J. Phys. Chem. 1996, 100, 2447-2455
2447
Enthalpy-Entropy Compensation in Hydrophobic Interaction Chromatography Anant Vailaya and Csaba Horva´ th* Department of Chemical Engineering, Yale UniVersity, New HaVen, Connecticut 06520-8286 ReceiVed: August 8, 1995; In Final Form: October 25, 1995X
Exothermodynamic relationships between thermodynamic quantities and molecular structure are employed to facilitate a molecular interpretation of enthalpy-entropy compensation (EEC). For hydrophobic interactions the compensation temperature TC is expressed in terms of the enthalpy and entropy change, both per unit nonpolar surface area of the molecules, and it is concluded that the utility of TC as a diagnostic tool for the mechanistic identity of processes rests on this simple dependence of TC on molecular parameters. Whereas classical EEC is observed only with processes involving no heat capacity change and TC is evaluated from the slopes of linear enthalpy Versus entropy plots of data measured at any temperature, this investigation shows that even when the heat capacity change is finite and constant or varies linearly with the temperature, EEC can occur with processes if they are subject to the same mechanism at a fixed temperature. In turn, the compensation temperature changes with the experimental temperature, reflecting mechanistic changes as expected with processes such as hydrophobic interaction chromatography that are governed by hydrophobic interactions and driven by entropy or enthalpy change at low or high temperatures. These compensating processes exhibit at least one isoenergetic temperature T*G, which marks the intersection point of curved van’t Hoff plots, where all species have the same free energy change in the same way as at TC in the case of linear van’t Hoff plots. In turn, the isoenthalpic T*H and isoentropic T* S temperatures mark the intersection points of the respective plots of enthalpy and entropy Versus temperature as described in the literature. The triad of isothermodynamic temperatures is characteristic for processes which can be represented by constant heat capacity change and evince compensation behavior.
Introduction Exothermodynamic relationships1 are empirical correlations of thermodynamic quantities that are widely used to examine the role of molecular structural parameters in chemical equilibria and rate processes. The most common are the so-called linear free energy relationships (LFERs) that are based on the assumption that the free energy of a process is the sum of free energy increments contributed by structural elements of the substances involved. One of the first and still most popular representatives of LFERs is the Hammett equation.2 Its applicability was later extended to include aliphatic reactivities by the Taft equation,3 which relates the ionization of aliphatic acid derivatives to the electronic effect of the substituents. A similar approach to correlate molecular structure with biological activity has made substantial advances with the formulation of quantitative structure-activity relationships.4,5 Another form of the LFER was introduced by Martin in chromatography,6 and it states that for two compounds that differ in a structural element the difference in the logarithmic partition coefficients is proportional to the free energy change for that structural element but not the rest of the molecule. This explains the remarkable separating power of chromatography for closely related biopolymers, e.g., large peptides or proteins which differ only in a single amino acid. Enthalpy-entropy compensation (EEC) is another exothermodynamic relationship manifested by a linear dependence of the enthalpy on the corresponding entropy upon changing an experimental variable of the process under investigation.1,7 Such invariance of the free energy in molecular associations or other * Author to whom all correspondence should be addressed: Prof. Csaba Horva´th, Department of Chemical Engineering, Yale University, P.O. Box 208286, New Haven, CT 06520-8286. Phone: (203) 432-4357. Fax: (203) 432-4360. Email:
[email protected]. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-2447$12.00/0
binding processes may be interpreted so that a greater enthalpy change implies stronger binding with a more restricted configuration in the bound state; hence, it results in a higher order with a concomitantly greater change in the entropy.8,9 When EEC occurs, plots of enthalpy Versus entropy are linear with slopes called the compensation temperature TC, which is regarded as a process characteristic. Chemical reactions or equilibrium processes having similar compensation temperatures are considered fundamentally related and called isokinetic or isoequilibrium processes, respectively.1,10,11 Traditionally, EEC has been extensively studied with processes exhibiting linear van’t Hoff plots.1 In a classical review7 on this subject, EEC was examined for a wide variety of processes involving small molecules in aqueous solutions and the values of TC were found to fall in a relatively narrow range between -13 and 42 °C. This was attributed to the mechanistic similarity of the processes due to the dominating role of water as the solvent, commonly referred to as the hydrophobic effect. EEC was found also for various interactions involving proteins in aqueous media, with TC falling in the same range. Hydrophobic interaction chromatography (HIC) is widely used for the purification and analysis of proteins.12,13 It is carried out with a weakly hydrophobic stationary phase and an aqueous salt solution as the eluent so that the integrity of the protein molecule is largely conserved in the chromatographic process.14-16 A comprehensive treatment of the salt effect on HIC retention is adapted from the solvophobic theory17-20 while others are based on Wyman’s linked functions.21-24 Retention and selectivity in HIC is governed by hydrophobic interactions, which are believed to play an important role in determining the architecture and dynamics of biological systems.25 The hydrophobic effect is mainly due to an unfavorable interaction of nonpolar substances or moieties of the molecules with water that is responsible for their low solubility.26 © 1996 American Chemical Society
2448 J. Phys. Chem., Vol. 100, No. 6, 1996 Processes, such as the dissolution in water of nonpolar liquid27,28 and gaseous29-31 and solid substances32-34 as well as protein folding,35-38 are driven by the hydrophobic effect, and according to thermodynamic analysis of calorimetric data they are accompanied by significant entropy and heat capacity changes at room temperature. Recently, the effect of temperature from 5 to 50 °C on the retention of dansyl amino acids in HIC was investigated on three stationary phases, and van’t Hoff plots were found to be nonlinear.39 The thermodynamic analysis of the chromatographic data yielded essentially the same results as those obtained with other processes based on the hydrophobic effect. EEC has been employed for the interpretation and organization of data in HPLC. Retention parameters were extensively studied in reversed phase chromatography on various nonpolar alkyl-silica stationary phases under a wide range of conditions as far as the eluites and the composition of the hydro-organic eluents are concerned, and the TC values were found to be indistinguishable.40-44 This led to the conclusion that, under the conditions employed, the intrinsic mechanism of the interaction of small eluite molecules with the bonded stationary phase was the same. Compensation behavior was also observed with chromatographic systems comprising polar stationary phases and nonpolar eluents, but the value of TC was significantly lower, indicating a different retention mechanism.40 Further, ion-pair chromatography45,46 and ion-exchange chromatography of alkali metal and alkaline earth metal ions with surface sulfonated resins47,48 were also found to exhibit EEC. Recent examination of gas chromatographic retention data revealed that the retention of eluites, which belong to a homologous series, exhibits compensation behavior.49 Exothermodynamic analysis of calorimetric data obtained at room temperature with processes driven by the hydrophobic effect has shown that plots of enthalpy (enthalpy per gram in the case of protein folding) or entropy Versus heat capacity are linear.34,38 From the slopes the isoenthalpic and isoentropic temperatures that are characteristic for the process have been evaluated. It has been shown that these temperatures arise from the linear dependence of the thermodynamic quantities on a single molecular property, such as the change in nonpolar wateraccessible surface area upon transfer.33,50 It is intriguing to investigate the existence of such exothermodynamic relationships in HIC and to examine EEC with the HIC retention data yielding nonlinear van’t Hoff plots. Comparison of the results with those of other processes based on the hydrophobic effect is expected to facilitate the understanding of the underlying physicochemical basis of HIC. Furthermore, HIC offers an alternative approach to calorimetry in the study of the hydrophobic effect.39 This unusual application of HPLC falls in the domain of molecular chromatography aimed at the extraction of physicochemical and molecular information from the retention data. In turn the versatility and high precision of this technique that requires minute sample quantities have prompted our use of HIC as a model process in the study of the physicochemical basis of EEC. In this paper EEC will be investigated theoretically for processes involving structurally related substances and characterized by heat capacity change that is either zero, or finite and constant, or temperature dependent. The linear relationships between thermodynamic quantities and molecular structure are used to elucidate the physicochemical basis of EEC and to express the compensation temperature in terms of molecular parameters. The results are expected to provide a unified exothermodynamic framework for the diagnosis, organization, and interpretation of experimental data.
Vailaya and Horva´th Generalized EEC and Other Exothermodynamic Relationships General Form of EEC. As mentioned earlier, EEC manifests in a linear dependence of the enthalpy change ∆H° on the corresponding entropy change ∆S° for the process under consideration that can be expressed in a generalized form
∆H°x ) TC∆S°x + ∆G°TC
(1)
where ∆G°TC is the free energy change at TC and the subscript x denotes the particular experimental variable employed to obtain a set of ∆H°x - ∆S°x pairs for a given process. The variable can be, for instance, the number of reoccurring structural elements in a group of molecules such as a homologous series, a solvent property such as the surface tension, the dielectric constant or the concentration of a component in mixed solvents, or the temperature. In addition, variation of a property of the interacting species such as the stationary phase in chromatography may also be employed to yield enthalpyentropy values in the study of EEC. In each case, the compensation behavior offers a different view of the same process under investigation in the range of the chosen variable. Traditionally, a set of structurally related substances,1,51 different solvents,7,52 or both is employed to study EEC, such as in the vaporization of various nonpolar substances from their pure states as well as from other nonpolar liquids.53,54 Compensation behavior was observed with either variable, when the other was kept constant, and the TC values obtained in the two cases were quite similar. From this finding it follows that the process of vaporization is mechanistically identical for the variety of nonpolar substances and nonpolar solvents investigated. Progress has also been made to establish a theoretical foundation for EEC with processes based on the hydrophobic effect by taking the temperature as the variable and assuming constant heat capacity change.55 It is desired to develop a framework of EEC to examine the experimental data obtained with HIC. Since retention in HIC is mainly governed by the properties of the hydrophobic molecules or moieties in aqueous media, the choice of a group of related substances, that differ from each other in the number of a certain quantifiable structural element, affords a very convenient means to investigate the mechanism of interaction. In this study the subscript m is taken for x to indicate the use of molecular property in the evaluation of the relevant ∆H°m - ∆S°m pairs for a given process. Linear Free Energy Relationships. The fundamental form of the LFER is written as z
∆G° ) ∑∆G°j
(2a)
1
which expresses the additivity of incremental free energy change ∆G°j assigned to the structural element j of a substance with z elements. For molecules such as homologues that have a varying number of reoccurring structural elements, eq 2a can be written as
∆G°m ) aN + b
(3)
where N is the number of these structural elements k in question, a ) ∆G°k, i.e., the corresponding free energy change of the ∆G°j, the free energy change due to all element, and b ) ∑z-N 1 nonreoccurring elements. Both a and b apply to all the substances in the set.
Hydrophobic Interaction Chromatography
J. Phys. Chem., Vol. 100, No. 6, 1996 2449
A similar linear exothermodynamic expression was recently proposed50 for the relationship between the free energy change of a molecule involved in hydrophobic interactions and a certain molecular property value, Xm. It holds for a group of structurally related substances that are subject to the same mechanism of interaction in the process. The relationship is expressed as50
∆G°m ) agXm + bg
(4a)
where ag is the free energy change per unit value of the molecular property Xm whereas bg is the free energy change when Xm ) 0. Both ag and bg, which may be called group molecular parameters, are constant for the set of substances under investigation. Experimental data obtained with processes involving the transfer of various groups of small molecules into water support eq 4a,56-59 and in these cases the nonpolar molecular surface area exposed to water, ∆Anp, has been taken for Xm. Then, the group molecular parameters for the free energy ag and bg are believed to measure hydrophobic and polar properties of the substances, respectively. For a set of homologues eq 4a, with ∆Anp for Xm, can be derived from eq 3. When the contribution to the free energy change ∆G°k by each reoccurring structural element of the molecule is proportional to its surface area ∆Ak, eq 3 can be written for the group of substances as
∆G°m ) Nak∆Ak + b
(5)
where ak is the free energy change per unit surface area of the reoccurring element. By using the total surface area ∆A of all reoccurring elements in the molecule, we obtain from eq 5 that
∆G°m ) ak∆A + b
(6)
With homologous substances methylene groups are often the reoccurring elements and therefore ∆A may constitute the nonpolar surface area ∆Anp of the compound. Then, eq 6 is similar to eq 4a with ak ) ag and b ) bg. Linear Relationships for Other Thermodynamic Quantities. In view of the Gibbs Helmholtz relation
∆G° ) ∆H° - TδS°
(7)
eq 2a can be written for the entropy, enthalpy, and free energy change as
∆S° ) ∑∆S°j
(2b)
∆H° ) ∑∆H°j
(2c)
∆C°p ) ∑∆C°p,j
(2d)
effect.50 By comparing eqs 4a-c we can express the group molecular parameter ag in terms of ah and as, and bg in terms of bh and bs in a way formally similar to the Gibbs Helmholtz relation. In the following, empirical relationships between the thermodynamic quantities and molecular structural parameters will be employed in combination with EEC to obtain a mechanistic and molecular interpretation of the processes. EEC and the Group Molecular Parameters. Whereas eq 1 predicts that the plots of ∆H°m Versus ∆S°m will be straight lines for the process showing compensation with the same slope TC, it does not carry explicit information on the effect of the molecular structure on either thermodynamic quantity. The quest for a molecular interpretation of EEC prompted us to examine the relationship between TC and ∆G°TC of eq 1 and the group molecular parameters of eqs 4a-c. In light of eqs 1, 4b, and 4c, TC can be expressed by the group molecular parameters, ah and as, as
TC )
ah as
(8a)
so that the free energy change at TC is given by
∆G°TC ) bh -
ahbs ) bh - TCbs as
(8b)
Equation 8a establishes a criterion for EEC to occur with processes driven by hydrophobic interactions. The values of ah and as can be readily evaluated from the slope of linear plots of enthalpy or entropy Versus the molecular property. The validity of eq 8a may be tested by comparing TC obtained from the molecular group parameters to that obtained directly from the slope of enthalpy Versus entropy plots. Equation 8b expresses the free energy change at TC in terms of the molecular parameters bh and bs, and since ag ) 0 at the compensation temperature, ∆G°TC is equal to bg. When Xm is the nonpolar water-accessible surface area of the molecule, then the free energy change at TC is simply given by the contribution of the polar moiety common to the structurally related substances in the group. Thus, both eqs 8a and 8b elucidate the physicochemical basis of EEC in processes involving structurally related substances. EEC and the Heat Capacity Change
Indeed, various estimation methods for the thermodynamic quantities of a substance involve such group contribution schemes.60 In analogy with the analysis in the previous section, entropy, enthalpy, and heat capacity change may be written for a group of substances differing from each other in the number of a certain quantifiable structural element as
Heat Capacity Change and Other Thermodynamic Quantities in Hydrophobic Interactions. The energetics of processes involving hydrophobic interactions are accompanied by large entropy and heat capacity changes that are believed to arise from a reorientation of the water molecules at the cavity around the solute molecule.54,61 More recently, it was shown that the heat capacity in such processes is determined by the change in the water-accessible nonpolar surface area of the molecules upon transfer.62 The free energy change of a process is related to the equilibrium constant K as
∆S°m ) asXm + bs
(4b)
∆G° ) -RT ln K
∆H°m ) ahXm + bh
(4c)
∆C°p,m ) acXm + bc
(4d)
and combining eqs 7 and 9, the dependence of the logarithmic equilibrium constant on the temperature is obtained as
where as, bs, ah, bh, ac, and bc are the group molecular parameters. Equations 4b-d have recently been successfully used in the analysis of processes involving the hydrophobic
ln K ) -
∆H° ∆S° + RT R
(9)
(10)
According to Kirchoff, the enthalpy and entropy changes for
2450 J. Phys. Chem., Vol. 100, No. 6, 1996
Vailaya and Horva´th
TABLE 1: Summary of the Features Associated with Enthalpy-Entropy Compensation with Related Substances Differing in the Number of a Certain Quantifiable Structural Element When in the Process the Heat Capacity Is Either Zero (Case A), or Finite and Constant (Case B), or Linearly Dependent on the Temperature (Case C), Wide Figure 1 intersection of plots of ∆C°p,m, ∆H°m, ∆S°m, and ∆G°m Versus T case A B C
temperature dependence of thermodynamic quantities ∆H° ∆S° ∆G° ∆C°p 0 const * 0 a′(T - TC0)
constant linear nonlinear
constant nonlinear nonlinear
van’t Hoff plots
linear nonlinear nonlinear
linear nonlinear nonlinear
isothermodynamic or compensation temps TCa b c d T* H, T* S, T* G T* e ) T* ) T* H S ) T* G
thermodynamic quantities at the characteristic temps ∆G°TC ∆H*, ∆S*, ∆G* ∆C* p ) 0, ∆H*, ∆S*, ∆G*
a Compensation temperature. b Isoenthalpic temperature. c Isoentropic temperature. d Isoenergetic temperature. e Temperature at which the heat capacity changes converge to zero.
the process are related to the corresponding heat capacity change ∆C°p as
∆H° ) ∫∆C°p dT
Let us assume that the heat capacity change is linearly dependent on the temperature, so that for a given substance ∆C°p is given by
(11a)
∆C°p ) a′(T - TC0)
(15)
and
∆C°p ∆S° ) ∫ dT T
(11b)
In this study, three cases are investigated, as shown in Table 1. In case A heat capacity effects are absent; i.e., ∆C°p ) 0, in the experimental temperature range so that the enthalpy and entropy changes are temperature invariant according to eqs 11a and 11b, and plots of ln K Versus 1/T (van’t Hoff plots) are linear. In such cases ∆H° and ∆S° are extracted from the slope and intercept of van’t Hoff plots, respectively. In case B, the heat capacity change is nonzero and temperature invariant for the process; i.e., ∆C°p ) const * 0, and the enthalpy depends on the temperature according to eq 11a as
∆H° ) ∆H°H0 + ∆C°p(T - TH0)
(12a)
where TH0 is a reference temperature, defined later according to convenience, and ∆H°H0 is the enthalpy change at that temperature. Similarly, the relation for ∆S° is obtained from eq 11b as
( )
∆S° ) ∆S°S0 + ∆C°p ln
T TS0
(12b)
where ∆S°S0 is the entropy change at TS0. By convention,36,37 TH0 and TS0 are equated to TH and TS, the temperatures at which ∆H° and ∆S° are zero, respectively. Thus, eq 12a yields
∆H° ) ∆C°p(T - TH)
(13a)
and eq 12b takes the form
( )
∆S° ) ∆C°p ln
T TS
(13b)
Combining eqs 10, 13a, and 13b we obtain for the logarithmic equilibrium constant that
ln K )
(
)
∆C°p TH TS - ln - 1 R T T
(14)
Equation 14 allows the evaluation of the three parameters, ∆C°p, TH, and TS, from nonlinear van’t Hoff plots by a least square fitting procedure. With the parameters from eq 14 the enthalpy and entropy changes are readily evaluated by eqs 13a and 13b, respectively. In case C, the heat capacity change is a function of the temperature, so that the van’t Hoff plots are again nonlinear.
where a′ is constant and TC0 is a reference temperature at which ∆C°p is zero. The logarithmic equilibrium constant is expressed according to eqs 10, 11a, 11b, and 15 as
ln K )
[( )
2 2 a′ T - TH + TC0(TH - T) + (TS - T) + R 2
TC0 ln
]
T (16) TS
and the thermodynamic quantities associated with the process may be evaluated by fitting eq 16 to the experimental data. EEC without Change in Heat Capacity. Classical EEC is observed when the enthalpy and the entropy of the process do not change appreciably with temperature. This holds for many gas phase reactions because the heat capacity change is often so small that, in a temperature interval of 100 °C or so, ∆H° is virtually constant.1 When enthalpy Versus entropy plots are linear, the compensation temperature TC can be evaluated readily in accordance with eq 1. If changes in all three thermodynamic quantities can be related to Xm by eqs 4a-c, then the ratio of ah to as determines the magnitude of the compensation temperature. In the case of classical EEC with linear van’t Hoff plots, both group molecular parameters ah and as are independent of the temperature. Consequently, experiments conducted at different temperatures yield the same TC value. The value of TC can also be directly obtained from van’t Hoff plots for the substances under investigation as follows. By combining eqs 4b, 4c, and 7, we obtain that
∆G°m ) (ahXm + bh) - T(asXm + bs)
(17)
A function linear in two variables always yields an intersection point when it is plotted against one of the variables, with the other as the parameter.50 Upon differentiating both sides of eq 17 with respect to Xm, we obtain that
∂∆G°m ) ah - Tas ∂Xm
(18)
Setting the left hand side of eq 18 to zero yields eq 8a. Thus, both eqs 8a and 18 imply that all the van't Hoff plots of the substances under investigation intersect at a single point when the process exhibits compensation. The coordinates of this intersection point are ∆G°TC and TC, which are related to each other by the heuristic expression in eq 1.
Hydrophobic Interaction Chromatography
J. Phys. Chem., Vol. 100, No. 6, 1996 2451
Enthalpy-Entropy Compensation with Constant Heat Capacity Change. Many processes which take place in aqueous solution, such as acid dissociation63 and other ionization reactions64 as well as dissolution of gaseous molecules in water,53 exhibit classical EEC behavior at room temperature.7 However, in a sufficiently wide temperature range the van’t Hoff plots for these processes and many others based on the hydrophobic effect are nonlinear; consequently the parameters ah and as and the compensation temperature depend on the temperature. Here we examine the case where ∆C°p is essentially constant or does not change much in the temperature interval of the experiments so that an average value can be used.39 It is noted that the approximation of constant ∆C°p by most investigators28,37,38,50,62 has been the basis for the characterization of processes involving hydrophobic interactions.33 If entropy change is the same for all substances subject to the process at a temperature T* S and is given by ∆S*, one has from eq 12b the relationship
()
∆S°m ) ∆S° + ∆C°p,m ln
T T*S
(19a)
where T*S is the isoentropic temperature.50 For a group of structurally related substances, a plot of ∆S°m Versus ∆C°p,m at a given temperature will be a straight line with the slope of ln(T/ T*S) and with ∆S* as the intercept when eq 19a holds. The significance of T* S has been recognized in studies on protein folding.36,37 An analogous relation for the enthalpy can be obtained from eq 12a as
∆H°m ) ∆H* + ∆C°p,m(T - T* H)
(19b)
where T*H is the isoenthalpic temperature and ∆H* is the enthalpy change at this temperature. In a given process all species have the same T*H and ∆H* values if the mechanism is invariant and the plot of ∆H°m versus ∆C°p,m is a straight line with the slope of T - T*H and with ∆H* as the intercept. Combination of eqs 7, 19a, and 19b yields for the free energy change the following expression
(
∆G°m ) ∆C°p,m T - T* H - T ln
)
T + ∆H* - T∆S* T*S
(20)
Upon differentiating eq 20 with respect to ∆C°p,m and then setting the left hand side of the resulting equation to zero, we obtain that
T)
(T - T*H) T ln T*S
()
TC,T )
(21b)
In analogy with T*H and T*S it is befitting to term T* G the isoenergetic temperature and all the three characteristic temperatures may be appropriately called isothermodynamic temperatures. It is also noted that both the isoenergetic temperature T*G and the compensation temperature TC, in the classical EEC,
ah (T - T* H) ) as T ln T* S
(22a)
For the free energy change ∆G°TC,T at temperature TC,T we obtain the expression
∆G°TC,T ) bh - TC,Tbg ) ∆H* - TC,T∆S*
(22b)
As TC,T is a function of the temperature, we call it the specific compensation temperature in order to distinguish TC,T from the traditional TC. Both eqs 22a and 22b are very similar to eqs 21a and 21b and show that TC,T is related to T* G. Enthalpy-Entropy Compensation with Temperature Dependent Heat Capacity Change. Recent studies on dissolution of gaseous alkanes in water,65-67 protein folding,68-72 and the retention of dansyl amino acids in HIC39 have shown that ∆C°p depends linearly on the temperature and converges to zero at high temperatures. In light of this observation the temperature dependence of the heat capacity change may be approximated by the following expression
∆C°p,m ) a′(T - T*)
(23)
where T* is the characteristic temperature at which heat capacity change for all the species in the group becomes zero and the coefficient a′ depends on the molecular property (cf. eq 15). Equations 4d and 23 are based on the assumption that ac and bc are dependent linearly on the temperature. The enthalpy change according to eq 11a is
a′ ∆H°m ) - (T - T*)2 + ∆H* 2
(24a)
and the corresponding entropy change is given by eq 11b as
(
∆S°m ) a′ T* - T + T* ln
(21a)
Equation 21a is an implicit function with the solution T* G, which is a characteristic temperature of the process and can be evaluated at different T*H and T* S values by an iterative procedure. At T*G the standard free energy change for all substances in the group attains a common value, ∆G*, that is given by
∆G* ) ∆H* - T*G∆S*
mark the common intersection point of van’t Hoff plots where the free energy change is the same for all structurally related substances. The group molecular parameters of eqs 4b-d can be written in terms of the characteristic temperatures and the corresponding thermodynamic quantities.50 When the heat capacity change is temperature invariant or assumed to be so, i.e., both ac and bc are constant, then ah and as are temperature dependent. Their ratio TC,T is the slope of the enthalpy Versus entropy plot at a given temperature T and is given by
T + ∆S* T*
)
(24b)
where ∆H* and ∆S* are the enthalpy and entropy changes at T*. Combining eqs 7, 24a, and 24b, we obtain for the free energy change
∆G°m ) -
a′ 2 T T* - T2 + 2T*T ln + ∆H* - T∆S* (25) 2 T*
(
)
By differentiating eqs 24a, 24b, and 25 with respect to a′ and setting the left hand side of the resulting equations to zero, we find all isothermodynamic temperatures to be the same, i.e. T*H ) T*S ) T*G ) T*, with the important consequence that at this temperature, T*, all species have the same enthalpy, ∆H*, entropy, ∆S*, and free energy, ∆G*, change. According to eqs 8a and 8b the specific compensation temperature in the case of linear temperature dependence of heat
2452 J. Phys. Chem., Vol. 100, No. 6, 1996
Vailaya and Horva´th
Figure 1. Illustration of enthalpy-entropy compensation with related substances differing in the number of a certain quantifiable structural element when in the process the heat capacity change is either zero (A), or finite and constant (B), or linearly dependent on the temperature (C), as listed in Table 1.
capacity change is given as
TC,T )
ah ) as
(T* - T) T T* ln T* 2 -1 T - T*
(
)
Figure 2. Illustration of enthalpy-entropy plots for two processes: (A) heat capacity effects are absent so that van’t Hoff plots are linear; (B) heat capacity change is constant in the temperature interval investigated, and van’t Hoff plots are nonlinear. TC,T1 and TC,T2 represent specific compensation temperatures for the data obtained at experimental temperatures T1 and T2.
(26a)
and the free energy change at this temperature is
∆GTC,T ) bh - TC,Tbs ) ∆H* - TC,T∆S*
(26b)
Table 1 summarizes the results of the above analysis. Plots of the thermodynamic quantities Versus temperature, elucidating the characteristic temperatures observed in the three cases, are illustrated in Figure 1. Characteristic Temperature Relations. Both TC and TC,T are the slopes of linear enthalpy-entropy plots of data obtained with substances which share the same mechanism in the process under investigation. Whereas the plot in Figure 2A represents data obtained in a process yielding linear van’t Hoff plots, Figure 2B shows plots arising from data when the heat capacity change is substantial and constant. Whereas classical EEC data measured at different temperatures fall on a single line with slope TC, with nonzero but constant heat capacity change, only data measured at a given temperature yield a straight line. The slope TC,T represents a compensation temperature that depends on the experimental temperature according to eq 22a. In Figure 2B the two straight lines with slopes TC,T1 and TC,T2 represent the specific compensation temperatures, for the data obtained at experimental temperatures T1 and T2 , respectively. The ∆H°m Versus ∆S°m plots are vertical and horizontal lines at temperatures T*S and T* H according to eqs 19a and 19b. On the other hand, the line at temperature T*G marks conditions where the differences between the molecular properties of substances, which dominate their behavior in the process, vanish as indicated by eq 21b.
Figure 3. Graph illustrating the temperature dependence of the compensation temperature TC (case A) and the specific compensation temperature TC,T (case B) according to eq 22a with T* H ) 100 °C and T*S ) 110 °C. At the maximum of TC,T both the abscissa and the ordinate values are T*G.
In this light EEC with related substances differing in the number of a certain quantifiable structural element can be generalized as
∆H°m ) TC,T∆S°m - ∆G°TC,T
(27)
where TC,T is given by ah/as. When ∆C°p ) 0, this ratio is independent of the temperature and given by TC. When ∆C°p is constant but nonzero or linearly temperature dependent, the ratio is given by eqs 22a and 26a, respectively. Figure 3 illustrates the dependence of TC,T on the experimental temperature in the range from -100 to 100 °C according to eq 22a for a process having T* H ) 100 °C and T* S ) 110 °C, characteristic temperatures typically observed with protein folding. Since upon differentiating both sides of eq 22a with respect to temperature and setting the left hand side of the resulting equation to zero we can also obtain eq 21a, it proves
Hydrophobic Interaction Chromatography
Figure 4. Dependence of the specific compensation temperature TC,T on the experimental temperature according to eq 22a with T°H as the parameter. The value of T*S is fixed at (a) 25 °C, (b) 75 °C, and (c) 110 °C. The equation TC,T ) T is shown by the dashed line.
that TC,T attains a local extremum at the isoenergetic temperature T*G. Indeed it is seen in Figure 3 that the specific compensation temperature increases, reaches a maximum, and then decreases with the experimental temperature. As predicted by eq 21a and shown in Figure 3, when the operating temperature is T* G, the specific compensation temperature is also T* G. It is seen that TC,T changes only slightly with temperature between 0.95T*G and 1.05T* G. Thus, under certain conditions, e.g. in such a small experimental temperature range, a process may yield quasilinear van’t Hoff plots and a compensation behavior similar to classical EEC even if it is accompanied by (constant) heat capacity change.7 The variation of TC,T with the experimental temperature at T*S ) 25, 75, and 110 °C with T* H as the parameter is illustrated in Figure 4. It is seen in Figure 4a that TC,T at any experimental temperature between -200 and T* S increases, whereas above T*S it decreases with the isoenthalpic temperature. Similar behavior is depicted also in Figures 4b and c for different values of T*S. In contradistinction, at any experimental temperature between -200 °C and T* S the value of TC,T decreases and above T*S it increases with the isoentropic temperature. The dashed lines in Figure 4 represent the equation TC,T ) T and mark the loci of the isoenergetic temperatures at different T* H values. When T*H < T* S, the dashed line intersects both lines of the TC,T versus T plots particular to a given T*H. Both intersection points represent isoenergetic temperatures that lie outside the temperature range bound by the isoenthalpic and isoentropic
J. Phys. Chem., Vol. 100, No. 6, 1996 2453
Figure 5. Dependence of the free energy change on the reciprocal temperature according to eq 20 with the respective ∆H* and ∆S* values taken as -40 kJ mol-1 and -50 J mol-1 K-1 and ∆C°p,m as the parameter. The ∆C°p,m values were taken as 100, 200, 300, and 400 J mol-1 K-1. The isothermodynamic temperatures are fixed as (a) T* H ) 100 °C and T*S ) 110 °C, (b) T* H ) T* S ) 110 °C, and (c) T* H ) 125 °C and T*S ) 110 °C.
temperatures. The lower T* G value increases whereas the higher T*G value decreases with increasing T* H. As T* H apvalues come closer to each other until proaches T*S the two T* G the condition T* G ) T* H ) T* S is reached. However, when T* H is larger than T*S, the dashed line TC,T ) T does not intersect the TC,T Versus T plots, and consequently, there is no isoenergetic temperature in this case. Another way of illustrating this situation is offered by the corresponding van’t Hoff plots for the three cases: T*H < T*S, T*H ) T* S, and T* H > T* S, shown in Figure 5. As expected, when T*H < T* S, the van’t Hoff plots intersect at two points and in the singular case of T* H ) T* S, they have only one common intersection point. In contrast, for T* H values greater than T* S the van’t Hoff plots do not intersect at all, as shown in Figure 5c. The dependence of TC,T on the experimental temperature suggests a mechanistic change in the process with temperature. Processes based on the hydrophobic effect and accompanied by a reasonably constant heat capacity change are entropy and enthalpy driven at low and high temperatures, respectively.28,73 Many other processes such as retention of enantiomers on chiral columns74 or that of conformers on reversed phase systems75 also yield nonlinear van’t Hoff plots, and the dependence of TC,T on the experimental temperature indicates that the mech-
2454 J. Phys. Chem., Vol. 100, No. 6, 1996
Vailaya and Horva´th
TABLE 2: Summary of Thermodynamic and Exothermodynamic Relationships for Processes That Are Driven by the Hydrophobic Effect and Exhibit Constant Heat Capacity Change thermodynamic parameters
exothermodynamic relationships
∆C°p ) const * 0
∆C°p,m ) acXm + bc
∆H° ) ∆C°p(T - TH0) + ∆H°H0
∆H°m ) ahXm + bh
∆H°m ) ∆C°p,m(T - T* H) + ∆H*
∆S° ) ∆C°p ln(T / TS0) + ∆S°S0
∆S°m ) asXm + bs
∆S°m ) ∆C°p,m ln(T / T*S) + ∆S*
∆G° ) ∆C°p[T - TH0 - T ln(T / TS0)] +
∆G°m ) agXm + bg
∆G°m ) ∆C°p,m[T - T* H - T ln(T / T* S)] +
∆H°H0 - T∆S°S0
∆H* - T∆S*
∆G° ) ∆H° - T∆S°
∆G°TC,T ) ∆H°m - TC,T∆S°m
group molecular parameters ac ) const bc ) const ah ) ac(T - T* H) bh ) bc(T - T* H) + ∆H* as ) ac ln(T / T* S) bs ) bc ln(T / T*S) + ∆S* ag ) ac[T - T* H - T ln(T / T* S)] bg ) bc[T - T* H - T ln(T / T* S)] + ∆H* - T∆S* ag ) ah - Tas bg ) bh - Tbs TC,T ) ah / as ∆GTC,T ) bh - TC,Tbs T* ∆G* ) ∆H* - T* G ) (ah / as)T* G∆S* G
anism of interaction is altered upon changing the temperature. Since in a process showing compensation behavior TC,T is the same function of the temperature for all the substances investigated, it signifies that the mechanism of the process is invariant with respect to the substances. Thus TC,T can be a useful diagnostic tool for the mechanism of a process with a group of substances at constant operating conditions and/or with one substance at different temperatures, solvents, or other changing experimental conditions. Conclusions A generalized framework is provided for enthalpy-entropy compensation with processes which are characterized by heat capacity change that is either zero or finite and constant or a linear function of the temperature and which involve substances with quantifiable structural differences. Table 2 summarizes the results with processes that are driven by the hydrophobic effect and assumed to have constant heat capacity change. The molecular interpretation of the process parameters is facilitated by the use of linear relationships,50 which have been introduced recently for the dependence of the thermodynamic quantities on certain temperature invariant molecular properties. Within the hermeneutics of EEC this approach was also employed with both linear and nonlinear van’t Hoff plots to elucidate common intersection points that represent the characteristic temperature T*G and the compensation temperature TC for a process showing EEC. As T*G is akin to T* H, the isoenthalpic, and T* S, the isoentropic, temperatures, which mark the common intersection points of the respective enthalpy and entropy Versus temperature plots, T*G is termed the isoenergetic temperature. Thus, T* G together with T*H and T*S completes the triad of isothermodynamic temperatures that has exothermodynamic significance mainly for processes driven by the hydrophobic effect. It is noted that there is an additional isothermodynamic temperature, represented by the intersection of heat capacity Versus temperature plots, when the change in heat capacity with the temperature may not be considered constant for a given process, as shown by recent calorimetric studies on protein folding68-70 as well as chromatographic studies on the retention of dansyl amino acids in HIC.39 Although most results presented here were obtained with substances having quantifiable structural differences, the analytical framework presented here is more general than that which has been customary in the treatment of EEC, as depicted in Figure 6 by plots of a thermodynamic quantity Y against an experimental variable x. The quantity Y may represent the free energy ∆G°x, enthalpy ∆H°x, or entropy ∆S°x change whereas the variable x could also be the temperature, a property of the
Figure 6. Schematic illustration of enthalpy-entropy compensation by plots of the thermodynamic quantity Y representing the free energy, enthalpy, or entropy against the experimental variable x.
solvent, or the stationary phase ligands in HIC. As often observed with various processes and shown in Figure 6, the dependence of the enthalpy or the entropy changes on a given experimental variable can be usually approximated by straight lines. In most instances, both thermodynamic quantities compensate each other; i.e., a decrease in enthalpy is accompanied by a concomitant decrease in entropy in the range of the variable or Vice Versa. Perfect compensation occurs when the variation of enthalpy and entropy with respect to the chosen experimental variable is such that the change in the free energy or the equilibrium constant is invariant in the experimental range of the variable. This is illustrated in Figure 6 by the parallel ∆H°x and T∆S°x lines and the horizontal ∆G°x lines that are observed when the experimental temperature is in the vicinity of the compensation temperature. Further work is required along these lines to extend the scope of the study to various variables and thus to relate other process characteristics to molecular parameters. In this endeavor, reversed phase chromatography, the pendant of HIC, offers a suitable multivariant system to investigate the molecular basis of compensation phenomena. Acknowledgment. This work was supported by Grant Number GM 20993 from the National Institutes of Health, US Public Health Service. References and Notes (1) Leffler, J.; Grunwald, E. Rates and Equilibria of Organic Reactions; Wiley-Interscience: New York, 1963; Chapters 6-9. (2) Hammett, L. P. Physical Organic Chemistry; McGraw Hill: New York, 1940. (3) Taft, R. W. J. In Steric Effects in Organic Chemistry; Newman, M. S., Ed.; Wiley: New York, 1959; p 556. (4) Leo, A.; Hansch, C.; Elkins, D. Chem. ReV. 1971, 71, 525. (5) Hansch, C. In Drug Design; Ariens, E. J., Ed.; Academic Press: New York, 1971; Vol. 1, p 271. (6) Martin, A. J. P. Biochem. Soc. Symp. 1949, 3, 4. (7) Lumry, R.; Rajender, S. Biopolymers 1970, 9, 1125.
Hydrophobic Interaction Chromatography (8) Everett, D. H. In Adsorption from Solution; Ottewill, R. H., Rochester, C. H., Smith, A. L., Eds.; Academic Press: New York, 1983; p 1. (9) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; Freeman: San Francisco, CA, 1960. (10) Leffler, J. J. Org. Chem. 1955, 20, 1202. (11) Leffler, J. J. Org. Chem. 1968, 33, 533. (12) Shaltiel, S. In Methods in Enzymology; Jakoby, W. B., Wilchek, M., Eds.; Academic Press: New York, 1974; Vol. 34, p 126. (13) Hofstee, B. H. J. In Methods of Protein Separation; Catsimpoolas, N., Ed.; Plenum Press: New York, 1976; Vol. 2, p 245. (14) El Rassi, Z.; Lee, A. L.; Horva´th, Cs. In Separation Processes in Biotechnology; Asenjo, J. A., Ed.; Marcel Dekker: New York, 1990; p 447. (15) Haidacher, D.; Horva´th, Cs. NATO ASI Series C: Mathematical and Physical Sciences, NATO ASI Proceedings on Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, S., Guiochon, G., Eds.; Kluwer Academic Publishers: Dordrecht, 1992; p 443. (16) Antia, F. D.; Fellegva´ri, I.; Horva´th, Cs. Ind. Eng. Chem. Res. 1995, 34, 2796. (17) Sinanogˇlu, O. In Molecular Associations in Biology; Pullman, B., Ed.; Academic Press: New York, 1968; p 427. (18) Horva´th, Cs.; Melander, W. R.; Molna´r, I. J. Chromatogr. 1976, 125, 129. (19) Melander, W. R.; Horva´th, Cs. Arch. Biochem. Biophys. 1977, 183, 200. (20) Melander, W. R.; Corradini, D.; Horva´th, Cs. J. Chromatogr. 1984, 317, 67. (21) Wyman, J. AdV. Protein Chem. 1964, 19, 223. (22) Wu, S.-L.; Figueroa, A.; Karger, B. L. J. Chromatogr. 1986, 371, 3. (23) Barford, R. A.; Kumosinski, T. F.; Parris, N.; White, A. E. J. Chromatogr. 1988, 458, 57. (24) Roettger, B. F.; Myers, J. A.; Ladisch, M. R.; Regnier, F. E. Biotechnol. Progress 1989, 5, 79. (25) Kauzmann, W. AdV. Protein Chem. 1959, 14, 1. (26) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley-Interscience: New York, 1980; Chapter 1. (27) Gill, S. J.; Wadso¨, I. Proc. Natl. Acad. Sci. U.S.A. 1976, 73, 2955. (28) Baldwin, R. L. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 8069. (29) Dec, S. F.; Gill, S. J. J. Solution Chem. 1984, 13, 27. (30) Dec, S. F.; Gill, S. J. J. Solution Chem. 1985, 14, 827. (31) Cabani, S.; Gianni, P.; Mollica, V.; Lepori, L. J. Solution Chem. 1981, 10, 563. (32) Murphy, K. P.; Gill, S. J. J. Chem. Thermodyn. 1989, 21, 903. (33) Murphy, K. P.; Gill, S. J. Thermochim. Acta 1990, 172, 11. (34) Murphy, K. P.; Gill, S. J. J. Mol. Biol. 1991, 222, 699. (35) Privalov, P. L.; Khechinashvili, N. N. J. Mol. Biol. 1974, 86, 665. (36) Privalov, P. L. AdV. Protein Chem. 1979, 33, 167. (37) Privalov, P. L.; Gill, S. J. AdV. Protein Chem. 1988, 39, 191. (38) Murphy, K. P.; Privalov, P. L.; Gill, S. J. Science 1990, 247, 559. (39) Haidacher, D.; Vailaya, A.; Horva´th, Cs. Proc. Natl. Acad. Sci. U.S.A., in press.
J. Phys. Chem., Vol. 100, No. 6, 1996 2455 (40) Melander, W. R.; Campbell, D. E.; Horva´th, Cs. J. Chromatogr. 1978, 158, 215. (41) Melander, W. R.; Chen, B.-K.; Horva´th, Cs. J. Chromatogr. 1979, 185, 99. (42) Melander, W. R.; Stoveken, J.; Horva´th, Cs. J. Chromatogr. 1980, 199, 35. (43) Melander, W. R.; Mannan, C. A.; Horva´th, Cs. Chromatographia 1982, 15, 611. (44) Melander, W. R.; Horva´th, Cs. Chromatographia 1984, 19, 353. (45) Riley, C. M.; Tomlinson, E.; Jefferies, T. M. J. Chromatogr. 1979, 185, 197. (46) Riley, C. M.; Tomlinson, E.; Hafkenscheid, T. L. J. Chromatogr. 1981, 218, 427. (47) Boots, H. M. J.; de Bokx, P. K. J. Phys. Chem. 1989, 93, 8240. (48) de Bokx, P. K.; Boots, H. M. J. J. Phys. Chem. 1989, 93, 8243. (49) Li, J.; Carr, P. W. J. Chromatogr. A 1994, 670, 105. (50) Lee, B. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 5154. (51) Butler, J. A. V. Trans. Faraday Soc. 1937, 33, 229. (52) Evans, M. G.; Polanyi, M. Trans. Faraday Soc. 1936, 32, 1333. (53) Barclay, I. M.; Butler, J. A. V. Trans. Faraday Soc. 1938, 34, 1145. (54) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. (55) Lee, B. Biophys. Chem. 1994, 51, 271. (56) Chotia, C. Nature (London) 1974, 248, 338. (57) Hermann, R. B. J. Phys. Chem. 1972, 76, 2754. (58) Hermann, R. B. J. Phys. Chem. 1975, 79, 163. (59) Reynolds, J. A.; Gilbert, D. B.; Tanford, C. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 2925. (60) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw Hill: New York, 1987; Chapters 5-6. (61) Frank, H. S. J. Chem. Phys. 1945, 13, 493. (62) Livingstone, J. R.; Spolar, R. S.; Record, M. T., Jr. Biochemistry 1991, 30, 4237. (63) Everett, D. H.; Wynne-Jones, W. F. K. Trans. Faraday Soc. 1939, 35, 1380. (64) Laidler, K. J. Trans. Faraday Soc. 1959, 55, 1725. (65) Naghibi, H.; Dec, S. F.; Gill, S. J. J. Phys. Chem. 1986, 90, 4621. (66) Naghibi, H.; Dec, S. F.; Gill, S. J. J. Phys. Chem. 1987, 91, 245. (67) Naghibi, H.; Ownby, D. W.; Gill, S. J. J. Chem. Eng. Data 1987, 32, 422. (68) Makhatadze, G. I.; Privalov, P. L. J. Mol. Biol. 1990, 213, 375. (69) Privalov, P. L.; Makhatadze, G. I. J. Mol. Biol. 1990, 213, 385. (70) Privalov, P. L.; Makhatadze, G. I. J. Mol. Biol. 1992, 224, 715. (71) Makhatadze, G. I.; Privalov, P. L. J. Mol. Biol. 1993, 232, 639. (72) Privalov, P. L.; Makhatadze, G. I. J. Mol. Biol. 1993, 232, 660. (73) Spolar, R. S.; Ha, J.-H.; Record, M. T., Jr. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 8382. (74) Pirkle, W. H. J. Chromatogr. 1991, 558, 1. (75) Melander, W. R.; Nahum, A.; Horva´th, Cs. J. Chromatogr. 1979, 185, 129.
JP952285L
