The enthalpy of hydration MYd of an ion (i.e. the enthalpy change which accompanies the dissolution of one mole of the ideal gaseous ion in an infinite volume of water a t a temper- ature of 298°K and a pressure of 1 atm) appears in the ther- modynamic analysis of several important topica in general and inorganic chemistry (13) . Available tahulations of experi- mental hydration enthalpies cover only about 20 or 30 cations and half a dozen anions. Hydration enthalpies for unstable ions, calculated from empirical formulas, are useful in the rationalization of periodic trends in chemical behavior (4,5). The purpose of this paper is to present acomprehensive tah- ulation of experimental hydration enthalpies, and to discuss the calculation of hydration enthalpies for unstable ions.

Cations

Derek W . Smith University of Waikato

Hamilton, New Zealand

Experimental Hydration Enthalpies Hydration enthalpies of ions (6-10) are readily extracted

from tahulations of thermodynamic data (11, 12) from the enthalpies of formation AH? of single aqueous ions. These are usually expressed on a scale relative to the hydrated proton H+ (aq), and refer to a standard state of unit molality and unit activity a t 298'K. For example, AH)" for Mg2+ is -462 kJ mole-'. This means that AHo for the process

Ionic Hydration Enthalpies

M g N + 2H+(aq) - Mg2+(aq) + Hdg)

is equal to -462 kJ mole-' a t 298'K. Thus AH)" for Mgz+(aq) is the sum of the standard enthalpy changes

Mg(s) - Mgk) AH,

Mgk) - Mg2+(g) AH2

Mg2+(g) - Mg2+(aq) A H 3

2Ht(aq) - 2Ht(g) AH4

2H%) - 2Hk) A H 5

2HW - Hnk) A H 6

AH3 is equal to miyd for Mg2+; the other enthalpy changes are available in standard tabulations, except for AH4, which is the negative of twice the hydration enthalpy of the proton. The absolute magnitude of AHEyd(H+) is estimated to he -1091 f 10 kJ mole-' (7). Hence AHfYd(Mg2+) is found to be -1921 f 20 kJ mole-'. AH)' is not always available in standard tahulations; however, AG)' for most cations is obtainable from redox potentials. To find AH? from AG)", we require the standard entropy change for

The absolute entropies of M(s) and H2(g) a t 298OK can he found experimentally; the absolute entropy of H+(aq) is es- timated (13) to be close to zero a t 29E°K, and the entropy of Ma+(aq) can be calculated from an empirical formula (14). Values of AH)" thus ohtained are less accurate than those de- termined experimentally, but the entropy change is only large in the case of small, highly charged cations, such as AB+; here AH? and A G ~ differ by 46 kJ mole-' but MEyd is -4665 kJ mole-'. Thus an error of 10% in estimating the TASO term produces an error of only 0.1% in the hydration enthalpy.

Calculation of Hydration Enthalpies

Fairly rigorous calculations have been performed on a few

cations and anions (15), though simple empirical formulae are clearly desirable. A popular (4,5) expression

where r is the crystal radius (in A) of Mz+, works very well for cations with the noble gas configuration, hut gives low values of -AH&, for d n cations. This is not surprising. The various collections of crystal radii (16-20) are meaningful in so far as they reproduce interatomic distances in crystals when added together, but the notion that (with the addition of a constant term) they represent the effective sizes of gaseous ions is questionahle. The hydration enthalpy is expected to depend on the polarizing power of the ion; however, the polarizing power of a cation does not depend on its charge and crystal radius alone. For example, Na+ and Cu+ have comparable crystal radii hut Na(1) compounds are much more ionic. This may he related to the higher electron affinity of Cu+. Clearly, a satisfactory empirical formula for AH!,, cannot depend on crystal radius and charge alone, and must somehow incorpo- rate the enhanced electron affinities of dn ions.

We suggest that the hydration enthalpy of MZ+ can be satisfactorily ohtained from the empirical equation

where r is the Pauling univalent'radi'us (16) in A of the cation, adjusted to incorporate the effects of an 'anomalously' high electron affinity. For cations having a noble gas configuration, and for lanthanide ions, r is equal to the Pauling univalent radius ra. For d n or 5fn ions, r is given by r ~ ( l ~ l I ) " ~ , where I is the relevant ionization e n t h a l ~ v of M(e) at 298'K. and In . ~~ ~ ~~ -

is the corresponding ionization inthalpy 'f;;r the formation OF the analoeous do or P cation of charre 2. For examnle in the case of z"~+, I is 2652 kJ mole-'. T K ~ analogous '1;oble gas' cation is Ca2+, for which lo is 1748 kJ mole-'. The univalent radius of Zn2+ is 0.88 A so that r in eqn. (2) is 0.88(1748/ 2652)'/2A, i.e. 0.71 A. We prefer Pauline univalent radii since they are inwnded to repiesent the r e h i w sires of isoelrr- tronic ions. Pauling (16) converted the univalent radius roof M2+ into a crystalradius r, by

rt = r,,-2/(n-1) (3)

where n is the Born exponent. Such a procedure seems inap- propriate for ions in solution. Empirical crystal radii (20) are unsatisfactory for 'soft' cations like Cu+ and Hg2+; the 'ap- parent' crystal radius of six-coordinate Cut varies between 0.6 and 0.9 A, and it has heen suggested (21) that the Pauling radius of 0.96 A represents an ideal value for Cu+ in a een- uinely ionic crystai. Thus there is some justification for-our use of 'contracted' Pauling univalent radii with such ions.

In Table 1 we present experimental hydration enthalpies for all cations MZ+ which can be studied in aqueous solutions. Alongside are values calculated from empirical formulae (1) and (2). Most of the experimental values were obtained as previously described from tahulations of AH)" for the aqueous ions (11, 12). Ionization enthalpies were taken from Moore (22), and atomization enthalpies from standard tabulations (11,12). For most ions, an error off 3z kJ mole-' is estimated to cover experimental uncertainties. A systematic error of f 102 kJ mol-' (arising from uncertainty in the hydration

540 I Journal of Chemical Education

Table 1. Experimental and Calculated Hydration Enthalpies for Cations in kJ mole-'.

Figures in parentheses for the experimental valuer indicate the PIoDable error in kJ mole-', without regard for the inherent system- atic error of +lo k~ mole-'. The calculated valuer AhPb..~ (ca1.3 121 and AhPhyd (calc) (1) refer, respectively, to valuer obt%&~i;&- ' eqs. (2 ) and (1).

enthalpy of the proton, -1091 kJ mole-') is not included in Table 1. Where the hydration enthalpy was taken from the experimental AG?, and a calculated AS?. the errors are , , greater. The relevant electrode potentials in such c a m wrre taken from Huheev (23). The third ionization enthaloies for most of the lanthanides were taken from thermochemical estimates (24-26). For the actinides, missing ionization enthalpies were estimated by interpolati'on and extrapolation, leading to larger errors in the experimental hydration enthakies. For the ammonium ion, the hydration &thalpy was obtained from the experimental AH? of NH4+(aq) and the estimated AH? (27) of NHl+(g).

Pauling univalent radii were used as far as possible for the calculated hydration enthalpies using eqn. (2). Where Pauling (16) did not give a univalent radius, we have used his empirical crystal radii in eqn. (3) to obtain univalent radii. For nd'O(n + l)s2 ions, the Born exponent was taken to be that appro- priate to the configuration (n + l)s2(n + l)p6, and the re- sulting univalent radius used directly in eqn. (2). Empirical crvstal radii not given bv Pauline were taken from Shannon and Prewitt, for Sx-~oordinatio~(20). Calculated hydration enthalpies from eqn. (1) use Pauling crystal radii. All calcu- lated hydration enthalpies are rounded off to the nearest multiple of 10. The agreement between exoerimental hvdra- tion enthalpies and t l k e calculated from eqn. (2) is striking; enthalpies calculated from eqn. (1) are less satisfactory, even for .nuhle 93s' cations. Thc mist serious discrepancies appear in the 3d and V series. Calculated values for trmsition metal ions are improved by corrections for crystal field stabilization energy (28). In the lanthanide M3+ ions, the experimental enthalpies show pronounced discontinuities between Eu and Gd and between Ho and Er, arising from 'breaks' in the third ionization enthaloies. These are not renroduced in our cal- culated enthalpies, hut the latter are wkhin about 3% of the experimental values.

Anions

Experimental Hydration Enthalpies The entbaloies of hvdration of anions are difficult to de-

termine expe&nentaliy. Enthalpies of formation of aqueous anions relative to the proton have been tabulated (11.12) but such enthalpies for gaseous anions are not exp&&entally accessible. However, the AH? for a gaseous anion can be es- timated from the lattice enthalpies of ionic crystals (29). Lattice enthalpies are known quite accurately (29) for ionic halides: for solids containine oolvatomic anions. thev can he ~.~ ~, ~~~ ~

estimal'ed (30) but values th;;; obtained are not very accurate and it is dit'fir~dt to estimate the error. Thus, apart from the

Table 2. Experimental and Calculated Hydration Entha1pi.s for Anions (in kJ mole-').

~ n i o n -AHDhyd Iexpt.) - A F I " ~ ~ ~ (caic.)

Volume 54, Number 9, September 1977 / 541

halide ions, experimental hydration enthalpies are more poorly defined for anions than for cations.

Calculation of Anion Hydration Enthabies The empirical equation

= -7m12 Yd -77- kJ mole-' (4)

. c

where r, is the Pauline crvstal radius in A. works rather well for halide ions (5). ~r;stai radii have ~it t le 'meanin~ for poly- atomic anions, hut thermochemical radii (30) may indicate their relative sizes. We suggest that for polyatomic anions, eqn. (4) is applicable with the denominator equal to (rr + 0.3)A, where r, is the thermochemical radius. In Table 2 we list ex- ~erimental and calculated hvdration enthal~ies for anions of

known crystal or thermoch&ical radius. The experimental values for F-, C1-, Br-, I-, OH-. and SH- were taken from tabulated enthalpies of hydration relative to the proton (8). For HF2-, SO4'+, Se042-, CN-, N3-, NO3-, BF4-, NCO-, CO?, and S2-, we used literature values (1,29) of enthalpies of formation of the gaseous ions, and determined the hydra- tion enthalvies from these and the enthahies of formation of the aqueous ions, relative to the proton'(ll,12). For other anions. we estimated the enthalvv of formation of the easeous . . ion from calculated lattice enthalpies (30) and experjmental enthalpies of formation (11.12) of sodium andlor potassium salts. The calculated hydration enthalpies were obtained from e m . (41. with the denominator eaual to the Pauline crvstal " - radius ri for monatomic anions, a i d rt + 0.3A for polyatomic anions. The agreement between calculated and experimental values is surprisingly good, considering the approximations involved. Unfortunately, it is difficult to estimate the ther-

mochemical radius of an unknown anion; such radii have yet to be determined for many known anions.

Literature Cited (1) Dasent, W. E.,"lnorganic Energetics," Penguin Baaks, Baltimore. 1970. 12) Myer8.R. T.. J. CHEM. EDUC.. 53, I7 (1976). (3) Lessley, S. D., and Ramdale, R. 0.3. CHEM. EDUC., 53.19 (1976). 14) Johnson,D.A.,"SomeThcrmodynamieAspeetsofInoqanieChemiatry,"Cambridge

University Press, 1968. (5) Phillips, C. S. G.. end Williams, R. J. P.. "lnorganie Chemistry." Oxford University

PI... IQfC . .. -. ... ". ,6. M . . ~ ~ , ~ , n F r , I L ~ ~ ~ ~ ~ . 4 6 3 119681 (1. Hsll8u~II.H F.and h y h u l l . 3 C .Trow Faroda) Yw. SY. 112 f i I IY iR) (8, H ,rwnbky.O R . Cnrm Re. . 65.4li7 1W6, (9, C.mnum.H F .and Krknr J O'\l .m " U o d c m ~ p ~ o f E I ~ n r h e m ~ ~ t r y " , t E d ! l o r

R ~ C L , ~ ~ . ~ o ' h ~ ~ , n . ~ t t ~ ~ ~ ~ t h . ~ . ~ ~ d . ~ ~ UCI 1.1951.~ 4: !I01 ih,.n,wm .I F .and J ,I.;ouer.C .m "~ lodem i\npd~dFlpnmhematry.'~Fd,!~.ri'

H . . L w . J O'M .andc'onaa$. 8. E . Hurtewurthr. Lnndon. Nu. 5. 1969.p. 1 11, Avlur r1 .O 11 .and h d l a ~ T .I \'. SI C h t m ~ a l Ilara."Wdey IAYIII&IIB .S)dnev.

1911 ~~~~

112) Wessf R. C. (Editor) '"Handboakofchemistryend Physics." Chemical Rubber Co.. 56th. Ed., 1915. Table D61-78.

(13) Noyes, R. M., J. A m ~ r Chem. Sm., 84.513 (1962). (14) Powell, R. E.. and Latimer, W. M., J. Cham. Phys.. l9.1139 (1951). (15) Goldman, S..snd Bates. R. G.. J Amer. Chom. Soc.. 94.1476 11972). (16) Psuling, L., "The Natureofthe Chemical Band," 3rd. Ed.. Cornell University Press.

lthara. N.V.. 1%"~

542 1 Journal of Chemical Education