Research: Science and Education
New Methods To Estimate Lattice Energies Application to the Relative Stabilities of Bisulfite (HSO3–) and Metabisulfite (S2O52–) Salts H. Donald B. Jenkins* Department of Chemistry, University of Warwick, Coventry, West Midlands CV4 7AL, United Kingdom; *
[email protected] David Tudela Departamento de Química Inorgánica, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Thermochemical cycles are very useful to understand some apparently surprising facts in inorganic chemistry. For example, the activity of group 2 metals increases down the group, but the opposite trend is found for group 12–14 metals, as well as transition metals, while for group 1 metals, the activity increases in the nonregular order Na < K < Rb < Cs < Li (1). The standard reduction potentials, which put the activity series of metals on a quantitative basis, can be related to the enthalpy change for the process M(s) → Mn+(aq). The enthalpy change may be divided into steps of atomization of the metal, ionization, and hydration of the ion, thus accounting for all the observed trends, provided that entropy effects, although important, are similar for all the metals in a series. Bond energy data are useful to rationalize the chemical behavior and stability of molecules containing covalent bonds (2), thus explaining many aspects of nonmetal chemistry. For ionic compounds, on the other hand, different thermochemical cycles, involving lattice energies, can be used to explain some apparently contradictory trends. For example, the stability of alkali metal fluorides decreases from LiF to CsF, while the stability of alkali iodides increases from LiI to CsI. The thermochemical cycle for the formation of MX(s) shows the metal-dependent steps of atomization, ionization, and lattice enthalpy. For the small fluoride salts, the stability follows the order of the lattice enthalpy, while for salts of low lattice enthalpy, such as iodides, the most stable salts are for metals of low atomization and ionization enthalpies. The solubility of ionic salts can be understood by means of thermochemical cycles involving the lattice energy of the salts and the hydration enthalpies of the gaseous ions. The appropriate thermochemical cycles can explain why the reaction of alkali metals with oxygen leads to the oxide for Li, the peroxide for Na, and the superoxides for K, Rb, and Cs. Some other reactions that can be understood by means of thermochemical cycles, are the decomposition of carbonates, the dehydration of hydroxides, and so forth (3). In general, any trend observed for ionic compounds can be explained by means of a thermochemical cycle that involves lattice energies. Also, the estimation of unknown thermochemical magnitudes or the prediction of the stability of new materials often demands such an approach. Nevertheless, the lack of lattice energy data for many compounds could limit the use of thermochemical cycles unless reliable and simple methods to estimate them are available. Fortunately,
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much progress has recently (4–8) been made in the estimation of lattice energies (3, 9, 10) and of standard entropies (11, 12). This article reviews some recent methods that involve simple calculations and can be easily used by any student. As a detailed example, we show how these methods can be applied to the case of the relative stabilities of bisulfite and metabisulfite salts of alkali metals (13). This system could not have been investigated in the past, in a quantitative way, because no estimates of lattice energies for bisulfites and metabisulfites had ever been made. Estimation of Lattice Energies Lattice (potential) energy, Upot, is the internal energy change occurring when 1 mole of a crystalline compound, MpXq, at 1 atm pressure, is converted into defined gaseous ions Mq+ and Xp−, which are infinitely removed from one another.
Mp Xq(c)
pMq+(g) + qX p−(g)
(1)
Upot cannot be directly measured experimentally, since, in reality, compounds form gaseous atoms or molecules (rather than gaseous ions) once their lattices are disrupted. However lattice energy can be indirectly determined if made the unknown limb of a thermochemical cycle otherwise based on experimental quantities. Traditionally, rigorous computational techniques employing purpose-written software (e.g., WMIN, ref 14; GULP, ref 15 and 16; LATEN, ref 17 and 18; etc.) have been established. These frequently involve the use of a Madelung summation (9). However, such computational approaches require complete crystal structure data to be available (including atomic coordinates). For lattices of topical interest such detail is frequently unavailable or else the necessary parameterization is uncertain. More popular methods to determine the lattice energy have been empirical equations. The well-established Kapustinskii equation (19) employing a spherical model for ions having defined “thermochemical” radii (20, 21) has been used successfully for over four decades. Very recent developments (4, 6) based on original work by Bartlett et al. (22), have made important advances and have simplified the estimation procedures for Upot enormously. For lattice energies up to 5000 kJ mol᎑1, Upot is inversely related to the cubic root of the molecular (formula unit) vol-
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
Research: Science and Education Table 1. Constants Used in Equations 2 (α, β) and 4 (γ, δ) To Estimate Lattice Energies Salt
Charge Ratio
α/(kJ mol᎑1 nm)
Ionic Strength
β/(kJ mol᎑1)
γ/(kJ mol᎑1 cm)
δ/(kJ mol᎑1)
1981.2
MX
1:1
1
117.3
M2X
1:2
3
165.3
51.9 ᎑ 29.8
8375.6
103.8 ᎑ 178.8
MX2
2:1
3
133.5
60.9
6764.3
365.4
MX
2:2
4
101.6
91.5
6864.0
732.0
ume, Vm, by − 13
Upot = 2I αVm
+ β
(2)
where Upot is in kJ mol᎑1, Vm in nm3, α and β are the appropriate fitted coefficients collected in Table 1 for different stoichiometries (4, 6), and I is the ionic strength of the lattice (23), simply calculated as I = 1/2 ∑inizi2, where ni is the number of ions of type i in the formula unit with charge, zi, and the summation extends over all ions in the formula unit. The molecular volume is calculated by dividing the unit cell volume (24), obtained from powder or single-crystal Xray diffraction data, by the number of formula units per cell: Vm = Vcell兾Z. The molecular volumes are approximately additive, as indicated for a compound MpXq:
(
)
(
Vm = pV Mq + + qV X p −
)
(3)
Equation 3 can be used to calculate ion volumes, provided that the molecular volume and the counter ion volume are known, and to calculate molecular volumes, to use in eq 2, from tabulated ion volumes such as those reported in ref 4. Bearing in mind the relationship between the molecular volume and the density of a material, ρ, lattice energies can be estimated from density data (6) by (for lattice energies up to 5000 kJ mol᎑1), Upot = γ
ρ Mm
1
3
+ δ
(4)
where ρ is in g cm᎑3, Mm is the chemical formula mass (in g) of the ionic compound, and the coefficients γ and δ are collected in Table 1 for salts of different stoichiometries. Density data are widely documented (a good source being ref 25, section 4-35–4-98: Physical Constants of Inorganic Compounds) and the nondestructive measurement requires only small quantities of material. Lattice energies of hydrates can be calculated using eq 2 (where Vm is taken to be the molecular, formula unit, volume of the parent, anhydrous salt) by adding the term 54.3n kJ mol᎑1 (26) where n is the number of moles of water of crystallization. For lattice energies higher than 5000 kJ mol᎑1, they can be calculated1 from molecular volume and density data, as shown in refs 5 and 6. When lattice energies are incorporated into a thermochemical cycle, the corresponding lattice enthalpy, ∆HL, for a salt MpXq is calculated as, ∆H L = Upot +
p
nM n − 2 + q X − 2 2 2
RT (5)
where nM and nX depend on the nature of the ions, Mq+ and Xp−, and are equal to 3 for monatomic ions, 5 for linear polyatomic ions, and 6 for nonlinear polyatomic ions. With the use of eqs 2 and 4, lattice energy can now be estimated, simply and easily, using readily available or experimentally measurable quantities. Moreover such estimates can be made for new or even for hypothetical ionic materials, provided that they are formed by ions of known ion volume.2 These new approaches possess the advantage that they avoid the need to model ions as spheres, as most ions of topical interest are patently nonspherical. The cube root of the molecular volume in eq 2 can be related to the interatomic distance in the Born–Landé and Kapustinskii equations (since it has dimensions of reciprocal length), but for nonspherical ions it seems more reasonable to talk about ion volumes than about ionic radii. This cube root dependency in eqs 2 and 4 serves also to minimize the effect on Upot of any errors arising in the Vm or density data used. These new methods allow us to make reliable estimates of lattice energy and standard entropy and hence lead on to estimation of the free energy change, ∆G, of reactions for almost all, if not all, ionic lattices, and they have rapidly been adopted and usefully applied to the study of a variety of problems (27–38). These include •
Exploration (27) of the solvent dependence of the reaction products of SnMe 2 Br 2 and Et 4 NBr in water and in CHCl3兾hexane mixtures.
•
Investigation (29) of the energetics of HEDM materials.3 These new high nitrogen, nonnuclear, energetic materials are important because of their insensitivity to friction or impact. These compounds have some of the highest reported enthalpies of formation values.
•
Comparison of experimentally measured enthalpies of combustion (30) with theoretical values derived from the use of eq 2.
•
The contribution from calculations of the type reviewed in this article to the current debate (28, 31, 32) concerning the stability of the polynitrogen species containing the pentazole cation N5+ and the N5− and N3− anions.
•
The derivation of a particularly simple quantitative measure to compare the efficacy of fluoride ion donors (33) shows that further synthetic efforts designed to increase the cation size of fluoride ion donors (as has been the trend recently) is hardly warranted. It transpires from this study that very little is gained from increasing the cation size past a certain level and that secondary factors, such as chemical and physical properties, become overriding considerations.
•
Exploration of the thermochemistry of noble gas salts (34) and of dithiazolium salts (35).
JChemEd.chem.wisc.edu • Vol. 80 No. 12 December 2003 • Journal of Chemical Education
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Research: Science and Education •
∆H 7, ∆S 7, ∆G 7
Quantification of the fluoride ion affinity values, which are difficult to obtain experimentally (36).
Our ion volume database (4) continues to provide a quantitative comparison of ion sizes (37) in its own right and this scale has recently been rationalized against other measures of size (38). The advantages and usage of the new lattice energy estimation procedures based on volume and density data are shown in the following example where eqs 2–5 are worked through in detail, including estimated errors, and used to calculate thermochemical magnitudes.
2ⴚ
O
O
O
S O
2ⴚ
H H
S
O O O
O S
O
S
O
+ H2 O
O
ⴙ
2 M + 2HSO3ⴚ(g)
∆H 7, ∆S7, ∆G7
∆H cond
(6)
Figure 1. Thermochemical cycle to examine relative stabilities of bisulfites and metabisulfites.
M
V(MHSO3)/ Upot(MHSO3)c/ V(M2S2O5)/ Upot(M2S2O5)c/ nm3 (kJ mol᎑1) nm3 (kJ mol᎑1)
Li
0.0842a
639(± 22)
0.1418a
1723(± 60)
Na
0.0861a
635(± 22)
0.1457a
1706(± 60)
K
0.0921
a
623(± 22)
0.1575
b
1658(± 58)
Rb
0.0961a
616(± 22)
0.1655a
1628(± 57)
Cs
0.1010b
608(± 21)
0.1754a
1593(± 56)
b
From experimental unit cell data.
Calculated using eq 2. Error assumed to be 3.5%.
(8)
Since this reaction is independent of the nature of M +, the thermodynamics of the system can be parameterized as a function of ∆Hcond, the enthalpy of condensation of two isolated HSO3− ions (eq 8). Available crystal structure data for the salts in question are limited to studies on two salts: CsHSO3 (rhombohedral, a = 0.46721 nm, α = 85.31⬚, Z = 1; ref 44 ) for which the 1484
Table 2. Lattice Energies (and Estimated Errors) for MHSO3 and M2S2O5 Salts Obtained from the Volume-Based Equation 2
c
M2S2O5(s) + H2O(l) (7)
S2O52−(g) + H2O(g)
ⴙ
2 M (g) + S2O52ⴚ(g) + H2O(g)
From combination of ion volumes (eq 3).
can be understood by means of the thermochemical cycle in Figure 1 by introducing the lattice enthalpy terms, related to Upot(MHSO3) and Upot(M2S2O5) by means of eq 5, and for which the closing loop is the following reaction: 2HSO3−(g)
∆H cond
a
These compounds have had a checkered history. Crystalline compounds, originally thought to be NaHSO3 and KHSO3, were identified by Foerster et al. (40) to be metabisulfites, Na2S2O5 and K2S2O5. Simon and Schmidt (41) reported synthesis of MHSO3 salts (M = Rb and Cs), but the contention that these salts contained HSO3− anions was challenged (42) on the grounds that they failed to react with disulfur dichloride in anhydrous tetrahydrofuran to form HCl and M2S4O6. Almost two decades later, Meyer et al. (43) provided convincing spectroscopic identification for the presence of HSO3−, establishing the salts as being RbHSO3 and CsHSO3, respectively. The fact that RbHSO3 and CsHSO3 can be isolated while attempts to crystallize MHSO3 (M = Li, Na, K) lead to the formation of M2S2O5,4
2MHSO3(s)
U pot(M2S2O5) + ∆vapH °(H2O, l)
2U pot(MHSO3) + RT
Stability of Sulfites and Metabisulfites Although the bisulfites, MHSO3 (M = Li, Na, and K) are widely believed to be stable compounds, they have never been obtained as solids (13). It appears that the cations, M+ are insufficiently large enough to stabilize the HSO3− ion relative to the metabisulfite, S2O52− ion. The latter ion results from the loss of water from the ion formed by the hydrogenbonding interaction of the two tautomers of HSO3− (39).
M2S2O5(s) + H2O(l)
2 MHSO3(s)
Table 3. Lattice Energies (and Estimated Errors) for M2S2O5 Salts (M = K, NH4) Obtained from Density-Based Equation 4 M
ρ/(g cm᎑3)
Mm(M2S2O5)/ g
UPOT(M2S2O5)/ (kJ mol᎑1)
K
2.3
222
1647(± 58)
NH4
1.8
180
1626(± 57)
NOTE: Error assumed to be 3.5%.
molecular (formula unit) volume, V(CsHSO 3)兾nm 3 = 0.10101, and K2S2O5 (monoclinic, a = 0.6936 nm, b = 0.6166 nm, c = 0.7548 nm, β = 102.62⬚, Z = 2; refs 45, 46) for which V(K2S2O5)兾nm3 = 0.1575. Using our database of ion volumes (4), by subtracting the relevant alkali metal ion volumes from the above volumes, according to eq 3, we conclude that the ion volumes are: V(HSO3−)兾nm3 = 0.0822 and V(S2O52−)兾nm3 = 0.1378.5 These data have been recombined in Table 2 with V(M+) data (4) to provide V(MHSO3) and V(M2S2O5) values for salts for which such values are experimentally unknown, in order to estimate Upot(MHSO3) and Upot(M2S2O5) using the volume-based eq 2.6 The experimental densities, ρ, of K2S2O5 (Mm = 222 g) and of (NH4)2S2O5 (Mm = 180 g), are reported to be 2.3 g cm᎑3 and 1.80 g cm᎑3 respectively (46). Using our new density-based eq 4 (6), the lattice energies (and their possible errors) listed in Table 3 are predicted. It should be noted that the agreement between Upot(K2S2O5) obtained from the independent density consideration is to within less than 1% of that derived from the volume-based results.
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
Research: Science and Education Table 4. ∆H7 (with Estimated Errors) for the Reaction: 2MHSO3(s) → M2S2O5(s) + H2O(l), Parameterized as a Function of ∆Hcond, the Enthalpy Change of Equation 8
we can either: (i) employ Latimer’s Rules (47) and, additionally, assume that,
S °298 (MHCO3 , s ) S °298 (MHSO3 , s) = S °298 (M2 SO3 , s ) S °298 (M2 CO3 , s)
∆H7/(kJ mol᎑1)
M Li
∆Hcond – 487(± 68)
Na
∆Hcond – 478(± 68)
K
∆Hcond – 454(± 66)
Rb
∆Hcond – 438(± 65)
Cs
∆Hcond – 419(± 63)
From the cycle shown in Figure 1 and taking ∆vapH ⬚(H2O, l)兾(kJ mol᎑1) to be 44.0 (25), ∆H7 can be parameterized,3 at 298 K, in terms of ∆Hcond,
∆H 7 = 2Upot (MHSO3 )
(11)
and so generate the results shown in Table 5, averaging to an overall, ∆S7兾(J K᎑1mol᎑1) = 21(±2) or (ii) assume that the entropy change of eq 7, ∆S7, is likely to be similar to the entropy change of a comparable stoichiometric reaction in which a single water molecule is lost, as for example in the dehydration of hydroxides to oxides, that is, ∆S12 (cf. eqs 7 and 12):7 ∆S12
2 MOH(s)
M2O(s) + H2O(l)
(12)
Using eq 13,
− Upot (M2 S2 O5 ) − ∆ vap H °(H2 O, l )
∆S12 = S °298 (M2 O, s) + S °298 (H2 O, l )
(9)
− 2 S °298 (MOH, s)
+ ∆H cond + RT as is shown in Table 4. We have assumed that the errors in ∆vapH ⬚(H2O, l) and RT will be negligible compared to the errors in Upot. Despite large estimated errors, Table 4 shows the trend of increasing stability of MHSO3 salts as the size of the cation increases, in agreement with the observation that while MHSO3 (M = Li, Na, and K) have never been obtained as solids, RbHSO3 and CsHSO3 are stable compounds. If this fact is due to thermodynamic causes, ∆G7 must be negative for M = K, but positive for M = Rb, thus allowing an estimation of ∆Hcond to be made.4 Estimation of ∆Hcond In the absence of entropy data for bisulfites S ⬚298(MHSO3, s) and metabisulfites S ⬚298(M2S2O5, s) we can adopt three strategies for the estimation of ∆S7.4 Since,
∆S 7 = S °298 (M2 S2 O5 , s ) + S °298 (H2O, l )
− 2S °298 (MHSO3 , s)
(13)
∆S12 (≈ ∆S7)兾(J K᎑1mol᎑1) averages to give 19(±3), which is in excellent agreement with the above estimate (Table 6). Finally, the third approach (iii) consists of employing our computational analogue equation for entropy (11, 12) to estimate the entropies of the solids involved using their molecular (formula unit) volumes, Vm, that takes the form, S °298 / ( J K −1 mol −1) = 1360 (Vm / nm 3 ) + 15 (14) and leads (Table 7) to an estimate of ∆S7兾(J K᎑1mol᎑1) of 19. Averaging our three estimates leads ∆S7兾(J K᎑1mol᎑1) of 20 and so accordingly, we can assume that T∆S7兾(kJ᎑1mol᎑1) ≈ 6(±1) at 298 K8 and ∆G7 can therefore be parameterized as a function of ∆Hcond as shown in Table 8. Again, Table 8 leads to results that explain that the dehydration of MHSO3 salts may be favorable for small cations such as Li+, Na+, and K+, but be unfavorable for large cations such as Rb+ and Cs+. If we assume that ∆G7 < 0 for M = K but ∆G7 > 0 for M = Rb,
(10)
∆H cond − 460 (± 66) /(kJ mol −1) < 0
(15)
Table 5. Estimation of ∆S7 Employing Latimer’s Rules and Comparison of MHSO3 Salts with MHCO3 Salts M Li
S⬚298(MHCO3)a/ (J K᎑1 mol᎑1)
S⬚298(M2CO3)a/ (J K᎑1 mol᎑1)
S⬚298(MHCO3)/ S⬚298(M2CO3)
S⬚298(M2SO3)/ (J K᎑1 mol᎑1)
S⬚298(MHSO3)a/ (J K᎑1 mol᎑1)
S⬚298(M2S2O5)b/ (J K᎑1mol᎑1)
∆S7c/ (J K᎑1 mol᎑1)
87
93
0.94
109
102
153
19
Na
104
126
0.83
142
118
186
20
K
111
141
0.79
157
124
201
23
Rb
123
163
0.76
179
136
223
21
Cs
130
177
0.73
193
142
237
23
a
Data from ref 46.
b
Taken to be S⬚298(M2SO3) + S⬚(S) + 2S⬚(O) where S⬚(S) = 36 J K᎑1 mol᎑1c and S⬚(O) = 4 J K᎑1 mol᎑1d; S⬚(S) and S⬚(O) being Latimer parameters.
S⬚298(H2O, l)/(J K᎑1 mol᎑1) = 70 (ref 25), eq 10.
c
∆S7/(J K᎑1 mol᎑1) averages to 21(± 2).
d
JChemEd.chem.wisc.edu • Vol. 80 No. 12 December 2003 • Journal of Chemical Education
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Research: Science and Education Table 6. Estimate of ∆S12 for Hydroxide Dehydration Reactions M
S⬚298(MOH, s)a/ S⬚298(M2O, s)a/ (J K᎑1mol᎑1) (J K᎑1mol᎑1)
∆S12 (≈ ∆S7)b/ (J K᎑1mol᎑1)
Li
42.8
37.6
22
Na
64.5
75.1
16
Tl
88.0
126.0
20
a
From ref 25.
b Sparse data only enables estimate to be made for M = Li, Na, and Tl. S˚⬚298(H2O, l)/(J K᎑1mol᎑1) = 70 (ref 25), eq 13. ∆S12 (≈ ∆S7)/(J K᎑1mol᎑1) averages to 19(± 3).
Table 7. Estimation of Entropy Change, ∆S7, Using the Computational Analogue Equation for Entropy in the Form: Sº298 = kVm + C M
Sº298(M2S2O5, s)a/ Sº298(MHSO3, s)a/ (J K᎑1mol᎑1) (J K᎑1mol᎑1)
Li
207.8
∆S7b/ (J K᎑1mol᎑1)
129.5
19
Na
213.2
132.1
19
K
229.2
140.3
19
Rb
240.1
145.7
19
Cs
253.5
152.4
19
a
Molecular volumes from Table 2. Sº298(H2O, l)/(J K᎑1mol᎑1) = 70 (ref 25), eq 10.
b
Table 8. ∆G7 (and Estimated Errors) Parameterized as a Function of ∆Hcond M
∆G7 /(kJ mol᎑1)
Li
∆Hcond – 493(± 68)
Na
∆Hcond – 484(± 68)
K
∆Hcond – 460(± 66)
Rb
∆Hcond – 444(± 65)
Cs
∆Hcond – 425(± 63)
Notes
NOTE: ∆G7 = ∆H7 –T∆S7 with T∆S7/(kJ mol᎑1) = 6(± 1).
and
∆H H cond − 444 (± 65)/(kJ mol −1) > 0
(16)
leading to the condition that 379 ≤ ∆H cond /(kJ mol −1) ≤ 526
(17)
and indicating that the most probable value of ∆Hcond is
∆H cond /(kJ mol −1) ≈ 453 (± 74)
(18)
Conclusions Thermochemical cycles are very useful in teaching and understanding inorganic chemistry, but the lack of lattice energy data for many ionic compounds could limit their use unless reliable and simple methods to estimate lattice energies are available. This article reviews some recent methods, 1486
based on the molecular volume or the experimental density, that involve simple calculations that can easily be done by any student. Equations 2 and 4 open up new possibilities, not previously available, using readily available starting data, and we have shown how they can be employed to study the bisulfite– metabisulfite system. Table 8 shows that the dehydration of bisulfites to metabisulfites is less favorable as the size of the cation increases, thus explaining that MHSO3 cannot be isolated for M = Li, Na, and K, but that it is stable when M = Rb and Cs. Furthermore, we have estimated the enthalpy change for eq 8, ∆Hcond, and shown that it corresponds to an endothermic process. Nevertheless, we can estimate ∆Hcond only with an uncertainty of 16% (i.e., 74 in 453 kJ mol᎑1), because this system is an unusual one in that it requires us to make three estimates of Upot [i.e., two Upot(MHSO3) and Upot(M2S2O5)]. Normally for more typical thermochemical cycles (requiring less lattice energy estimates to be made) errors would not accumulate to the same extent. With eqs 2 and 3, as well as tabulated ion volumes (4), many interesting questions can be posed to students. For example, why the hypothetical allotropic form of oxygen, dioxygenyl superoxide (O2+O2−), is unstable with respect to O2 (the ionization energy and electron affinity of O2 are also necessary); or why ionic ammonium hydroxide would be unstable with respect to gaseous water and ammonia molecules (the proton affinities of NH3 and OH− are necessary). For new ions, whose ion volumes are unknown, there are two approaches by which single ion volumes can usually be estimated. The first involves use of extrapolation–interpolation techniques (for examples of this see footnotes 176 and 177 in ref 7 and footnotes 121 and 129 in ref 48). Secondly, correlation of volumes with other size parameters (e.g., covalent radii) can be an option (4).
1. The relevant volume-based and density-based equations (5) are as follows: Upot = AI(2I兾Vm)1/3, where A = 121.39 kJ mol᎑1 nm (note error in formula, eq 4 given in ref 24 in Scheme 1) and Upot = B[I 4(ρ兾Mm)]1/3 where B = 1291.7 kJ mol᎑1 cm and ρ is the density (in g cm᎑3) and Mm is the chemical formula mass (in g). 2. One of the advantages of the database of ion volumes established in ref 4 and further ratified in the studies described in ref 8 is that it permits estimation of unknown volumes of ions. For example, no anhydrous salts of S2+ currently exist, yet we can conjecture that V(SN+) < V(S2+) < V(S2N+). Taking values form our database (4) (Table 6): V(SN+)兾nm3 = 0.032 and V(S2N+)兾nm3 = 0.060, We can predict that V(S2+) should not be far from 0.046 nm3. More recently Liebman et al. (49) have devised an “Isozahlic Rule” employing the fact that isomeric salts should have approximately the same molecular (formula unit) volume, Vm, (e.g., since V(S2N+ O2−) ≈ Vm(S2+ NO2−) by the Isozahlic Rule and we know V(S2N+) (Table 6; ref 4 ), V(O2−) and V(NO2−) (Table 5; ref 4 ) from our database, then using eq 5 of ref 4, we have V(S2+) ≈ V(S2N+) + V(O2−) − V(NO2−) ≈ 0.051 nm3, so confirming the earlier conjecture as to the approximate magnitude of V(S2+). 3. Note that in the publications (29, 30) of Klapötke’s Munich group eq 2 is incorrectly transcribed although correctly employed mathematically so that conclusions are sound. 4. The enthalpy, entropy, and free energy changes that arise from the reaction in eq 7 are designated with the subscript 7. This
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
Research: Science and Education form was adapted for convenience as these terms appear in many equations and often in the text. 5. A crystal structure has been reported (46) also for orthorhombic (NH4)2 S2O5 (a = 0.7133 nm, b = 0.6085 nm, c = 1.5062 nm, Z = 4) for which V[(NH4)2S2O5]兾nm3 = 0.1634, leading to V(S2O52−)兾nm3 = 0.1214. Since the NH4+ ion possesses one of the most variable volumes from compound to compound, the value of V(NH4+) (4) is regarded as unreliable. 6. Since the maximum error anticipated in these predictions is of the order of 3.5 % (see table 7, ref 4), this value is selected and used to estimate the possible errors. (See tables of results). 7. The entropy change that arises from the reaction in eq 12, which approximates that involved in reaction 7, is designated with the subscript 12. This form was adapted for convenience. 8. It should be noted here that in forming the T∆S7 term from ∆S7 at 298 K, changing units from (J K᎑1 mol᎑1) to (kJ mol᎑1) necessitates multiplication by 298兾1000 or 0.298, so considerably reducing the contribution of any error present to the overall thermodynamics as measured by ∆G.
Literature Cited 1. Tudela, D. J. Chem. Educ. 1996, 73, A225. 2. Kildahl, N. K. J. Chem. Educ. 1995, 72, 423. 3. Johnson, D. A. Some Thermodynamic Aspects of Inorganic Chemistry, 2nd ed.; Cambridge University Press: Cambridge, United Kingdom, 1982. 4. Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J; Glasser, L. Inorg. Chem. 1999, 38, 3609. 5. Jenkins, H. D. B.; Glasser, L. J. Am. Chem. Soc. 2000, 122, 632. 6. Jenkins, H. D. B.; Tudela, D.; Glasser, L. Inorg. Chem. 2002, 41, 2364. 7. Brownridge, S.; Krossing, I.; Passmore, J.; Jenkins, H. D. B.; Roobottom, H. K. Coord. Chem. Rev. 2000, 197, 397. 8. Marcus, Y.; Jenkins, H. D. B.; Glasser, L. J. Chem. Soc., Dalton Trans. 2002, 3795. 9. Waddington, T. C. Adv. Inorg. Chem. Radiochem. 1959, 1, 157. 10. Dasent, W. E. Inorganic Energetics—An Introduction, 2nd ed.; Cambridge University Press: Cambridge, United Kingdom, 1982. 11. Jenkins, H. D. B.; Glasser, L. Inorg. Chem., in press. 12. Glasser, L.; Jenkins, H. D. B. Thermochim. Acta, submitted for publication. 13. Tudela, D. J. Chem. Educ. 2000, 77, 830. 14. Busing, W. R., WMIN Program; ORNL-5747, Oak Ridge National Laboratory, TN, April 1981 (revision of March 1984). 15. Gale, J. D. GULP—General Utility Lattice Program, Imperial College兾Royal Institution of Great Britain, 1997. 16. Gale, J. D. J. Chem. Soc., Faraday Trans. 1997, 93, 629. 17. Pratt, K. F. Rationalization of Lattice Energy Calculations with Particular Reference to Complex Salts. Ph.D. Thesis, University of Warwick, United Kingdom, 1978. 18. Jenkins, H. D. B.; Pratt, K. F. Comp. Phys. Commun. 1980, 21, 257. 19. Kapustinskii, A. F. Quart. Rev. Chem. Soc. (London) 1956, 10, 283.
20. Jenkins, H. D. B.; Thakur, K. P. J. Chem. Educ. 1979, 56, 576. 21. Roobottom, H. K.; Jenkins, H. D. B.; Passmore, J.; Glasser, L. J. Chem. Educ. 1999, 76, 1570. 22. Mallouk, T. E.; Rosenthal, G. L.; Muller, G.; Busasco, R.; Bartlett, N. Inorg. Chem. 1984, 23, 3167. 23. Glasser, L. Inorg. Chem. 1995, 34, 4935. 24. See footnote 2 in Jenkins, H. D. B.; Tudela, D.; Glasser, L. Inorg. Chem. 2002, 41, 2364. 25. Handbook of Chemistry and Physics, 80th ed.; Lide D. R., Ed.; CRC Press LLC: Boca Raton, FL, 1999. 26. Jenkins, H. D. B.; Glasser, L. Inorg. Chem. 2002, 41, 4378. 27. Tudela, D.; Diaz, M.; Alvaro, D. A.; Ignacio, J.; Seijo, L.; Belsky, V. K. Organometallics 2001, 20, 654. 28. Vij, V.; Wilson, W. W.; Vij, V.; Tham, F. S.; Sheehy, J. A.; Christe, K. J. Am. Chem. Soc. 2001, 123, 6308. 29. Hammerl, A.; Klapötke, T. M.; Noth, H.; Warchold, M. Inorg. Chem. 2001, 40, 3570. 30. Klapötke, T. M.; Rienäcker, C. M.; Zewen, H. Z. Anorg. Allg. Chem. 2002, 628, 2239. 31. Dixon, D. A.; Feller, D.; Christe, K. O.; Wilson, W. W.; Vij, A.; Vij, V.; Jenkins, H. D. B. J. Am. Chem. Soc., in press. 32. Christe, K. O.; Vij, A.; Wilson, W. W.; Vij, V.; Dixon, D. A.; Feller, D.; Jenkins, H. D. B. Chem. in Britain 2003, 39, 17. 33. Christe, K. O.; Jenkins, H. D. B. J. Am. Chem. Soc. 2003, 125, 9457. 34. Lehman, G. J.; Elliott, H. St. A.; Jenkins, H. D. B.; Schrobilgen, G. J., to be submitted for publication. 35. Du, H.; Jenkins, H. D. B.; Passmore, J.; Schriver, M. J., to be submitted for publication. 36. Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J. Inorg. Chem., 2003, 42, 2886. 37. Fortes, A. D.; Vocaldo, L; Brodholt.; Wood, I. G.; Jenkins, H. D. B. J. Chem. Phys. 2001, 115, 7006. 38. Marcus, Y.; Jenkins, H. D. B.; Glasser, L. J. Chem. Soc., Dalton Trans, 2002, 3795 39. Maylor, R.; Gill, J. B.; Goodall, D. C. J. Chem. Soc., Dalton Trans. 1972, 2001. 40. Foerster, F.; Brosche, A.; Birberg–Scholtz, C. Z. Phys. Chem. 1924, 110, 435. 41. Simon, A.; Schmidt, W. Z. Elektrochem. 1960, 64, 737. 42. Schmidt, M.; Wirwoll, B. Z. Anorg. Allg. Chem. 1960, 303, 184. 43. Meyer, B.; Peter, L.; Shaskey-Rosenlund, C. Spectrochim. Acta 1979, A35, 345. 44. Johansson, L-G.; Lundquits, O.; Vannerberg, N-G. Acta Cryst. 1980, 36B, 2523. 45. Lindquist, I.; Mörtsell, M. Acta Cryst. 1957, 10, 406. 46. Landolt–Börnstein, New Series, Group III /7b 3; Crystal Structure Data of Inorganic Compounds; Key Element S, Se, Te; Springer Verlag: Berlin, Heidelberg, New York, 1982. 47. Latimer, W. M. Oxidation Potentials, 2nd ed.; Prentice Hall, Inc.: Englewood Cliffs, NJ, 1961. 48. Cameron, T. S.; Deeth, R. J.; Dionne, I.; Du, H.; Jenkins, H. D. B.; Passmore, J.; Roobottom, H. K. Inorg. Chem. 2000, 39, 561 49. Liebman, J. L.; Lee, J.; Glasser, L.; Jenkins, H. D. B., to be submitted for publication.
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