kind of stagnant layer at the surface, saturated with gas. Gas diffuses across this surface layer and into the bulk liquid (7). The difference between various mechanisms is in the interpretation of the exchange coefficient ktr. The simplest mechanism, and the one we use here, is based on our observations (6) of the effects of increasing the liquid stirring rate in a closed system in which the gas pressure is being monitored. We found that the transfer rate increases to a maximum as stirring is increased, and then stays constant until vigorous stirring deforms the surface. On this basis we proposed that molecular transport in the well-stirred region can be viewed as a simple relaxation process between two phases whose composition is uniform at all times (i.e., with no surface layer). The starting point for the treatment is then an equation similar to reaction 1: CO 2(g)
CO2(soln)
(6)
The kinetic treatment, given in full elsewhere (6), involves the volumes of the two phases and the interfacial area. Corresponding to eqs 3 and 4 above, the equations relating k1 and k2 are: τ = [Vg /A]/[k1 + (k2Vg / Vs)]
(7)
κRT = k1 /k2
(8)
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In the Laboratory where Vg and Vs are gas and solution volumes, and A is interfacial area. The “equilibrium constant” κ is the known Henry’s law constant (in mol L{1 atm{1 ). Experimentally, τ is determined as the reciprocal of a first-order rate constant kexp (in s{1), found by measuring the pressure changes above the solution: ln [( p∞ – p) / ( p∞ – p o)] = {kexp t
(9)
Significantly, the kinetic treatment also shows that if the simple relaxation scheme is valid, then the rate constant k2 for desorption is identical to the mass transfer coefficient ktr. Experimental Details In the student experiment, the transducer (an Omega PX160)1 measured pressure changes up to a few torr around atmospheric pressure, and was therefore suitable for studying the relaxation close to equilibrium between CO2 gas at 1 atm and its saturated solution. The best arrangement and procedure at any particular site will be determined by the available equipment, especially by the range of the transducer. However, we have previously shown (6) that the kinetics developed above applies equally well far from equilibrium, over more than 4 half-lives, and gives consistent results when Vg , Vs , and A are varied over more than an order of magnitude.
The Apparatus Gas absorption took place in a 200-mL Erlenmeyer flask containing 100 mL of water (Vs ) and a 3-cm magnetic stirrer (Fig. 1). The gas was absorbed from a 500-mL filter flask, which together with connectors gave Vg ~ 800 mL. Both flasks were immersed in temperature-controlled water baths. Tank CO2 was obtained via a ballast of about 10 L containing a few hundred milliliters of CO 2-saturated water, so that the experiment was conducted with gas saturated with water vapor. Voltage from the transducer was recorded on a conventional chart recorder. The transducer was checked for linearity using a U-tube water manometer, but for the method of data treatment used, absolute calibration was not required. Procedure Distilled water is introduced into the absorption flask, and with stopcocks A, B, and C open, and D closed, air is flushed from the apparatus by passing a steady stream of CO2 for about 10 min with the stirrer off. While gas is still flowing, the stirrer is then turned on rapidly to cause as much vortexing as possible, in order to almost saturate the solution in about 1 min. The gas flow is turned off, stopcocks A and C are quickly closed, and the transducer and recorder are started. The stirring control is adjusted downwards to give the maximum possible rate without distorting the surface. The pressure drop is then recorded as the system relaxes towards equilibrium. A final uninterrupted rate curve is collected for about 20 min. A typical record, taken from a student experiment, is shown in Figure 2. It is not necessary to continue the run until equilibrium is reached. The volume Vg is found by filling the two flasks with water and estimating geometrically the smaller contributions to Vg from connecting tubing. The surface area is found to within about ± 5% by estimating the interior diameter of the small flask at the water level, or by sacrificing an identical flask to measure the inner diameter more exactly. The chief problem encountered by our students was getting the solution to the correct degree of saturation be-
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D C
CO tank 2
,, ,, ,, , ,, ,,,, ,,,,,,,, A
B
transducer
Vs
Vg
Figure 1. Experimental arrangement for measuring the absorption of carbon dioxide by water.
fore starting a run. A solution insufficiently saturated gave a pressure drop which quickly went out of range of the transducer response. When encountered, this difficulty was solved by increasing the stirring rate for a further 15 s, and bringing the system back to atmospheric pressure by momentarily opening stopcock C. This could be repeated several times until a measurable trace was obtained. The problem would not arise when using a transducer with a larger range, say ± 0.5 atm. We had students perform the experiment in pairs, each pair carrying out at least two runs. Most pairs finished well within the 4-hour lab period, even with one or more aborted runs. If only one apparatus is available, as was the case at Simmons, the experiment has to be part of a rotation or done on a “sign up” basis. A final noteworthy feature of this experimental arrangement is that unlike most other kinetics experiments involving gases, a vacuum line is not required. S AFETY: Although the gas pressures developed in the apparatus described are close to atmospheric, carbon dioxide from the tank is delivered at a much higher pressure. Conventional safety precautions such as eye protection must be especially observed when manipulating equipment at any pressure different from atmospheric.
Figure 2. Typical recorder trace of the pressure drop during carbon dioxide dissolution. The total pressure change shown is about 10 torr.
Journal of Chemical Education • Vol. 74 No. 12 December 1997
In the Laboratory Data Treatment Curves like those in Figure 2 have to be measured by hand unless some form of automated data collection and processing is available. If xt (cm) is the pen displacement measured from any convenient reference point on the chart at time t (s), then a tabulation of x t versus t is made selecting points one minute apart, to give, typically, 20 data points. These are treated by the Guggenheim Method (11), which is a standard method for extracting an experimental rate constant for a system approaching an equilibrium. It avoids the tricky “infinite time” pressure measurement, p∞, and does not require the measurement of any pressure in absolute terms. Pairs of points are selected at times t and t + δt, where δt is a fixed time interval, and the difference in displacement calculated: ∆x = | xt – x(t + δt) | Then according to the Guggenheim approach, eq 9 can be cast into the form: ln ∆x = {kexp t + constant
(10)
Thus a plot of ln ∆x versus t should be linear, with slope kexp. In practice, the method works best when the fixed time interval δt is as long as possible: for 20 data points at 1-min intervals it is convenient to choose δt = 10 min. Thus ∆x1 = x1 – x11, and so on, giving 10 points for the plot. Figure 3 shows such a plot made from the rate curve in Figure 2. The value of kexp is 2.2 × 10{3 s{1 , corresponding to a relaxation time of 450 s with this geometry. Of course, more sophisticated methods of data treatment are possible: in our original experiments with these systems (6), the transducer voltage was collected by a digital voltmeter and the results directly fed into a computer that calculated kexp and its statistical reliability directly, without any intervening graphs. However, for instructional purposes, it may be that more insight is gained by starting with a recorder trace. The constants k1 and k2 can be calculated from kexp and κ by combining eq 7 with eq 8 (recalling τ = 1/kexp ) : k1 = [kexpVg /A] / [1 + (Vg / κRTVs)]
(11)
Values of the other quantities in eq 11 for the sample experiment shown were: Vg = 820 cm 3; Vs = 100 cm 3; A = 66 cm2; T = 296 K; κ = 0.036 M atm{1 . This gives k1 = 2.6 × 10{3 cm s{1, and k2 (= ktr) = 3.0 × 10{3 cm s {1 .
Discussion
Validity of the Simple Mechanism All quantitative deductions above, made from rate curves like that show in Figure 2, are based on the premise that the kinetics can be interpreted as exchange between two homogeneous phases, with no significant concentration gradients on either side of the interface. The rationale for this assumption comes chiefly from the observations on the effects of stirring: when kexp is measured as a function of stirring speed a plateau is reached, which we believe occurs when mechanical mixing has become faster than molecular diffusion, and no stagnant surface layer exists. In the plateau region, our interpretation of kexp is valid. Increased stirring has little effect until the surface starts to be distorted. In developing the student experiment, we verified the plateau behavior for our apparatus and obtained the result shown in Figure 4. Similar behavior has been observed for the systems N2 /water, air/water, and CO/sulfuric acid (12). We did not have each student pair verify the existence of a plateau,2 but found that reasonably consistent results were obtained simply by setting the stirrer just below the surface distortion speed, as judged visually. Reproducibility of the Experiment Results for this experiment are moderately reproducible. The class average for kexp was 3.5 (± 1.5) × 10{3 s-1 (n = 17). However, k1 and k2 were several times higher than when the measurements were originally made at the University of Oregon (6). Possible reasons for this include questions about the validity of the simple mechanism, or the presence of a surface layer of foreign material (such as detergent), which is known to have a large effect on gas transfer rates (8). Interesting student work in the future might include studying transfer rates in water of varying degrees of purity. The Mass Transfer Coefficient ktr The value found for k2 ( ~10{3 cm s{1 ) using our arrangement falls about midway within the range of literature values reported for ktr. These are generally direct experimental field or laboratory measurements (for example where the pressure of CO 2 is maintained constant, and its buildup in solution is monitored) and not based on relaxation in a closed system and associated mechanistic assumptions.
0
4
VORTEX FORMATION
- ln kexp
ln( ∆p)
2
3
4
6 INCOMPLETE MIXING
2 0
5
10
0
5
10
stirring speed (arbitrary units)
time / min Figure 3. Guggenheim plot of rate curve shown in Figure 2.
Figure 4 Effect of stirring speed on rate constant for carbon dioxide dissolution.
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In the Laboratory For an unstirred system, some kind of surface layer of dissolved gas undoubtedly exists. In the “stagnant layer” model, dissolved gas is in equilibrium with the gas phase at the surface, and its concentration decreases downwards within the layer, at the bottom of which it is the same as in the bulk solution. Transfer is rate-controlled by the diffusion across the layer, and the treatment predicts that ktr ~ D/z, where D is the diffusion coefficient and z is the thickness of the layer. Values of ktr at the low end of the range (~10{4 cm s{1) allow z to be estimated at about 100 µm. In the discussion of the existence of surface layers, it is important to distinguish between disagreements between experimental results made under different conditions, and different interpretations of experimental results based on different models. Thus ktr as defined in eq 5 clearly depends on the degree of agitation of the solution (among other factors), but not on the geometry. In our experiment, under well-stirred conditions, we believe that the simple model is valid, and therefore ktr can be identified with k2. Other values of ktr obtained under different conditions (such as no agitation) require a different interpretation.
The Mass Accommodation Coefficient The value of k1 can be used to estimate an important quantity called the mass accommodation coefficient (γ), which is simply the probability that a molecule hitting the surface will be absorbed rather than bounce off. The coefficient is found by comparing the experimental k1 to its maximum possible value calculated from kinetic theory (1) and assuming every colliding molecule dissolves. The expression for γ is: γ = k1[2 π M / R T]1/2
(12)
Using k1 = 3.5 × 10 {3 cm s{1, M = 44 g mol{1, T = 296 K, and R = 8.3 × 107 erg (deg-mol){1 , we obtain γ ~ 4 × 10{7. That is, only about 2 molecules for every 5 million that hit the surface actually dissolve. We would not expect γ to be particularly dependent on stirring conditions: even if a CO2 molecule hits a saturated solution (as in the stagnant-layer model), the surface forces it experiences must be almost entirely due to solvent, because at saturation the surface mole fraction of CO2 is only ~ 10{4. However, in an unstirred solution it is not possible to calculate γ because kexp cannot be related to k1 in any simple way. A knowledge of accommodation coefficients is important environmentally, such as in the formation of aerosols. The coefficient to some extent correlates with the solubility of a gas, because both are partly determined by the magnitude of solute–solvent interactions. There are not many measurements to compare, but the value for N2/water is ~ 10{8–10 {9 (12), while for relatively miscible solutes it is much larger (~10{2 for both SO2 and methanol in water [13]). The best measurements for a single substance (that is, for
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condensation and evaporation equilibria) suggest that accommodation coefficients in those systems may be much closer to unity (14). Concluding Comments An environmentally important process, the dissolution of CO2 in water, can be can be studied using a arrangement that is simple and inexpensive to set up. The procedure is straightforward and does not require the use of a vacuum line. Students become familiar with the concept of relaxation to equilibrium, kinetic treatment of reversible reactions, the special problems of interphase transfer, and the idea of mass accommodation coefficient. The experiment is also interesting for students because the mechanism of interphase transfer is still incompletely understood. Notes 1. Omega Engineering Inc., One Omega Drive, Box 4047, Stamford, CT 06907. 2. We wish to thank a reviewer for suggesting that since stirring effects are so important, they might be further investigated as a class project.
Literature Cited 1. Atkins, P. W. Physical Chemistry, 5th ed.; Freeman: New York, 1994. 2. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 2nd ed.; Houghton Mifflin: Boston, 1995. 3. Noggle, J. H. Physical Chemistry, 3rd ed.; Harper Collins: New York, 1996. 4. Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper & Row: New York, 1987. 5. Salzberg, H. W.; Morrow, J. I.; Cohen S. R.; Green, M. E. Physical Chemistry Laboratory, Principles and Experiments; Macmillan: New York, 1978. 6. Noyes, R. M.; Rubin, M. B.; Bowers, P. G. J. Phys. Chem. 1996, 100, 4167–4172. 7. Danckwerts, P. V.; Gas–Liquid Reactions; McGraw-Hill: New York, 1970. 8. Skirrow, G. In Chemical Oceanography, 2nd ed., Vol. 2; Riley, J. P.; Skirrow, G., Eds.; Academic: London, 1975; Chapter 9. 9. Doney, S.C. J. Geophys. Res. 1995, 100, 8541–8554 (see also for earlier refs). 10. Holmen, K; Liss, P. Tellus 1984, 36B, 92–100. 11. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism, 3rd ed.; Wiley-Interscience: New York, 1981. 12. Noyes, R. M.; Rubin, M. B.; Bowers, P. G. J. Phys. Chem. 1992, 96, 1000–1005. 13. Worsnop, D. R.; Zahniser, M. S.; Kolb, C. E.; Gardner, J. A.; Watson, L. R.; Van Doren, J. M.; Jayne, J. T.; Davidovitz, P. J. Phys. Chem. 1989, 93, 1159–1172. 14. Niknejad, J.; Rose, J. W. Proc. Roy. Soc. 1981, 378A, 305– 327.
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