The Born Equation and Ionic Solvation P. W. Atkins and A. J. MacDermott Physical Chemistry Laboratory, South Parks Road, Oxford, England
The starting point for most discussions of the thermodynamics of ion solvation is the Born expression for the Gihhs function of solvation. Born's original calculation' is succinct, but has heen replaced in a variety of introductory texts by a simplified version. The point of the present note is to emphasize that this widely used simplification is incorrect (although i t leads to the correct answer), and to present an alternative version. Born ascribes the chanae in energv to the modification of the energy rnntent of the&rroundhR mpdinm (vacuum initiallv, a~ntinuousdielecrrirof ~ermittivitv6 finsllyJ, and uses the kipression for the energy bf an electric field u(f) = 1/2fJ drE2
(1)
Since the electric tield, E, at a distanre r f'rnm an ion of rharge ze (and radiusr,) has maznitud~2e11atr2, nnd d r = r2rlrdl!. this gives
for the energy of the electric field of the ion in a medium of permittivity E . I t follows that the molar Gihbs function of solvation is the change in energy of the electric field on going from a vacuum to a dielectric:
I t follows immediately that AG, = we (discharge,f,
+ we (charge, E , )
= 1)
= -%(N/4aro)(z2ezlri){1- (l/r,)l
(5)
in accord with the Born equation, eqn. (3).AH, and AS, are then obtained from the relations G = H - TS and S = -(dGIaT),. The false step in the calculation is the use of the classical electrostatic expression, which is based on the gradual growth of charge. If the charge on a sphere is to be increased, a definite number of electrons have to be attached. If the final charge is to he ze, then n. electrons have to he supplied (where n, = - 2 ) . In the case of z = -1, one electron has to be added to an uncharged sphere. The work of attachment is zero, because there is no charge on the sphere to repel the incoming electron, and an electron cannot repel itself. (The electron affinity carries through the calculation and cancels in the final expression: we ignore it for the present purpose. Questions of the self-energy of the electron are irrelevant.) This is contrary to the classical result, eqn. (4). Likewise, the work of withdrawing a unit charge (an electron) from an otherwise uncharged sphere is also zero. Therefore, the Gihbs function of solvation of a sinelv charged ion should. on this basis. be zero. in conflict withe& (5). This analvsis can he eeneralized to the calculation of the work of charging an ion to ze by bringing up (or taking away) a succession of n, electrons. As the sphere is initially un-
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where N is Avogadro's constant and f, = J t a is the relative permittivity (dielectric constant). In elementary expositions attempts to circumvent eqn. (1) have generated erroneous simplifications. These calculations run as follows. First, the Gibhs function is identified with the work required to discharge an ion in a vacuum plus the work required to recharge it when it is immersed in the dielectric, Figure 1. The ion is regarded as a sphere of radius ri and classical electrostatic arguments are used to calculate the work required to increase the charge from zero to ze
= %(1/4r~)(ze)~/rj
' Born, M., Z. Phys., 1, 45 (1920).
(4) Figure 1. lation.
Conventional scheme f w the simplified version of the Born calcu-
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May 1982
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charged, and therefore at apotential 4o = 0, no work has to be done to bring up the first electron. When it has been attached, the sphere is a t a potential = -el4mor;. Work has to be done to bring up the second electron, and the amount involved to bring it from infinity is -e&. The potential on the sphere is now $2 = (-2e)/4morJ as a result of the presence of the new electron. The third electron requires an amount -e@* of work, = and when it is present the potential of the sphere is (-3e)/4nror;. I t follows that the amount of work required to attach all n, electrons is Figure 2. Modified scheme for the simplified version of tion.
the Born calcula-
charge of the ion + local medium is q + q~ (q and q ~will , have opposite signs). The work involved is therefore When a large number of electrons have been brought up in succession in order to achieve a large charge (as in the usual applications of classical electrostatics), n, is so large that n, (n, - 1) = n:. In this case in agreement with eqn. (4). In other words, when n, >> 1the addition of a sequence of small charges may he approximated by the continuous charging process. In contrast, when n, is not large it is essential to use eqn. (6). Then we see that if only one charge is to be added, the work involved is zero. In chemistry we are almost always dealing with cases for which n. n 1. The work of dischareing a monovalent ion in a vacuum is zero. The work of recharging it after its immersion in the dielectric is also zero. Therefore AG is zero. However, this result is absurd, as salts of monovalent ions do dissolve, and so clearlv some asoect of the calculation is misconceived. where the simplified version diverges from Born's is in the ascription of the energy change to the ion instead of the medium. What we shall now do is to reconstruct the simplified version, but consider the charges in the medium surrounding the ion. Consider the sequence of steps again. First there is the discharge of the original ion in the vacuum. The work involved is zero. Then the ion is charged by the addition of one unit of charee. The work involved is zero. Now we arrive a t the crux of the calculation. The ion nquires its charge instanraneously. hut the dielectric oolariziition armnd it crows unlv nl'trr rhe charge has been attached. (It is irrelevant to the argument to discuss how fast the ~olarizationgrows: the seauence charee followed hy polariration is the csi(.ntinl p i n t . I The growrh of charge on the suriace uf the dielectric cavitv surroundinr the chnqrd ion is ;I amrinuou process hecalls~,for exon~ple. it ran be pictured as the result of the rotation d the dipoles constituting the medium into energetically favorable &entations. There is work involved in this step, and the Gibhs function of solvation is in fact the work of establishing this polarization charge. This sequence is illustrated in Figure 2. Since the charging process is r m t i n u o ~ ~rqn. s , [.I,may he used to cnlcularv rhr work invoked. The initial chdree on thc cavity is -e (the singly charged ion), hut the same argument applies to all charge types, and we shall denote i t q. The polarization of the medium results in the appearance of a charge q~ on the surface of the dielectric cavity, and so the final u
360
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Journal of Chemical Education
The value of q~ can he established as follows. The force between two equal charges separated by a distance R in a vacuum is When the charges are immersed in a dielectric medium there are two ways of expressing the force. Either it is expressed in terms of the original charges through F'
= q2/4s6R2
(10)
or note is taken of the polarization charge of the medium F' = (q + ~ n ) ~ / 4 a c & ~
(11)
These alternative ways of treating the effect of the solvent, one allowing E, to deviate from unity and the other modifying the effective ionic charge, must give rise to the same value of the force, and so we require and so Substitution of this result into eqn. (8) gives As this is the only contribution to the work of solvating a monovalent ion, i t follows that
AG, = -1/2(Ne2/4r~ori)ll- (l/r,)J
(15)
which is the Born equation for the case z = f1. The same kind of arguments apply when l z I > 1, hut the work of discharging and the work of recharging, are no longer zero. The important point, however, is that the work of recharging the ion in a dielectric is equal and opposite to the work of discharaina . .. it in a vacuum, and so the two contrihulions mnrel; t h ~ sis becaus(. the reiharging may b t . rrgnrded for the p!lrpt,ee o f rhe calculi~tionn i oct wring before the dielectric~mediumresponds to the presence of the charge, and so E, is effectively unity. The sole contribution to AG is thus the work of subsequently establishing the polarization charge. This is given by eqn. (14) with q = ze; hence AG is given in these cases too by the Born expression.
