J . Chem. SOC., Faraday Trans. I , 1985, 81, 579-596 Thermodynamics of Transfer of Noble Gases in Hydrophobic Solvents and in Phospholipid Membranes BY YEHUDA KATZ The National Physical Laboratory of Israel, Hebrew University Campus, Givat Ram, Jerusalem, Israel Received 2nd April, 1984 A theory of gas solubility has been formulated and tested using solubility data of noble gases in several hydrophobic solvents and in a phospholipid membrane of dimyristoyl lecithin. The theory describes solubility in terms of two independent processes: hole creation in the solvent and solute adsorption in these holes. It gives the standard thermodynamic functions of solution in terms of four independent microscopic parameters. Two parameters reflect the nature of the pure solvent and the others the nature of the solute and its interactions with the solvent.Examination reveals good agreement between theory and experiment for the solvents tested, paving the way for comparisons between membranes and bulk solvents which are more significant than the existing correlations. The Characteristics of the theory and its agreement with experiment suggest an interesting method for evaluating, from the solubility data of gases, the physical properties of hydrophobic regions of membranes and of other solvents which cannot be measured directly. Guided by our desire to engage in membrane research a theory which is both physically sound and biologically relevant, we develop here a solution theory for simple solutes and test it for bulk solvents and for phospholipid membranes. The theory is needed to evaluate the behaviour of solutions, correlating the macroscopic properties with the molecular features of the constituents. The correlations are then used to deduce solvent characteristics from measured solubilities and vice versa.The definition, characterization and analysis of biological membranes is often done by analogy with model bulk sol~ents.l-~ The long established procedure consists of measuring the extent to which solubilities in selected solvents correlate with the behaviour of the membrane investigated, assuming that the solvent which correlates best with a given property of the membrane is the one which most resembles the molecular features responsible for the pr~perty.~ This method has contributed significantly to our understanding of biological membranes but it also leads to ambiguities and controversies, because the empirical correlations were gross over- simplifications, sweeping the complexities involved in determining the behaviour of solvents into a limited number of empirical parameters.s-8 The best way to overcome this difficulty is to base the comparison between solvents on the extent to which they can be described by a given theoretical formulation instead of using the closeness of fit of an empirical correlation as a criterion.Only when a theory like this is found can the correlations be analysed in terms of the molecular parameters involved and the biological meaning of the experimental findings discussed. The formulation of a theory of solution and its test on membrane and bulk solvents, which is made here, is therefore a step toward introducing physical preciseness into the analysis of membrane phenomenology .Out of the many possible formulations of solution behaviour9 we choose a model 579 20-2580 TRANSFER OF NOBLE GASES which explains solubility as an adsorption of solute molecules in holes made in the solvent because of the thermal motion of its molecules. The choice is dictated by the simplicity of the model, by its ability to relate separately to the pure solvent and to its interactions with the solutes dissolved in it and by the fact that it is related to other successful existing models of solution. The fact that the parameters used here are closely related to some properties which are of special interest in membrane research also contributed to our choice.We limit our discussion here to the solubilization of the simple monoatomic noble gases in simple solvents. This limitation removes major theoretical obstacles and establishes firm ground on which the analysis of more complex solutes can be based. The advantages follow from the fact that the noble gases are the simplest of all solutes and yet their solubilities are determined by factors which also affect the solubilization of all known solutes.10 Simplicity is what makes the noble gases liable to quantitative theoretical analysis by simple models. The existence of a quantitative theory for the solubility of the noble gases enables one to analyse the behaviour of complex molecules in solution by comparing the experimental results for these solutes with predicted values.Noble gases absorb preferentially in the hydrophobic region of the membrane," thus making the noble gases important probes for the investigation of a membrane region which cannot otherwise be separated from the rest of the membrane and explored independently from the other parts. The theory of solution for the noble gases proves, therefore, to be beneficial for an understanding of the hydrophobic region as well as for an understanding of transport through membranes. To test the model we use literature data1'? l2 for the solubility of noble gases in bulk hydrocarbons and in a phospholipid membrane. The available data on the solubility in bulk hydrocarbons are much more abundant than those for solubilities in phos- pholipid membranes. This means that the testing of the model is better when dealing with the bulk solvents.THE MODEL The model we use considers solubility as the adsorption of a solute molecule into one of the holes which the thermal movement of the solvent molecules has created in the medium. We assume that the solute molecule recognizes the hole as a gas space in which it moves under the influence of the adhesive potential field of the environ- ment. We also assume that the two steps which constitute the process of solubility, namely hole creation and solute adsorption, are independent of each other. To simplify the treatment we limit our discussion to dilute solutions, thus neglecting interactions between the solute molecules in the solution.Consider first the number nh and the total volume & of holes which appear in one mole of a liquid whose molar volume is 5. The maximum number of holes a molar volume can accommodate is Nh. The number of holes is determined by the relative magnitudes of the cohesive energies holding the solvent molecules together and the kinetic energy, associated with the thermal fluctuations in the medium, which pulls the solvent molecules apart from each other. According to a basic law of statistical mechanics nh = Nh(vh) exp [-fll(vh)/kT] (1) where T is the absolute temperature, k is Boltzmann's constant and fll(Vh) is the reversible work required to create a hole of volume of at least vh. This work is equal in magnitude and opposite in sign to the cohesive free energy which holds the solvent molecules together.The distribution of holes and their average magnitude were calculated long and for our purpose it is sufficient to note that there is an averageY. KATZ 58 1 value of hole size and that the distribution around this size is a sharp one, meaning that Nh(Uh) andfl,(vh) refer to specific parameters determined by the average hole size. Obviously, by definition which, together with eqn (I), gives the total vacant volume, i.e. the volume free for solute adsorption vf as a function of the molar volume V, : v, = Vl exp ( - f , , / k T ) . (3) Consider now the adsorption of the solute in the free volume. The adsorption is characterized by an equation which relates the concentration of the solute in the free volume of solution n,/ V, to its concentration in the gas phase above the solution ng/ Vg.According to this equation 3 = 3 exp ( - f , , / k ~ ) v g v, where ng and n, are the number of solute molecules in the gas volume Vg and in the free volume vf, respectively, -fsl is the free energy of molecular adhesion, being the free energy gained when a solute molecule adsorbs to a hole, andf,, reflects the tendency of a solute to stick to the solvent. Combination of eqn (3) and (4) gives the expression (4) n exp [Vll -fs,)/k T3 ( 5 ) - _ 5 3 -n, v g Vl linking the experimentally measurable distribution of the solute to the microscopic parameters which determine the behaviour of solutions. Inspection of eqn ( 5 ) reveals that two independent microscopic free energies determine the phenomenological standard free energy of solution A p e .The first of the two,f,,, is independent of the solute and reflects the behaviour of the pure solvent. The other parameter, fsl, describes the interaction of a solute molecule with the solvent, dependent therefore on both solute and solvent properties. The derivation of the standard free energy of solution from eqn ( 5 ) is simple and straightforward. (a) Divide both sides of eqn ( 5 ) by Avogadro’s number, converting the number of molecules, ng and n,, into the number of moles, Ng and N,: 5 = 3 exp [Uil -f,,)/kT]. v g K (b) Substitute p/RT, where p is the partial pressure of the solute in the gas phase, for the concentration of the solute in the gas space Ng/Vg: pVg = NgRT. (7) ( c ) Insert mole fractions X , to replace N,, since in dilute solutions X , * N,.The result = x, exp [Ull -fs,)/kTl RT Vl leads directly to the standard free energy of solution ApO = RT In (RT/ V,) + NUl, -fsl) (9) since by definition N = R/k, and the standard free energy is given by A p e = RT In (p/X,). (10)582 TRANSFER OF NOBLE GASES We now turn our attention to the standard entropy A P and the standard enthalpy A H 0 of solution. These two important thermodynamic functions correlate with the chemical potential through the definition A/@ = AH@- T A P . (1 1) We analyse these functions here to understand how these are related to the microscopic details of the membrane. A review of the basic assumptions of the model reveals that we consider that the holes are detached from each other and that the solute adsorption is independent of hole formation.These features imply that each hole contributes separately to the entropy and enthalpy of the system and that this contribution has two independent parts. The two parts describe the independent contributions of hole formation and adsorption. It follows from these features of the model that the macroscopic thermo- dynamic functions are determined by four independent microscopic parameters : E,, and oll characterize hole formation and E , ~ and oS1 describe solute adsorption in holes. The parameters are characteristic entropies and energies which obey, on the microscopic scale, relations similar to the general eqn (1) : f l l = E l l - To11 (12) f S l = Es1- T%l (13) where the subscripts 11 and sl refer to hole formation and to adsorption, respectively.A combination of eqn (lo), (12) and (13) gives a relation which describes the standard free energy as a function of the microscopic entropies and energies. When the general thermodynamical relations are applied to this relation we obtain explicit relations between the microscopic and the macroscopic entropies and enthalpies : A H 0 = N(E,, - eS1) - RT (16) A S 0 = N(oll - osl) - R[ 1 + In (RT/ V,)]. The experimental finding that the dependence of entropies and enthalpies of solution on temperature are small strongly suggests that the microscopic entropies and enthalpies are also independent of temperature to a good approximation. A comment deserves to be made concerning the mechanism of hole formation in our treatment.Holes appear in our treatment as independent free entities which are formed or destroyed or interact with each other and perform Brownian motions because of the irregular thermal movement of the solvent molecules. As such they are not supposed to contribute to the free energy and to the other thermodynamic functions of the solution. These functions will then be determined by the process of solute adsorption in the holes. The reason that the work involved in hole formation contributes to the free energy of solution is that the holes disappear when adsorption of solute takes place. This disturbs the equilibrium which exists between the number of potential sites for hole formation Nh and the actual number of holes nh. The result is that a new hole must be added to the solvent to restore the disturbed equilibrium.Y.KATZ 583 This formation of a hole demands the expenditure of work against the cohesive forces of the solvent, leading to the equations given above. There is nothing new in using the convenience of considering the process of solution as consisting of two steps: (1) the formation of a hole in the liquid and (2) the accommodation of a gas molecule in this hole. The novelty is in the interpretation given to these steps and in the way of deriving them. The statistical treatment of the liquid in terms of free holes carries the concepts introduced by Uhlig14 and Eley15 into free-volume theories, allowing the simultaneous use of both: e.g. to calculate step (1) from Furth's theoryls and to evaluate step (2) using free volume considerations and obtain the properties of the pure solvent from the solubilities of the inert gases.The existence of such possibilities seems attractive, especially when analysing membranes, because it is there where holes in the form of kinks are assumed to play a dominant There should be no surprise that the equations given here appear also in other papers. The concept of splitting the process of solution into two parts has a dominant effect on the appearance of theories and ours is no exception. One must remember, however, that these steps relate to a different mechanism of solution from those commonly employed and must be considered accordingly. The test of the evaluation must be done by analysis of the thermodynamic functions of solution of the simple gases in the non-polar solvents and by independent calculations of the parameters given by the theory. Of special importance is the calculation of hole volumes, since holes play a basic role in the development of the theory and are considered as independent entities whose magnitudes depend only on the nature of the solvent.Indeed an evaluation of hole size along these lines is given in table 5 (vide infra). The relations developed thus far, although useful, are not sufficient for the description of membranes. To obtain membrane characteristics from the measured thermodynamic functions we need two additional equations. Inspection of eqn (1 6 ) and (17) shows that only two relations connect the four independent microscopic parameters which characterize the behaviour of solutions.To make useful applications of the parameters we also need an explanation which correlates them with the detailed molecularity of the solution, so as to make it possible to understand the detailed behaviour of the solution from the thermodynamical measurements. We devote the rest of this paper to the development of these two aspects. We consider in turn the enthalpies and entropies and the relation between the two, describing some of the molecular mechanisms involved in the determination of the parameters. We also develop the two extra equations needed for a complete definition of the microscopic parameters. One of the equations shows that the measured thermodynamic enthalpy of solution can be expressed in terms of the enthalpies of the pure components, and the other shows that the entropies,and the enthalpies of solution are related to each other, thus correlating the four microscopic parameters.r01e.17-19 ENTHALPIES There exists a well known relation which expresses the interactions between the solute and the solvent molecules in terms of the solute-solute and the solvent-solvent interactions. This relation XSl = O C l l X S S ~ ~ where xsl, xll and xss are energies of interaction between solute and solvent, solvent and solvent and solute and solute molecules, respectively, is well founded theoretically20 and is widely used in solution theories.21 We use it to obtain an expression which reduces the number of unknowns, because it correlates the solvent-solute interactions584 TRANSFER OF NOBLE GASES with the solvent-solvent interactions. The relation makes it possible to obtain the energy of a solvent from the measured enthalpies of the solution and of the pure solutes.This outcome is of special importance in membrane research, because membranes are heterogeneous and the different regions cannot be separated and analysed independently. The problem is solved, however, if solutes are adsorbed selectively onto a given region using the results to evaluate further the energy of the region. Consider first the energies E,, and E~~ and their relationship with the pair interaction energies xsl, xll and xSs. Inspection shows that these microscopic parameters, which specify the cohesive and adsorbing adhesive powers of the medium, respectively, reflect in their magnitude the number and strength of the pair interactions between the close neighbours which are involved in the process of solubilization. The picture leads immediately to the equations Es1 = 2zxs1 E l l = ZXll where z is the number of contacts between solvent pairs that is is made and 22 is the number of solutesolvent interactions molecule is adsorbed in a hole. abolished when a hole gained when a solute It is easy to show that similar considerations apply to the cohesive energy of the Ess = Z X S S .(21) pure solute E,, when the liquid state of the solute is considered: Combining eqn (18)-(21) and using simple algebra we obtain the expression ESl = 2(&11&ssP. (22) (23) It is through using this result with eqn (16) to give AH0 = N[E,, - 2 ( ~ , , E,,):] - RT that we discover the very important relation which links the molar energies of vaporization Ul and Us of the pure solvent and solute, respectively, to the standard molar enthalpy of solution AH*.The quantities in eqn (24), with the exception of Us and A H e , are independent of the solute. To derive eqn (24) we consider the process of evaporation in the pure liquid and compare it with the solubility of this pure entity in itself. Note that these two processes are identically opposite to each other, because evaporation is the desorption of a molecule from a medium identical to itself when the adhesive and cohesive forces are of the same nature. The energy content of a hole is because the creation of Nh holes is proportional to evaporation of a mole of solvent (note that for the solute Nh = N).We would like to stress again that eqn (24) is valid only for a solution in which both solute and solvent are hydrophobic. This is so because only for the interaction between hydrophobic molecules can we apply the assumption in eqn (18) of the geometric mean, which is characteristic of London interactions.20 Inspection of eqn (24) shows that if our theory is correct then it is possible to obtainY. KATZ 585 the cohesive energy of the membrane from solubility data and to predict the solubilities once the cohesive energy is known. This is an important and useful conclusion, because there are experimental difficulties in obtaining results which do not depend in one way or another on the solubilization of probes in the membrane.To justify further use of the model and verify its basic assumptions we measure here the extent of realization of eqn (24) using solubility data for noble gases in bulk hydrocarbons and in a phospholipid membrane. We also include solubility data for other solutes which are not as simple as the noble gases. The solvents we select are the four hydrophobic solvents n-hexane, n-dodecane, cyclohexane and benzene. The phospholipid membrane selected is a dimyristoyl lecithin lamella. This membrane can be divided into two definite regions, one hydrophilic and the other hydrophobic. The hydrophobic region is the space in the membrane into which the noble gases dissolve preferentially.ll The bulk solvents resemble each other and the hydrophobic region of the membrane in their hydrophobicity but differ from each other in the size, shape and value of the cohesive energy which holds the liquid molecules to each other.The hydrophobic region of the membrane, being a liquid crystal, differs appreciably from the bulk solvents. Solubility data in several polar solvents are also given here to establish an upper limit to deviations which may be caused by using the assumption of the geometric mean [eqn (22)] in systems which are affected by other factors beside the London forces mentioned above. The data on which we base our calculations have been collected from the literature.l1>l2 We present the results of our test on bulk solvents in fig. 1 and table 1. An examination of the validity of the theory for the description of the hydrophobic region of the phospholipid membrane is presented in fig.2 and in tables 2 and 3. The testing made for the membrane is less extensive than that for the bulk solvents because fewer experimental data are available for membranes. First we test the prediction, made in eqn (24), that there is a linear relationship between enthalpies of solution of simple solutes and the square root of their energies of vaporization. Our examination for the four hydrophobic solvents n-hexane, n-dodecane, cyclohexane and benzene is presented in fig. 1. The findings show good agreement between observation and prediction, provided that the data for helium and for hydrogen are excluded from the calculations. Comparing the calculated results with those found experimentally, we see a standard error of the mean that ranges from 1.2% in benzene to 3.2% in cyclohexane.These errors correspond to standard deviations of estimation of -t 93 cal and & 21 2 cal, respectively. One can see that the predicted linear relationship holds within experimental error, since the deviations are smaller than the experimental errors commonly found in enthalpy measurements. l2 Similar linear correlations are found when the solvents are polar solvents. These results, which are theoretically less interesting, are not presented in fig. 1 in order to prevent confusion which may result from too many data points. Inspection shows that agreement between the estimated results and the experimental results in hydrophobic solvents manifests itself in correlation coefficients of 0.99 (only for n-hexane do we find a lower correlation coefficient at 0.81).The model predicts that the slopes in fig. 1 are twice as large as the square roots of the corresponding intercepts. This prediction is seen in eqn (24) and it follows directly from the assumption eqn (18) of the geometric mean. Our examination, depicted in table 1 , validates the theory and justifies its use to describe solubilities in bulk solvents. Table 1 gives the values of the slopes, the square roots of the corresponding intercepts and the ratio between them. The theory predicts that the ratio between the two parameters is one. We find that this theoretical expectation is fulfilled for solutions in hydrophobic solvents. Larger deviations from this expectation are found for polar solvents.586 TRANSFER OF NOBLE GASES 0 He 12 H2 Ne 24 N2 36 02 CH4Kr 48 Xe C0zC#1660 (o'>:/(J rnol-')f A r Fig.1. Relations between the energies of vaporization, Us, of solutes and their standard enthalpies of solution for the solvent systems n-hexane (a), n-dodecane (O), cyclohexane (X) and benzene (0). Although data for He and H, are also given, they are not used in the calculation of the linear relations. The correlations of the lines with the experimental results are 0.81, 0.98, 0.99 and 0.99, respectively. Table 1. Values of solvent energy parameters derived from fig. 1 for different solventsa n-hexane n-dodecane cyclohexane benzene perfluorocyclohexane nitro benzene ethanol acetone 122.4 122.4 126.5 141.6 81.5 107.7 96.7 83.4 16 593 16 895 18644 23 056 7 535 18 377 11 553 14232 128.8 130.0 136.6 151.8 86.8 135.6 107.5 119.3 1.05 1.06 1.08 1.07 1.07 1.15 1.11 1.43 a The cohesive energy NE,, is in units of J mol-I.Experimental results are from ref. (12). The solvents perfluorocyclohexane, nitrobenzene, ethanol and acetone are not shown in fig. ' 1 ; the values for these solvents ar,e derived in the same way as the data for the hydrophobic solvents- l)fntercept/(NEIl)~lope *Y. KATZ 9000 - I - 8000 +7 < Q 7000 587 - - - He 0 6000 I 1 1 1 I I 10 20 30 40 50 U / J mol-' Fig. 2. Test for the applicability of the theory to solubilities of noble gases in the dimyristoyl lecithin (DMPC) membrane. A linear relation between the energies of vaporization, Us, of the solutes and their standard free energies of solution in DMPC is expected theoretically. Table 2.Energy parameters of the DMPC membrane. A test of the ability of the theory to describe the behaviour of membranesa He Ne Ar Kr Xe CHF) CHP) 33526 36125 32023 30976 28549 29553 29804 @ 7.08 40.7 76.1 89.6 l06.2 85.3 85.3 Ape A 41023 a -103.8 Nc,,(from A) 30453 B 123.7 b 0.00217 N&,,(from B) 32714 a A p e are standard free energies of solution in DMPC from ref. (1 1). Values in the last two columns are the free energies in DPPC + DPPA and DPPC + DPPA +cholesterol mixtures, respectively.22 Vaporization energies of the solutes are from ref. (2 1) and the Barclay-Butler constants a and b are from ref. (8). Units are SI. A and B are the intercept and the slope of fig. 2, respectively. The theory demands that A = [(Nell-RT)(l -bT)-aT] and B = [2(N~,,):(l -bT)].This follows from the simultaneous use of eqn (24) and (28) and the general thermodynamic equation Ap* = A H e - TAP. The molar cohesive parameters Neil from A and B agree only if the combining rule [eqn (22)] and the Barclay-Butler relation are valid in the membrane. To make sure that the special structure of the bilayer has no effect on the applicability of the theory to membranes, we have examined the validity of the combining rule eqn (22) for membranes. We have also examined the consistency between two independent calculations: one based only on energy relations and another using the Barclay-Butler correlation between entropies and enthalpies of solution.* We first test if there is a linear relationship between the free energies of solution of the noble gases in the bilayer and the square roots of the solute energies of vaporization. The results of these tests appear in fig.2 and table 2. The existence588 TRANSFER OF NOBLE GASES Table 3. Energy parameters of the DMPC membrane from enthalpies of solution of argon and krypton in the membranea solute AHe argon 795 5810 76.1 1 29430 171.55 krypton - 2470 8 040 89.60 32 280 179.50 a Enthalpies of solution AH* and vaporization energies of the pure solutes are from ref. (1 1) and (21), respectively. Units are SI. Molar cohesive energies NE,, are calculated by solving eqn (24) as a quadratic with ( N E ~ , ) ~ as the unknown. E,, = vh U,/V, is obtained from eqn (25). of such a linear relation demands the validity of eqn (24) as well as the validity of eqn (28).This means the applicability of the combining rule eqn (22) as well as the existence of a linear relation between the entropies and enthalpies of solution. Our examination is done in this way because the data on enthalpies of solution of the noble gases in phospholipids which appear in the literaturell are not sufficient for the construction of a graph, as in fig. 1. Fig. 2 shows that the theory also holds for the hydrophobic region of the membrane. It shows the linear relation between the free energies of solution of the noble gases in the bilayer and the square roots of the energies of vaporization of these gases, as from theory. Further investigation, using the Barclay-Butler coefficients for this system,8 demonstrates that the numerical values of the slope and the intercept of fig.2 correlate in accordance with the combining rule. The results which show this agreement, between the theoretical expectations and the experimental findings, are given in table 2. More support for the applicability of our theory to membranes comes from inspection of the enthalpies of solution of the noble gases argon and krypton in the bilayer. Using experimental data given in the literature for the enthalpies of solutionll and energies of vaporizationz1 we solve eqn (24) as a quadratic equation with N E ~ ~ as the unknown. Inspection of the results in table 3 shows their agreement with the results in table 2. Only if the combining rule of the geometric mean, given by eqn (22), holds will the results for argon and krypton agree with each other and with the results of table 2.ENTROPIES There are many ways to arrange the molecules of a liquid so that from the outside it looks the same. The logarithm of that number is the entropy of the liquid.23 It follows that the standard entropy of solution measures the reduction in the possible number of arrangements occurring upon solution. We express the entropy of solution by two parameters, a,, and a,,, which appear in eqn (17). These parameters represent the microscopic entropies of hole formation and of solute adsorption to the holes, respectively. We assume that the holes are rigid spherical entities, practically impermeable to either solute or solvent. The model portrayed in fig. 3 is naive and unrealistic and its success must be determined by its ability to produce results which are consistent with the experimental findings.The model asserts that the confinement of the solute molecule to a hole reduces the volume available to the movement of its centre. This happens because the centre can come no closer than one radius from the wall. It is easy to show that the volume uhs free to solute movement in a sphericalY. KATZ 589 volume in which the centre of a solvent molecule cannot be found (1) solvent (I hich the centre o f a molecule I cannot enter solvent molecule \ molecule / - \ solvent ( 2 ) molecule hole of radius fi ( 3 ) ‘cavity R in the solvent medium space from which the centre of the solute is excluded solute molecule the solute can of radius r be found Fig. 3.Schematic diagram of the model showing the effect of excluded volumes on the entropy parameters. We define an excluded volume as a volume in which centres of molecules cannot be found. We describe an excluded volume of a sphere surrounding the point of contact between two molecules having radius a. (1) The radius of the excluded sphere is a. (2) When a hole of radius r is made the radius of the excluded sphere increases to r+a. (3) The volume available to the movement of the solute molecule of radius r in a hole of radius R is VhS = @(R - r)3 hole of volume uh depends on the volume of the solute us. The dependence for a spherical solute molecule of volume us islo The reduction in volume is a constraint which causes the entropy of the system to change.24 The amount of change is To test the applicability of the model to bulk solvents we calculate hole volumes Vh from measured enthalpies of solution using eqn (25) and insert the values so obtained into eqn (27) t o evaluate the entropic parameters oSl. We then compare these calculated parameters with the experimentally measured entropies of so1ution,12 examining the agreement between the theoretical expectations and the experimental findings.The calculated parameters, the experimental values and the outcome of the comparison are given in table 4. The comparison is also shown in fig. 4. The solvents used in the examination are the four hydrophobic solvents n-hexane, n-dodecane,00'1 28'09- 19'Z€- EI'SS- €9'82- €€'PS- OZ'SZ- 8L'€€- 8€'81- LI'SP- €0'91- 8'S9 68 09€1€ 090€Z auazuaq €0.1 6L.19- 8P'ZC- 0€'9S- IS'8Z- 9Z'LS- ZI'SZ- PZ'IS- €€'81- 86'oP- 66'SI- Z'99 601 O€OO€ oP981 auexaVPd~ 86'0 LS'09- OO'€€- Z6'PS- €6'82- 8L'IS- SP'SZ- PL'€P- PS'81- 6L'SP- 91'91- L'P9 9'822 OOL8S S6891 auEmPop-u 9S'I IE'99- 90'0€- 81'LS- 8S'9Z- 98'8s- 8P'CZ- 08'9P- €€'LI- 69'IP- SI'SI - P'PL ZCI 09062 E6S91 auexay-uY. KATZ 59 1 cyclohexane and benzene, and as solutes we use the noble gases.The molecular volumes v, are values taken from the l i t e r a t ~ r e . ~ ~ The theory predicts that the entropy change caused by changing solute in a given solvent must be equal to the change in the value of the osl parameter, the reason being that the only entropic factor which is solute dependent is the parameter osl. The comparison which we present in fig.4 shows that the theoretical expectations agree well with the experimental findings. An exception is found with n-hexane, where the slope found is much larger than the slope expected theoretically. A simple application of this test to phospholipid membranes cannot be made, since it demands the use of data on the energy of evaporation of the solvent in calculating eqn (25). Because evaporation of the hydrocarbon chains of the hydrophobic region of the membrane is not possible, it follows that their energy of vaporization is not available experimentally. Estimations of the energy of vaporization of the hydrophobic region of the membrane can, however, be made using theoretical calculations of the energy of interaction between hydrocarbon chains.26 We refrain here from this kind of evaluation, because of both the complexity of the calculations and the number of assumptions involved in them.THE BARCLAY-BUTLER RULE A LINEAR RELATIONSHIP BETWEEN ENTROPIES AND ENTHALPIES Barclay and Butler found that the enthalpies of solution in a given solvent correlate with the corresponding entropies. Their empirical rule describes the correlation as a linear relation between standard entropies and enthalpies of A P = a+bAHe (28) where a and b are characteristic constants. Barclay and Butler have shown that there is a value for a and another for b which are common to many solutions and many pure liquids. Here we explore this useful relation and explain its meaning in terms of our model. Combination of eqn (28) with eqn (16) and (17) reveals that the solute-dependent parameters E , ~ and osl correlate with each other.Mathematically we formulate this correlation by the differential a is a constant which includes all the solute-independent parameters a = No,, - R[ 1 +In (RT/ V1)] - b(NE,, - RT) a = A S 0 (solute independent) - bAHe (solute independent). (30) (31) which means that Comparison between eqn (29) and (30) shows that it is much easier to understand and interpret the b constant. The numerical values found empirically for the two Barclay-Butler constants of the bulk solvents are a x 26.4 cal mol-1 K-l and b x 0.001 34 K-1.8 These coefficients constitute the ‘normal’ curve that correlates the enthalpies and entropies of the bulk solutions.2s~ 29 The numerical values of the two Barclay-Butler coefficients, found from solubility data for the noble gases in the dimyristoyl lecithin membrane, are a x 24.8 cal mol-1 K-l and b x 0.002 18 K-1.8 Inspection shows that whereas the a constant of the Barclay-Butler coefficients in the phospholipid does not differ significantly from the ‘normal’ values, the b coefficient characterizing the phospholipid is appreciably higher than ‘ normal ’.TRANSFER OF NOBLE GASES 592 -9 - 10 r( -11 - ; -12 r( & -13 3 -14 -1 5 -1 6 -1 7 b I I I I I I / / I I I I I I l l I I 1 1 L -8 -7 -6 -5 -4 -3 -8 -7 -6 -5 -4 -3 -8 -7 -6 -5 -4 -3 -8 -7 -6 -5 -4 - 3 NuSl/J K-' mol-' Fig.4. Comparison between calculated No,, parameters and experimentally found entropies of solution A 9 for the four solvents (a) n-hexane, (b) n-dodecane, (c) cyclohexane and (d) benzene.The experimental values are from ref. (1 2). Inspection of derivations of the enthalpy and the entropy shows how the correlation between these apparently independent functions occur. First notice that osl is a function of the hole volume uh [eqn (27)]. The hole volume is determined by the energy parameter E,, and by the entropy parameter o,, [eqn (2), (3), (12) and (13)]. The parameters el, and E,, relate to each other through the assumption of the geometric mean [the combining rule is given in eqn (22)]. This chain of relations establishes a correlation between the entropy parameter osl and the energy parameter E,,. This correlation reflects itself in the Barclay-Butler relation. Mathematically the argument goes as follows: osl = 41(Vh), uh = 42(~11, o,,) and E,, = 4J.sl1) E,, = 4; l(cSl), where 41, t$2 and q43 are functions and 4; is the inverse function of & Combination of these (32) into one relation gives which shows the functional relationship between the entropy and the enthalpy.This result is rather surprising because entropies and enthalpies have different origins in this model. We treat the molecules as hard spheres when calculating entropies but apply the combination rule of the geometric mean when calculating energies. It means that we derive entropies from the repulsive part of the Lennard-Jones intermolecular potential whereas to calculate the energy we apply the attractive part of the Lennard-Jones potential. Since the parts of this potential are independent, we expect the entropies to be totally independent of the energies. The difficulty vanishes when we note that the cohesive energy of the solvent affects the dimensions of the hole in which the solute moves thus affecting the entropy of solution.Employment of the osl parameters, obtained through the use of the Barclay-Butler relation, to the evaluation of the hole volumes of the solvents shown in table 5 gives values which for the bulk solvents agree with the hole volumes calculated from energy consideration using eqn (27). Inspection shows that the estimated hole volumes from entropies of solution of argon, krypton and xenon agree to within 20% or better with the values calculated from energy considerations, and that the results from entropy measurements on helium and neon deviate appreciably from the rest.This means that at least for the bulk solvents the conclusions obtained from entropic considerations are consistent with the results deduced from energy considerations. Since entropy and enthalpy arise in our discussion from different parts of the intermolecular potential U s 1 = 41 { 4 2 [ 4 , l(&sl)l>Y. KATZ 593 Table 5. Calculation of hole volumes from solubility data of noble gases in hydrophobic solvents Nu, helium neon argon krypton xenon from 13.8 83.3 156.1 183.4 217.3 solvent eqn (25) 6.9 9.2 16.7 20.7 25.3 n- hexane 74.4 21.60 12 274 165 12 230 164 11 180 170 7 574 115 93 1 n-dodecane 64.7 21.80 cyclohexane 66.2 23.10 benzene 65.8 2.55 DMPC 5.45 13.20 138.14 I .86 13.20 138.46 2.13 13.45 128.20 1.94 15.45 94.80 1.44 3 1.20 25.48 25.25 69.34 0.93 25.50 69.23 1.07 25.30 65.50 0.99 28.90 52.20 0.79 58.90 22.60 28.60 65.55 0.88 29.00 65.35 1.01 29.60 62.26 0.94 33.90 50.80 0.77 69.15 25.24 34.00 62.03 0.83 34.30 62.11 0.96 35.10 59.30 0.90 32.70 49.70 0.76 88.90 28.47 The numbers beneath the solute’s name are the square root of its vaporization energy [ref.(21)] and its molar volume [ref. (25)]. For each solvent-solute interaction we calculate the entropy parameter Nosl, the molar hole volume [from eqn (27)] and the ratio between this hole volume and hole volume obtained from energy measurements [eqn (25)]. Nos, is calculated from the data in fig. 1 and 2 using eqn (29) for the calculation. Units used are J and cm3 mol-l. they are independent of each other.The fact that similar numerical values result from the independent evaluations of the hole volumes indicates that assigning the entropy to the hard part of the intermolecular potential and the energy to its soft part agrees with the experimental findings. Further support for the validity of eqn (27) for a description of entropy in hydrophobic solvents comes from the comparison of the hole volumes obtained using different solutes in the evaluation. The comparison shows that the volumes calculated for the holes in the bulk solvents from solubility data for argon, krypton and xenon give the same result to within 5 % . Even better agreement is found for the bilayer. We find that the deviation of hole size calculated from solubility data for neon, argon, krypton and xenon does not exceed 4%.The oS1 parameters, characterizing noble-gas solubility in the membrane, are larger than the corresponding parameters found in bulk solvents by 4-12 cal mol-l K-l and they span a larger range, which increases the sensitivity of the test in this medium. Thus our findings show not only that eqn (27) is able to describe the entropy of solution but also, more importantly, that the same theoretical formulation can be applied to the description of either bulk solvents or hydrophobic regions in membranes. A possible explanation for the deviation of the results obtained from helium solubility measurements from the theoretical expectation is that the small dimensions of helium make it possible for the solute to use vacancies which are too small to be occupied by the larger solutes.Consequently, any two holes connected by a series of small vacancies will behave as one hole when the solubility of helium is considered but with larger solutes. This leads to holes which are much larger for helium than for the other solvents. If this explanation is correct then we expect smaller deviations to594 TRANSFER OF NOBLE GASES occur in the denser solvents and for the larger solutes. The first expectation is borne out by comparing the hydrophobic region with the bulk solvents. The density of the hydrophobic region of the membrane is much larger than that of the bulk giving smaller deviations in the calculated hole volumes. We also find that the deviation diminishes as the solute increases in size.DISCUSSION Our model, which describes solubilization as consisting of two independent processes, hole creation and solute adsorption in holes, conforms well' with the experimental findings on the solubility of noble gases in hydrophobic solvents and in phospholipid membranes. These features make the theory simple and flexible and offer interesting possibilities. Most of this discussion will be concerned with applications to analysis of membranes. First, note that a theory which describes bulk solvents as well as membranes constitutes a basis for comparison of the various systems. Comparison between solubilities in bulk model solvents and the behaviour of the given solutes in membranes plays an important role in the analysis of membrane phenomena.'? However, this method is based on empirical correlation^.^ The finding of a theory which describes both bulk solvents and membranes provides a theoretical basis to the correlations made.We also note that our model paves the way to the derivation of a physical interpretation of the two most important aspects of membrane transport, namely membrane permeability and membrane selectivity. Inspection shows that the model describes solubilization in terms of hole creation, which depends only on the characteristics of the solvent medium, and of solute adsorption in holes, which depends on the nature of the solvent-solute interaction. Selectivity, which describes the differences in behaviour towards different solutes, is given by the second factor. Permeability, which is strongly dependent on membrane solubility,' is defined by a combination of hole-creation and adsorption parameters.The theory offers the possibility of deducing the characteristics of the pure solvent from the properties of the solution. The ability to obtain solvent features from solubility measurements follows from the existence of relations such as eqn (24), which correlate between the properties of the pure components of the solution and the characteristics of the solution itself. This becomes important when analysing hetero- geneous systems such as the phospholipid membrane. Solutes of known properties, such as the noble gases, are adsorbed preferentially to the hydrophobic region measuring the thermodynamic functions of transfer. The thermodynamic parameters which characterize the pure solvent are then evaluated using relations such as eqn (24).Inspection shows that there are consistent deviations between experiment and theory for the small solute molecules helium and hydrogen. The deviations are seen when looking at fig. 1 and 2. We attribute this deviation to our neglect of quantum-mechanical effects on the vaporization energies of these small solute molecules.21 Further inspection8 shows that the theory applies only when the solutes under consideration are noble gases. This is especially true when considering membranes. It can be shown, however, that the construction of a theoretical framework which describes the behaviour of the simple solute is an essential step in the formulation of more general theories. We note similarities between our theory and other existing theories;l6T 31-33 this provides flexibility and the possibility of extending our treatment by combining it with an existing one.The success of our calculations of the behaviour of inert gases in non-polar liquids permits a few comments to be made concerning the solution process and the methodsY. KATZ 595 for evaluating it. Thermodynamic functions of solution of these entities have been calculated many times by considering that the process of introducing the solute consists of two consecutive steps: (1) formation of a hole in the solvent and (2) introduction of the solute into the hoie.l59 3 4 v 35 Our treatment shows that the order of these steps can be reversed : first adsorption to an existing hole and then a formation of a new hole in the solvent medium.It also shows that, contrary to ref. ( 3 9 , free holes in the liquid are determining factors in the thermodynamics of gas solubility. This means that from the point of view of thermodynamics a mechanism based on the adsorption of the solute in an existing hole is as valid as the mechanism in which the dissolving gas has to ‘dig its own hole’ in the solvent. That our approach to the solubility of gases in liquids, based on a Furth statistical treatment of the thermodynamics of liquids by the theory of holes,ls leads to a consideration made by other methods is interesting. This brings together the scaled-particle theory, as used by P i e r ~ t t i , ~ ~ and such treatments which assume the existence of holes in the liquid and treat it by a statistical theory analogous to that used for the treatment of gases,36 thus offering the possibility of relating one to the other.Indeed, our calculations give results which agree quite well with experiment. This was only achieved with difficulty using one of the existing free-volume On the other hand, one of the most significant features of the hydrophobic region, namely its high adhesive entropy parameter osl, would not have been noticed if another mechanism for hole formation [e.g. the mechanisms proposed in ref. (15), (34) and (35)] was used. Finally we define the standard states of the thermodynamic functions employed. The standard functions A S e , AH0 and A p e describe changes which occur when one mole of solute is transferred to a solution having the properties of an infinitely dilute solution.Standard free energy and standard entropy refer to a hypothetical solution obeying Henry’s rule and having the mole fraction of the solute approaching unity. I thank Professors S . Alexander, A. Ben-Naim and B. 2. Ginsburg for many helpful discussions. E. M. Wright and J. M. Diamond, Annu. Reu. Physiol., 1969, 31, 581. S. H. Roth and P. Seeman, Biochim. Biophys. Acta, 1969,255, 207. R. Collander, Acta Chem. Scand., 1950, 4, 1085. W. D. Stein, in Membrane Transport (Elsevier, Amsterdam, 1981), vol. 2, pp. 1-28. J. M. Diamond and Y. Katz J. Membrane Biol., 1974, 17, 121. A. Finkelstein, J. Gen. Physiol., 1976, 67, 45. J. M. Wolosin, H. Ginsburg, W. R. Lieb and W. D. Stein, J. Gen. Physiol., 1978, 75, 427. Y. Katz, M. E. Hoffman and R. Blumenthal, J. Theor. Biol., 1983, 105. A. Ben-Naim, Water and Aqueous Solutions (Plenum Press, New York, 1974). York, 1965). lo J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory ofGases and Liquids (Wiley, New l1 Y. Katz, Biochim. Biophys. Acta, 1981, 647, 119. l2 E. Wilhelm and R. Battino, Chem. Rev., 1973, 73, 1. l3 R. C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, 1950). l4 R. Furth, Proc. Cambridge Philos. SOC., 1941, 252. l5 D. D. Eley, Trans. Faraday SOC., 1939, 35, 1281. l6 H. H. Uhlig, J. Phys. Chem., 1971, 41, 1215. H. Trauble, J. Membrane Biol., 1971, 4, 193. J. F. Nagle, J. Chem. Phys., 1973,58,252. M. B. Jackson, Biochemistry, 1976, 15, 2555. 2o F. London, Trans. Faraday SOC., 1937,33, 8. 21 J. H. Hildebrand and R. L. Scott, The Solubility ofNonelectrolytes (Dover, New York, 1964). 22 K. W. Miller, L. Hammond and E. G. Porter, Chem. Phys. Lipids, 1977, 20, 229.596 TRANSFER OF NOBLE GASES 23 R. P. Feynman, Lectures on Physics (Addison-Wesley, Reading Mass., 1965), vol. I, chap. 46. 24 I. M. Klotz and R. M. Rosenberg, Chemical Thermodynamics (Benjamin Cummins, New York, 1972). 25 A. Bondi, J. Phys. Chem., 1964,68,441. 26 L. Salem, J. Chem. Phys., 1962, 37, 2100. 27 I. M. Barclay and J. A. V. Butler, Trans. Faraday Soc., 1938, 34, 1445. H. C. Longuet-Higgins, Proc. R. SOC. London, Ser. A, 1951,205, 247. 29 H. S. Frank, J . Chem. Phys., 1945, 13, 493. 30 H. Lecuyer and D. G. Dervichian, J. Mol. Biol., 1969, 45, 39. 31 I. Langmuir, Colloid Symp. Monogr., 1925, 48. 34 S. Glasstone, K. J. Laidler and H. Eyring Theory of Rate Processes (McGraw-Hill, New York, 1941). 33 H. Reiss, Adv. Chem. Phys., 1966, 9, 1. 34 P. Meares, J. Am. Chem. Soc., 1954, 76, 3415. 35 R. A. Pierotti, J. Phys. Chem., 1963, 67, 1840. 36 H. Eyring, D. Henderson, B. Jones Stover and E. M. Eyring, Statistical Mechanics and Dynamics (Wiley, New York, 1964). (PAPER 4/541)