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RALPHK. BIRDWH~STEU University of West Florida Pensamla. FL 32504
What Is the Physical Significance of the Pictures Representing the Grotthus H+ Conductance Mechanism? H. G. Hertz, B. M. Braun, K. J. Miiller, and R. Maurer lnstitut fOr Physikalische Chemle und Elektrochemie der Universitst Karlsruhe, Kaiserstr. 12. D-7500 Karlsruhe, W. Germany
What's in a name? That which we call a rose by any other name would smell as sweet. Shakespeare In the article "The Molecular Structure Conundrum: Can Classical Chemistry Be Reduced to Quantum Chemistry?" (I)published in this Journal, S. J. Weininger concludes that it is helpful to show students the prohlem associated with the handling of a quasi-rigid molecular structure model, especially in condensed phases. We show that also in the case of acid solutions, such as HCI-H20 mixtures, we run into difficulties if we hold on to the picture of distinguishable H+ ions in the solution. I t is not possible to explain the high extraconductivity of the strong acid solutions while distinguishing between H+ and H in HzO. The difficulty has its manifestation in the graphical schemes occurrine in manv treatments includine textbooks: the correliponding protun jumping mechanism is called "Grotthus mechanism". For this reason we have daced this classical picture of the Grotthus mechanism i n t i focus. We offer a basic critique as well as constructive argument in order to arrive a t the correct understanding of the prohlem.
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The Dynamlc Deflnitlon of the H+ ion Can the H+ ion or the H30f ion in aqueous solution he reduced to quantum chemistry? Clearly, in accordance with the article by Weininger this question must be left open. So, let us ask the more modest question, can the H+ ion be defined on a level of classical mechanics? Before we quote such a definition, which we shall call the dynamic definition, we characterize the general situation as follows: All material svstems consist of Drotons. neutrons and electrons. Then an iinicsptern is defined to besuch that the number of protons is different from the number of elertrms. For a system in which the number of protons is equal to the nimber of electrons it is not obvious what the species "ion" is meant to he. Usually one refers t o a local nonequality of the number of protons and electrons; for example, for a cation of charge 1, one writes
where p, and pel are the number densities of protons and electrons, respectively, and AV is an appropriate region around the nucleus of the cation. A species defined in this way is difficult to identify experimentally, in particular if the ion and the solvent contain the same nucleus, that is, the same element so that solvent and ion may he indistinguishable (2). Therefore, in the case of the H+ ion one may try to take recourse to an operational or dynamic definition in the
following way: The H+ ion is exactly that particle which responds to the presence of an electric field strength V E with a drift velocity u+, where
with the Faraday constant F, XH+ being the equivalent conductance. Now the object of our analysis is to give an answer to the question, Are the diagrams representing the Grotthus mechanism compatible with the dynamic definition of the H+ ion? Some Bask Facts of the Electric Conductance of an Acld Soiutlon: Two Conflicting Polnts of Vlew To see the basic point let us carry out an experiment in which we have an electrolysis of hydrochloric acid. We ignore the anode reaction, since the oxygen is not involved. Provided that we use an ideal reversible cathode, the electric current running in our electrolysis cell delivers a certain amount of hydrogen that will he collected above the cathode. According to Faraday's law this amount of hydrogen is strictly proportional to the amount of the electric charge transported. Each proton (hydrogen nucleus), if it contributes to the electric current, transports its mass as well as its positive charge to the cathode. In other words, we cannot separate the mass transport from thepositiuecharge transport. Experimentally one finds a high conductivity of the solution, which may he understood to mean that there must he a high mobility of the H+ ions. There is an unambiguous relation between the specific conductivity and the analytical concentration of the acid constituent. This had led to many analytical applications of conductance measurements. In this sense effectively the "H' ion" is defined by the high conductivity. This we called the dynamic definition of the "H' ion" in the previous section. I t implies the need to distinguish between protons in the state HC and in the state
H"n .=-.
I t is a fundamental principle that the trajectories of the particles that transport the mass must he closed. Thus the trajectories of the positive electric charee must also be cloied. More preciseiy, we should say, the component of the trajectories in the direction of the electric current must he closed. Let the direction of the current represent the z component. In this direction nomissing links are allowed to exist in the protons' path through the solution (see Fig. 1). From other physical evidence, such as the enthalpy of hydration, the hydrodynamic radius of the H+ ion is supposed to be as large as that of K+ (3),leading to the model of the "H30+ ion". How is this comparatively large radius compatible with the high drift velocity, that is, the dynamic definition of the H+ ion? Volume 64
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Five 2. (a) The usual picture of the Orollhus mechanism. (b) Scheme w'W I ~ x ' l l r= 0 (see text). The tlme me system needs to change from the leH side to the right ride we call At.
Figure 1. (a1 A trajectwy rIq that is dosed wimin the interval shown. (b) A trajectwy ft0 that is not closed; it is imenupted. (c) A trajectory 40, the z component of whioh is closed hhis component has no interruptions).
The Grotthus Mechanlsrn and its Dlfflculties
T o solve this discrepancy, several authors have constructed microscopic pictures (4-8). The mechanism that these pictures are based on is called the "Grotthus mechanism". Figure 2 gives two examples. All of these pictures, for a given instant, pretend the definition of two classes of protons in the acid solution: the H+ ion (including possibly the H30+ ion) and the water proton. In fact, in the course of time the individual assignment is not considered to he preserved. The protons jump from an H+ state to a water state. A first glance a t those pictures gives the impression that we do not have to deal with fast motion of H+ ions. The hydrogen bonds of the water seem t o allow some "short cut" for the distance that the nroton has to move in order to transport the electric current. Is the Grotthus mechanism an ineenious scheme to cirrumvent the necessity of fast H* ion motion? We say no! It needs a closer look at these diaerams. If thev are ~ h v s i c a h correct, the "Hf ions" alone carry the burden o f m & and charae transport. We state again the basic reauirement: Bv any means of transportation-through the acid solution the H+ ions have to reach the cathode. Otherwise we could not explain the amount of evolved hydrogen we observe in the experiment. If we only allow the H+ ions t o move from the anode to the cathode. we have to claim fast motion. According to the diagrams (Fig. 2) the velocity is AzIAt. It is not the individual H+ that starts a t the left which arrives on the right. The H+ may in some way dip into the H-bonded system as shown in the diaeram. But in anv case the z r&nponent of the trajectory f i r the positive charge and thus the hvdroeen mass has to be closed. This is not the case for pictu;es like Figure2a. Sometimes the gaps interrupting the traiectories are supposed to he bridged bv the motion uf electrons. But the kiectrons not belonging to the anions on the average are localized and do not contribute to the electric current. Thus, i t seems that in fact we do require the dynamic definition of the H+ ion as a fast-moving particle.
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Figure 3. The corresponding picture of the Qotthvs mechanism in the velocity space. The heavily drawn arrows indlcate that there is an exchange of protons between states a and b. (a) The partial sum of the vectors of the water protons equals zero. (b) The partial sum of the veloclly vectors of the H+ ions correspondsto the observed mcductivily.
Another difficulty with the usual diagrams is the positions of the water protons during the jump mechanism, as may be seen in Figure 2a. Evidently only H bonds are rearranged in the hope of short-cutting the path from the anode to the cathode. Such a procedure is physically not acceptable because there is another important point that is usually overlooked: the water protons have to be treated as being in a resting state because the transport numbers refer t o the solvent at rest, and this condition isnot fulfilled in Figure 2a. However, we must now turn to an important point that again makes i t doubtful whether a n H+ ion in the dynamic sense can be defined. It has been shown in exnerimental that work on NMR relaxation and neutron scattering ?9,10) one does not observe any fast motion of protons in hydrochloric acid. Thus the question remains In what other way we can arriveat an understandineof what h a.~.n e uin s an acid solution during the flow of electr% current? Descrlptlon of Conducllvlty In Veioclty Space
The conductivity basically implies a velocity value, so we should construct the diagrams corresponding to Figure 2 in a space referring to velocities only. We shall later return to the "trajectory representation". Now the usual figures contain the water molecules as H-bonded species, the positive hydrogen ion is formulated as H+ (HzO) ion. If we change to a description using only velocities, we have difficulties descrihine the water molecules. Thev are in a resting state. First let us deal only with protons, that is, hydrogen nuclei. nropertv We consider Firure 3. This diaeram must have the . that the partiaisum of velocit;vectors of the water protons has t o be zero (Fig. 3a). Simultaneously, the partial sum of velocity vectors of the H+ ions has to correspond exactly to the observed conductivity (Fig. 3h). Now, as time goes on, the points belonging to class a, where the vector sum equals zero, exchange with those of class b, for which the total
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Figure 4. We construct a sphere around each oxygen nucleus with the radius 0.96 A. The protons we will find on these spheres belong to the water molecules. Around these O-H vectors we construct a torus with the condition that each torus only contains one proton, which is Vle second water proton. Ring radius and position of the torus may be taken from classical H20 geometry. The proton lhst is outside of the torus and not located on the O-H vector is the Ht ion.
Figure 5. Velocity vectors of the water protons. Dashed arrows Mnrespond to AX/r.. solid arrows are the velocity of the water protons that contribute to the electric current.
velocity does not vanish. Thus, class a represents a reservoir of velocities that makes the conductivity high, and still the sumot the \ectorsofall \.elocities hrlonging ~ o u , a t eprotons r is zero. How can this he acromplished'! First we can ask, is it possible to divide the protons-in an acid solution into such two classes? T o say which protons belong t o the water molecule and which belong to the "H+ ion" state is primarily a question of the electron distribution pel (r) in ordinary space. However, the electron distribution is not known precisely, thus it is easier to proceed by means of an ad hoc geometrical classification without loss of generality. For this reason we return to configuration space. Here we create boundaries that separate the two classes of protons, the "H+ ion" and the H in H20. This could he done, for example, by a torus construction (see Fig. 4) (11). However, velocity means a change of nlace in a time interval. Since the two classes are defined by surfaces in normal space, a part of the velocity vectors necessarily correspond t o a change from one class of protons to the example, one proton of the other one and vice versa. class "water" leaves its class; then this class is depleted of a proton. Thus, as the number of water molecules has to remain constant, a short time later a proton of the class "Ht ion" has to convert to the "water" class. In the first case we have an exit point on the torus surface while the second event corresponds to an entry point in the torus. The distance between these two points we call AX. The first event hannens at t = 0. the second a t t = r p .If we look at the states of ;roton at t = -dt and t = 7. d t , k e do not know that the individuals have chanced. The effect is that one proton apparently has moved from the exit point to the entry point of the torus surface with the velocity -AXIT.. The total rate of position changes of the protons in the state "water" is
In the equilibrium state the value of ZAX,/r. equals zero, 0. This hut in the presence of the electric field XAXjlr, proves the fact that we have a contribution of the water protons to the mass and positive charge transport in acid solutions. In Figure 5 we show again a part of the velocity vectors of the water protons, hut now the vectors that correspond to Z ~ A X J Thave ~ been drawn as dashed arrows. The remainine. fully . drawn arrows are the velocities of the water pnmrns that contribute to the electric current. Theauantitv YIAX,/r,mav heconverwd roan expression containing the mean lifeti&e of the proton in the water molecule, 7.This yields the formula for the partial equivalent conductivity of the protons in acid solutions (12,131.
where the superscript 1indicates that eq 1refers to 1mL; the sum comprises all empty tori that are contained in 1mL of the sample. As was explained above we have to require that the water is in a resting state. Consequently we have dRhldt = 0 (11). With eq 1we arrive a t the result,
with k = 1.06 X Thus we obtain
or
+
+
With the electric field strength V E and the lifetime of a proton in the water molecule 7 . NH+is the number of H+ ions per unit volume, and vh+ is the total drift velocity of all protons in the state H+ ion per unit volume. AX1 = AX'.NH, with ZlAXi = AXf.N,, N, and N H ~ O are the numbers of empty tori and water molecules, respectively, per unit volume, AX' is the mean value of the exit-entrance separation. 'I'hs srnmd term on the right-hand side nf eq 3 corresponds to a motion of the protons while they are in the water state; that is., thev have certain drift velocitv. and a t the same time they are within the water molecule. As a consequence, to the second term on the right-hand side of ea 3 corres~onds the totality of the fullyUdrawn arrows i n - ~ i g u r e6, if this figure is understood to refer to 1 mL. How great are the relative contributions of the first and second terms in ea 3? At 25 " C the experimental value for X+IFis -3.6 X 10-icm2N.s. The 117is known from NMR proton exchange measurements (14,151. We have
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lo-"
mL s-I, NHZoIV= 3.33 X
loz2mL-I.
and if we set IAX'IIIVEI = 0.5 X 10-l4 cmZN,we find that the second term on the right-hand side,of eq 3 is about equal to Volume 64
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Figure 6. Extended picture of the Grotthus mechanism containing the contribw lion of the rswientatii of the watw m o k u i e s (see te*). The heavily haw" curves represent the reomponent of the trajectory of the proton. Note that this quantity has no interruption
electric current by some mechanism. I t becomes effective, because it is connected with the proton exchange, which is given by the second term in eq. 3. Since this rotation of the water molecules restores the mean position of the water protons, the transport of the mass and the charge of the protons is distributed among a greater number of water protons. So, Figures 6 and 7 reconcile the two demands: high electric conductivity of acid solutions and the experimental finding of slow motion of the H constituent. We found only one textbook (4a) in which the corresponding diagram fulfilled the condition I ~ X ' l l r= 0. This is reproduced as Figure 2h. Here the z components of the trajectories are closed. All AX vectors are perpendicular to the direction of the electric current. Water rotation to reinstate the proton positions is also perpendicular to the direction of the current, thus does not contribute. From the very small mean value of \AX1\ and eq 3 it follows that in the strict sense Figure 2at, 6, or 6 with 7 formally are the correct representations, and Figure 2bgives agood picture of the real situation. Water Molecule Reorlentatlon as a Rate-DeterminingStep
I t might appear that the reorientation of the water molecules has already been incorporated in many figures of the Grotthus mechanism (46, 8, 16-18). The important difference between our description and that of other treatments lies in the fact that in the former case the water rotation is not merely the rate-determining step of the mechanism hut represents the contribution of the water protons to the amount of transported charge and mass. However, in those cases where the water reorientation is considered to be the rate-determining step, water rotation as a transportation mechanism of mass and positive charge must also be considered. We start by modifying eq 3:
I Figure 7. This picture replaces step c ) in Figure 6. It now indicates that the second step reorientation of the water molecules can be performed by any water molecule of the solution and rotation types a) and b) can be combined.
the first one. In fact, since the diameter of the water molecule is about 3 X 10-6 cm, the distance between the exit and the previous entrance in the water molecule must be extremely small in order to make the first term dominant. This has the important consequence that even a dynamic definition of the H+ on a classical level as described above cannot be given. The observed quantity always contains a certain contribution from proton motion in the water molecules.
wherevh is the mean total velocity of all protons (i.e., hydrogen nuclei) in 1mL. Let CHCL he equal to 1 mol/L, then vh/ NH+consists of 2 X 55.5 1single velocity vectors of hydrogen nuclei. Per added H+ there are 1000118 = 55.5 water molecules, because c ~ c = l 1mol/L. Each H20 contains two protons. Thus we have to add to the one H+ proton 2 X 55.5 water protons. Thus we have
+
How To Draw Proper Plctures of the Grotthus Mechanism
As we have seen that a dvnamic definition of the H t ion cannot he given, it remains t o be demonstrated in what way diaarams renresentina the Grotthus mechanism should he drawn to obtain a correct picture. In order to fulfill the requirement that the water protons are at rest we must complete the "Grotthus" picture of Figure 1by a third step, in which the necessary rotation of the water molecule is performed (see Fig. 6). The thick lines of Figure 6 represent the closed total trajectory to which the corresponding proton hascontrihuted, partly as a proton in the H+ ion state, partly as a proton in the water state. The reorientation of the water molecules can he performed by any water molecule of the solution, not only by those involved directly in the jump mechanism (Fig. 7). Of course, such a water rotation must he correlated with the
' Apart from lacking reinstatement d water molecule positions. 780
Journal of Chemical Education
Figure 8. A typical representation of the H&:
ion.
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F gdre 9 la) G ves a dem ed p c t x e of tne scheme F gLre 8 n wn cn the cross sections of the l a r ~defmng r m e water molecules are drawn and protons are num berw 1-9 A ftnh water mo ecu e madded. lnemtat,on of uhlch (5 the rate-aetermmmg stepaf Ihaprotontransfer beyond tnssynem d,0: (b) m s f l n h water mole. cule hasaccepteda proton in the state Ht ion. (c)A rotationof the water hasoccurredto reinstall all previous hydrogen positions in the water molecules. Arrowsrepresent partial or total trajectories of mass and positive charge of the hydrogen nucleus. Far further details see text.
where R = R(t) is the "total" position vector of the 2 X 55.5 1protons, that is,
+
The "total trajectory" R(t) is the sum of the trajectories of all single hydrogen nuclei. Let us first consider hydrogen nuclei 1-9 only, and let them be arranged as shown in Fieure 9. which corresnonds to the ;irrilngernent in Figure8 \,cry oftengiven in the literature l.7-71. T h r w e n circles in Finurp 9 11 reoresent the cross sections of &e tori that def&e the water molecules. The rotation of the water molecule a t the right-hand bottom of Figure 9a-lla (drawn with dashed lines) is supposed to he rate-determining for the H+ transport. We ask, How long is the time At that appears in eq 6? I t is given by the mean reorientation time of the "dashed" water molecule. We have
Fiaure 1 0 ..b.- l.d. The same as Flaure 9a-c. the onlv difference beino that now the thickness of the torus del ning the water molsc~lenar osan chosen to be smaller. The heavily drawn a r r o w n part c a r e lrajectorier of Me positive charge and mass in the Ht ion state
(An2) is the mean square solid angle that has been attained by a fixed vector in the water molecule after a time At when the Hz0 molecule performs rotational diffusion. D, is the rotational diffusion coefficient. Let us set An = %4a = 2a. Then it follows from eq 8 that
and with eqs 5-7
Figure 11. The same as Figure $a-c; however, now Me distance AX between exit and entrance of the proton in the torus defining Me water molecule has becamesmallsr. Heavily drawn arrows: trajectory of proton in thestate Ht ion.
Next we consider the first term on the right-hand side of eq 9, that is,
the water is in the resting state, as above we must of course have
= RIA(&)
- R(O ,)
I t is the change of the total position vector of protons 1-9 after a time interval At. At the end of the interval At particle 7 has attained the position close to the dashed water molecule as shown in Figure 9h, and the total process may be repeated, where now the dashed water molecule has the function of the upper left molecule in Figure 9a. But we did not yet give the proper attention to the water protons. Since
ARH,".~ = 0 (10) Thus, the water protons have to moveuntil the state as given in Figure 9a is reached. This is the final state; in fact, water protons are at all initial positions, hut the individuals are not the same. The arrows in Figure 9c show the total displacement of the hydrogen nuclei, and i t is clear that the motion of the proton in the water molecule partly is what one calls a rotation of the molecule. Let us denote the separation he-
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tween the exit and the previous entrance into the torus as AX' (see Fig. 9 4 . Then we see from Figure 9c that ARI.~= 3AX'. So far we have assumed that the electric field has the direction of the vector connecting position 1 in Figure 9a with position 7 in Fieure 9h. which is eiven hv the ratedetermining water molecule. o f course,
