Cohesive Energies and Enthalpies: Complexities, Confusions, and

Article pubs.acs.org/IC

Cohesive Energies and Enthalpies: Complexities, Confusions, and Corrections Leslie Glasser*,† and Drew A. Sheppard‡ †

Nanochemistry Research Institute, Department of Chemistry, Curtin University, GPO Box U1987, Perth, Western Australia 6845, Australia ‡ Hydrogen Storage Research Group, Fuels and Energy Technology Institute, Department of Physics and Astronomy, Curtin University, GPO Box U1987, Perth, Western Australia 6845, Australia S Supporting Information *

ABSTRACT: The cohesive or atomization energy of an ionic solid is the energy required to decompose the solid into its constituent independent gaseous atoms at 0 K, while its lattice energy, Upot, is the energy required to decompose the solid into its constituent independent gaseous ions at 0 K. These energies may be converted into enthalpies at a given temperature by the addition of the small energies corresponding to integration of the heat capacity of each of the constituents. While cohesive energies/enthalpies are readily calculated by thermodynamic summing of the formation energies/ enthalpies of the constituents, they are also currently intensively studied by computational procedures for the resulting insight on the interactions within the solid. In supporting confirmation of their computational results, authors generally quote “experimental” cohesive energies which are, in fact, simply the thermodynamic sums. However, these “experimental” cohesive energies are quoted in many different units, atom-based or calorimetric, and on different bases such as per atom, per formula unit, per oxide ion, and so forth. This makes comparisons among materials very awkward. Additionally, some of the quoted values are, in fact, lattice energies which are distinctly different from cohesive energies. We list large numbers of reported cohesive energies for binary halides, chalcogenides, pnictogenides, and Laves phase compounds which we bring to the same basis, and identify a number as incorrectly reported lattice energies. We also propose that cohesive energies of higher-order ionic solids may also be estimated as thermodynamic enthalpy sums.

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INTRODUCTION The cohesive or atomization energy of an ionic solid is the energy required to decompose the solid into its separate and independent atoms, each at 0 K; cohesive enthalpy adds the small extra energy to convert each of the materials to a desired temperature, such as 25 °C (298 K).1 From a thermodynamic standpoint, the cohesive energy/enthalpy is the sum of the negative formation energy/ enthalpy of the solid plus the formation energies/enthalpies of the independent atoms.2 This summation is illustrated as a Born− Haber−Fajans cycle in Figure 1. (It should be noted that the formation energy of the solid will include any effects of covalency.3) Provided the formation enthalpy of the solid is known, its cohesive enthalpy may readily be calculated since the formation enthalpies of the gaseous atoms are generally well-established. For example,4−6 at 298 K:

However, instead of this simple thermodynamic relation, current determinations of cohesive energy generally focus on theoretical calculations,9 based on atom−atom potential functions10−13 or quantum methods14 such as density functional theory (DFT),15 thereby developing insight into the interactions within the solid material which macroscopic thermodynamics cannot provide. In order to confirm the accuracy of the theoretical calculations, it is also customary to list “experimental” cohesive energies for comparison. Consequently, the literature contains many lists of cohesive energy. Unfortunately, these values are couched in many disparate ways, with energy listed in the form of various atom-based (Rydberg, Hartree, or eV) or calorimetric units (kcal mol−1 or kJ mol−1). Added to this motley collection of units, the values may be listed per mole of atoms, per mole of formula units, per mole of oxide ion in a formula, and doubtless on other bases. Furthermore, it is often not possible to determine whether the values are actually energies (referred to 0 K) or enthalpies (referred to 298 K). Thus, there is no common basis by which the energies of different materials may be compared with one another.

Δcoh H(NaCl, s)/kJ mol−1 = −Δf H(NaCl, s) + Δf H(Na, g) + Δf H(Cl, g) = 411.1 + 107.5 + 121.3 = 639.9

Additionally, von Szentpaly has found7,8 linear correlations between the standard cohesive enthalpies of gas-phase diatomic and triatomic molecules and their corresponding solids. © XXXX American Chemical Society

Received: April 29, 2016

A

DOI: 10.1021/acs.inorgchem.6b01056 Inorg. Chem. XXXX, XXX, XXX−XXX

B

983.0 1061.9 911.8 929.6 1 1 1 1 983.0 1061.9 911.8 929.6 983.0 1061.9 911.8 929.6 249.2 249.2 249.2 249.2 O(g) O(g) O(g) O(g) Ba(g) Ca(g) Co(g) Fe(g) BaO CaO CoO FeO 3079 3475 3996 3928

−553.8 −634.9 −237.9 −265.1

180.1 177.8 424.7 415.3

1.15 797.0 1 797.0 918.9 249.2 O(g) Ni(g) NiO 8.26

−239.7

430.0

0.91 845.7 1 845.7 771.8 249.2 O(g) Na(g) Na2O 0.6442

−415.1

107.5

0.98 0.99 0.99 0.99 0.99 0.99 869.0 771.9 743.0 698.4 645.9 651.1 1 1 1 1 1 1 869.0 771.9 743.0 698.4 645.9 651.1 855.6 763.5 735.7 688.9 639.9 647.0 79.4 79.4 79.4 121.3 121.3 121.3 F(g) F(g) F(g) Cl(g) Cl(g) Cl(g) 159.3 107.5 89.0 159.3 107.5 89.0 Li(g) Na(g) K(g) Li(g) Na(g) K(g) −616.9 −576.6 −567.4 −408.3 −411.1 −436.7

enthalpy sum/n·cohesive E n·cohesive E n cohesive E enthalpy sum

value/kJ mol‑1

ΔfH(X, g) X(g) ΔfH(M, g) M(g) ΔfH/kJ mol‑1 material

LiF NaF KF LiCl NaCl KCl 0.331 0.294 0.283 0.266 0.246 0.248

ref 14 Hartree Hartree Hartree Hartree Hartree Hartree ref 17 Rydberg ref 18 eV ref 19 kJ kJ kJ kJ

LATTICE ENERGY AND ENTHALPY An additional complication occurs through rather common and unfortunate confusion between cohesive energy and the related concept of lattice energy. Lattice (potential) energy, Upot, of an ionic solid is the total energy required to disrupt the solid to form independent gaseous ions. This confusion exists because the nomenclature was not firmly established in the earlier literature. Unlike cohesive energies, values of Upot cannot be calculated directly from experimental data, because free ions are not produced

cohesive E

■

ref and energy unit

In order to remedy this problem, we have collected a large set of “experimental” cohesive energies, and converted them to the common the basis of “kJ per mole of formula units” (Table 1). We regard this basis as more appropriate than “kJ per mole of atoms” which hides the individual atomic contributions in an amorphous average. Supporting Information Table S1 uses the same data, sorted into alphabetical order of the material’s chemical formula, permitting the desirable ready comparison. We also report the cohesive energies of the alkaline-earth halides calculated from the thermodynamic sum (Table S3) so that they are available for easy comparison to computational approaches. It should be noted that although cohesive energy (and lattice energy) should technically be reported at a temperature of 0 K, it is commonly reported at 298.15 K. In these instances the enthalpy changes between 0 and 298.15 K of the reactants and products are assumed to be equivalent and, hence, to cancel each other. Technically, the zero point energy of the solid should also be accounted for but is usually assumed to have a negligible contribution. This assumption becomes less valid as the atomic mass of the constituents decreases, such as for hydrides. Since low-temperature heat capacity data and zero point energies of the alkaline-earth halides are generally unavailable, the cohesive energies listed in Table S3 do not include these effects. Figure 2 plots the cohesive energies listed in Table 1 against the corresponding thermodynamic enthalpy sum. The close correspondence between the values is clear.

atom contributions/kJ mol‑1

Table 1. Published “Experimental” Cohesive Energies (Column 2) in the Energy Units Listed in Column 1, Followed by the Formation Enthalpy of the Ionic Solid and the Enthalpies of the Contributing Atoms, Finally Summed in the Column Headed “Enthalpy Sum”a

Figure 1. Cohesive enthalpy Born−Haber−Fajans cycle for the decomposition of a solid, MpXq, formed from the elements pM(s) plus q /2X2(g), involving sublimation (subl) and dissociation (diss) into gaseous atoms.

1.00 1.00 1.00 1.00

Article

Inorganic Chemistry


MgO MnO NiO SrO GaAs GaN TiO2 VO2 RuO2 SnO2

Na2O MgO Al2O3 K2O CaO TiO2 TiO Ti2O3 V2O3 Cr2O3 CrO2 FeO Fe3O4 Fe2O3 SrO Nb2O5 ZrO2 MoO2 BaO ZnS ZnSe ZnTe CdS CdSe CdTe AlP AlAs AlSb

3898 3813 4083 3262

6.52 8.96

19.9 18.17 15.07 14.4

9.1 10.34 10.65 8.17 11.01 9.93 12.88 11.11 10.35 9.26 7.74 9.68 8.69 8.28 10.41 9.51 11.4 9.05 10.18

146.6 124.5 106.3 131.6 113.6 95.8 198.0 178.9 165.0

kJ kJ kJ kJ ref 20 eV eV ref 21 eV eV eV eV ref 22 eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV ref 23 kcal kcal kcal kcal kcal kcal kcal kcal kcal

material

cohesive E

ref and energy unit

Table 1. continued

Ga(g) Ga(g) Ti(g) V(g) Ru(g) Sn(g) Na(g) Mg(g) Al(g) K(g) Ca(g) Ti(g) Ti(g) Ti(g) V(g) Cr(g) Cr(g) Fe(g) Fe(g) Fe(g) Sr(g) Nb(g) Zr(g) Mo(g) Ba(g) Zn(g) Zn(g) Zn(g) Cd(g) Cd(g) Cd(g) Al(g) Al(g) Al(g)

−71.0 −156.8 −944.7 −713.6 −305.0 −577.4 −415.1 −601.6 −1675.7 −363.2 −634.9 −944.7 −542.7 −1520.9 −1218.8 −1137.3 −581.6 −265.1 −1115.5 −824.8 −592.0 −1898.3 −1100.3 −587.9 −553.8 −203.0 −177.0 −117.0 −161.9 −144.8 −100.8 −111.7 −116.3 −50.4

M(g) Mg(g) Mn(g) Ni(g) Sr(g)

−601.6 −385.2 −239.7 −592.0

ΔfH/kJ mol

‑1

C

130.4 130.4 130.4 111.8 111.8 111.8 330.0 330.0 330.0

107.5 147.1 660.0 178.0 177.8 469.9 469.9 939.7 1028.4 794.3 397.1 415.3 1246.0 830.7 164.0 1445.7 608.8 659.0 180.1

469.9 514.2 642.7 301.2

272.0 272.0

147.1 282.4 430.0 164.0

ΔfH(M, g)

S(g) Se(g) Te(g) S(g) Se(g) Te(g) P(g) As(g) Sb(g)

O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g) O(g)

O(g) O(g) O(g) O(g)

As(g) N(g)

O(g) O(g) O(g) O(g)

X(g)


278.8 227.1 209.5 278.8 227.1 209.5 316.4 278.4 262.3

249.2 249.2 747.5 249.2 249.2 498.3 249.2 747.5 747.5 747.5 498.3 249.2 996.7 747.5 249.2 1245.9 498.3 498.3 249.2

498.3 498.3 498.3 498.3

278.4 472.7

249.2 249.2 249.2 249.2

ΔfH(X, g)

612.2 534.5 456.9 552.5 483.7 422.1 758.1 724.7 642.7

771.8 997.9 3083.2 790.4 1061.9 1913.0 1261.7 3208.1 2994.7 2679.2 1477.1 929.6 3358.2 2403.0 1005.2 4589.8 2207.4 1745.2 983.0

1913.0 1726.1 1446.1 1376.9

621.4 901.5

997.9 916.8 918.9 1005.2

enthalpy sum

613.4 520.9 444.8 550.6 475.3 400.8 828.4 748.5 690.4

878.0 997.7 1027.6 788.3 1062.3 958.1 1242.8 1072.0 998.6 893.5 746.8 934.0 838.5 798.9 1004.4 917.6 1100.0 873.2 982.2

1920.1 1753.2 1454.1 1389.4

629.1 864.5

997.9 916.8 918.9 1005.2

cohesive E

value/kJ mol‑1 n

1 1 1 1 1 1 1 1 1

1 1 3 1 1 2 1 3 3 3 2 1 4 3 1 5 2 2 1

1 1 1 1

1 1

1 1 1 1

613.4 520.9 444.8 550.6 475.3 400.8 828.4 748.5 690.4

878.0 997.7 3082.8 788.3 1062.3 1916.2 1242.8 3215.9 2995.9 2680.4 1493.6 934.0 3353.9 2396.7 1004.4 4588.0 2199.9 1746.4 982.2

1920.1 1753.2 1454.1 1389.4

629.1 864.5

997.9 916.8 918.9 1005.2

n·cohesive E

1.00 1.03 1.03 1.00 1.02 1.05 0.92 0.97 0.93

0.88 1.00 1.00 1.00 1.00 1.00 1.02 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 0.98 0.99 0.99

0.99 1.04

1.00 1.00 1.00 1.00

enthalpy sum/n·cohesive E

Inorganic Chemistry Article


InSb InAs GaSb GaAs InP

64 66 69 76 77

142.5 543.6 284.7 464.3 434.9 411.1 660.3 351.0 657.5 517.5 531.7 353.4 412.0

ref 25 kJ kJ kJ kJ kJ kJ kJ kJ kJ kJ kJ kJ kJ

D

−10.9 −15.5 −11.7 −10.8 −26.7 −54.4 −73.8 −73.2 −6.9 −7 −9 −18.8 −11.2 Mg(g) Nb(g) Mg(g) U(g) Sc(g) Ce(g) Ce(g) Ca(g) La(g) Nb(g) Ta(g) Mg(g) Sc(g)

In(g) In(g) Ga(g) Ga(g) In(g)

−30.5 −58.6 −41.8 −71.0 −69.3

M(g) Ga(g) Ga(g) Ga(g) In(g) In(g) In(g)

−114.6 −71.0 −41.8 −69.3 −58.6 −30.5

ΔfH/kJ mol

‑1

147.1 722.8 147.1 533.0 377.7 423.0 423.0 177.8 425.9 722.8 782.5 147.1 377.7

243.3 243.3 272.0 272.0 243.3

272.0 272.0 272.0 243.3 243.3 243.3

ΔfH(M, g)

Zn(g) Fe(g) Cu(g) Fe(g) Co(g) Al(g) Ir(g) Al(g) Ir(g) Cr(g) Cr(g) Ni(g) Fe(g)

Sb(g) As(g) Sb(g) As(g) P(g) Laves Phases

P(g) As(g) Sb(g) P(g) As(g) Sb(g)

X(g)


130.4 415.3 338.0 415.3 424.7 330.0 665.3 330.0 665.3 397.1 397.1 430.0 415.3

262.3 278.4 262.3 278.4 316.4

316.4 278.4 262.3 316.4 278.4 262.3

ΔfH(X, g)

418.8 1569.0 834.8 1374.5 1253.8 1137.4 1827.3 911.0 1763.3 1524.1 1585.8 1025.9 1219.6

536.1 580.3 576.1 621.4 629.0

703.0 621.4 576.1 629.0 580.3 536.1

enthalpy sum

142.5 543.6 284.7 464.3 434.9 411.1 660.3 351.0 657.5 517.5 531.7 353.4 412.0

267.8 276.1 288.7 318.0 322.2

727.2 647.3 579.9 663.6 603.8 537.6

cohesive E

value/kJ mol‑1 n

3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2

1 1 1 1 1 1

427.5 1630.8 854.1 1392.9 1304.7 1233.3 1980.9 1053.0 1972.5 1552.5 1595.1 1060.2 1236.0

535.6 552.3 577.4 636.0 644.3

727.2 647.3 579.9 663.6 603.8 537.6

n·cohesive E

0.98 0.96 0.98 0.99 0.96 0.92 0.92 0.87 0.89 0.98 0.99 0.97 0.99

1.00 1.05 1.00 0.98 0.98

0.97 0.96 0.99 0.95 0.96 1.00


a This result is to be compared with the cohesive energy converted16 to “kJ mol−1 of formula unit” in the penultimate column. The integer factor n corrects from the particular formula unit used for column 2. The final column compares the thermodynamic enthalpy summation with the published “experimental” cohesive energy. The references (listed in the initial column) often derive their data from earlier sources. bConversion factors applied:16 1 hartree = 2 Ry = 2625.5 kJ mol−1; 1 eV = 96.4869 kJ mol−1; 1 kcal mol−1 = 4.184 kJ mol−1.

MgZn2 NbFe2 MgCu2 UFe2 ScCo2 CeAl2 CeIr2 CaAl2 LaIr2 NbCr2 TaCr2 MgNi2 ScFe2

GaP GaAs GaSb InP InAs InSb

173.8 154.7 138.6 158.6 144.3 128.5

kcal kcal kcal kcal kcal kcal ref 24 kcal kcal kcal kcal kcal

material

cohesive E

ref and energy unit

Table 1. continued

Inorganic Chemistry Article


Article

Inorganic Chemistry

Table 2. Lattice Energies, Upot’s, Derived from Tables of Incorrectly Identified Cohesive Energies29,31−34a value/kJ mol‑1 apparent cohesive E/kcal mol‑1

material

apparent cohesive E

932 839 796 786 896 764 720 679 780 726 693 660 679 667 629 247 203 194 180 220 187 179 167 194 170 163 154 186 164 157 149 147 125 106 132 114 96 198 179 165 174 155 139 159 144 129 173 169

MgO CaO SrO BaO MgS CaS SrS BaS MgSe CaSe SrSe BaSe CaTe SrTe BaTe LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI ZnS ZnSe ZnTe CdS CdSe CdTe AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb TlCl35 TlBr35

3899.5 3510.4 3330.5 3288.6 3748.9 3196.6 3012.5 2840.9 3263.5 3037.6 2899.5 2761.4 2840.9 2790.7 2631.7 1033.4 849.4 811.7 753.1 920.5 782.4 748.9 698.7 811.7 711.3 682.0 644.3 778.2 686.2 656.9 623.4 615.0 523.0 443.5 552.3 477.0 401.7 828.4 748.9 690.4 728.0 648.5 581.6 665.3 602.5 539.7 724.6 707.2

Figure 2. Plot of published cohesive energy versus thermodynamic enthalpy sum using the data in Table 1 (blue ▲) for 81 ionic solids (some duplicated). The very close linear relation, ΔcohH = 1.001 × Sum + 16 with correlation coefficient R2 = 0.997, demonstrates that the published “experimental” cohesive energies are indeed simple thermodynamic sums. The data labeled + are for the ternary ionic solids considered in detail below (see Table 3).

Figure 3. Lattice enthalpy Born−Haber−Fajans cycle for the decomposition of an ionic solid, MpXq, formed from the elements pM(s) plus q/2X2(g), involving sublimation (subl), dissociation (diss), ionization to the cation (IP = ionization potential), and electron capture to form the anion (EA = electron affinity). The cycle is drawn for an enthalpy (ΔH) determination, where the generally small term mRT must be added to convert to lattice enthalpy, Upot, from energy.1 The ionization enthalpy is positive, but electron affinity is generally negative; hence, the formation enthalpies of cations are positive, but those of anions may be negative.

Upot(calc)

Upot(calc)/ apparent cohesive E

3795 3414 3217 3029

0.97 0.97 0.97 0.92

3071 2858 2736 2611 2721

0.94 0.94 0.94 0.95 0.96

1030 834 788 730 910 769 732 682 808 701 671 632 774 680 651 617

1.00 0.98 0.97 0.97 0.99 0.98 0.98 0.98 1.00 0.99 0.98 0.98 0.99 0.99 0.99 0.99

a

The authoritative Upot values in the penultimate column are from Jenkins and Roobottom.30

in the gas phase, and are rather found via a Born−Haber−Fajans cycle (Figure 3), where the energy to form a solid from its elements is compared with the energy required to form the gaseous ions from those same elements. (It should be noted that the formation energy of the solid will include any effects of covalency.) E


Article

F

0.95

0.83

1080.7 1

1

1080.7

1172.8

418.9 243.3 In(g)

Te(g) 338.0 Cu(g) −25.6

177.8

258.3 kcal

CuInTe2

338.0

Ca(g)

280.3 kcal

CuCaTe2

−40.2

Cu(g)

Te(g)

418.9

1025.8

974.9

1302.1 1099.4 454.2 243.3

Se(g) 338.0

In(g)

311.2 kcal

CuInSe2

−63.9

Cu(g)

338.0

272.0 Ga(g)

Cu(g) −75.7 CuGaSe2 330.4 kcal

ref 36

1172.8

0.84 1302.1 1

1382.4 1 1382.4 1139.9 454.2 Se(g)

1495.0 Ta(g)

O(g) 177.8

1565.0

Ca(g) −2728.2 CaTa2O6 66.77 eV

ref 38

61.92 eV

ref 37

0.82

0.93 6442.4 1 6442.4 5966.1

5974.5 5793.7 O(g) 177.8

1445.7

Ca(g) CaNb2O6

−2675.2

Nb(g)

X(g) ΔfH(M, g) M(g) ΔfH/kJ mol‑1 material

Table 3. Cohesive Enthalpy Sum per Formula Unit for Ternary Ionic Solidsa

(It will be noted that the value obtained is similar to, but larger, than the value of ΔcohH(NaCl, s) = 639.9 kJ mol−1 reported above.) This thermodynamic enthalpy sum for ΔLH(NaCl, s) may be compared with a computed value of 769 kJ mol−1,4 where it is likely that the thermodynamic enthalpy sum is the more reliable value since it is based on fewer assumptions about the interaction mechanism in the solid. However, not all ions are susceptible to evaluation as independent gaseous species. The most significant are complex ions such as sulfate, SO42−, as well as the fluoride anion, F−, and the oxide anion, O2−, the latter of which is metastable27 and so not subject to exact calculation. The high charge densities of F− and O2− strongly affect their interactions with their neighbors and may be the cause of the unreliability of the evaluations for these ions. Where the formation enthalpy of an ion is not available, it is possible to calculate a value by difference if the lattice enthalpy has been calculated for a solid with known formation enthalpy. As with cohesive energies, Upot may be determined computationally using some energetic potential (such as a Buckingham potential including Coulomb interactions among the charged species28) which more or less fully describes the interactions among the species present, or by a suitable quantum mechanical procedure such as density functional theory (DFT). Such computational procedures have the advantage of providing detailed insight into the interactions present, so they can account for such effects as differing pressures and temperatures and allow for some covalency, perhaps offset by the expense of requiring a known crystal structure.

cohesive E

= +411.1 + 609.4 + ( −233.1) = 787.4

ref and energy unit

+ Δf H(Na +, g) + Δf H(Cl−, g)


ΔL H(NaCl, s)/kJ mol−1 = −Δf H(NaCl, s)

1495.0

ΔfH(X, g)

As an example, we calculate ΔLH for NaCl(s), using a Born− Haber−Fajans enthalpy cycle, as the sum of the (negative) of its formation enthalpy plus the formation enthalpy of the gaseous ions26 at 298 K:

enthalpy sum

value/kJ mol‑1

Figure 4. Plot of apparent “cohesive energies” versus authentic lattice energies30 using the data in Table 2 for 25 ionic solids. The very close linear relation, “Ecoh” = 1.06Upot + 49.2 with correlation coefficient R2 = 0.999, demonstrates that these published listed “experimental cohesive energies” are correctly lattice energies.

cohesive E

n

1

5974.5

n·cohesive E/kJ mol‑1

0.97


Inorganic Chemistry


Article

Demonstrating that the published theoretical cohesive energy (column 2) closely matches the cohesive energy (penultimate column), often to within about 90% (final column). The integer factor n corrects from the particular formula unit used for column 2 (cf. Table 1). bThis value is reported to be aberrant. cThis value has been corrected from −38.9 kJ mol−1 to match the original source.

1100.0 1003.3 556.8 As(g) 130.4

301.2 Sn(g)

Zn(g) −14.9 ZnSnAs2 262.9 kcal

The older literature is rather confusing in its nomenclature, resulting in many lattice energies (producing free ions) being incorrectly described as cohesive (producing free atoms). As an important and much-referenced example, Ladd29 has produced tables of “crystal energy” which are lattice energies but have subsequently been treated as cohesive energies.

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RESULTS AND DISCUSSION Unfortunately, the difference between lattice and cohesive energies may be small for materials with 1:1 charge ratios, such as the alkali halides, so the values are easily confused. However, for ionic solids with larger charges, such as MgO, the discrepancies become obvious. Table 2 contains some data which has been incorrectly classified as cohesive energy together with authoritative lattice energies from Jenkins and Roobottom.30 Figure 4 plots these incorrectly identified cohesive energies against the known lattice energies, and the close linear relation confirms that these data are, in fact, lattice energies. Cohesive Energy and Thermodynamic Enthalpy Sums for Higher-Order Ionic Solids. It is notable that the tables above contain only binary materials. We have been able to find a few calculations by independent authors of cohesive energy for higher-order, such as ternary, ionic solids. Table 3 demonstrates that the thermodynamic enthalpy sum remains a valid measure of the cohesive energy, against which theoretical calculations can be measured, even beyond binary materials. Some of the discrepancies which remain (cf. Figure 2) may be attributed to use of older values of the gaseous atom formation enthalpies in the calculations. For example, Aresti et al.36 use 302.5 (for As(g)), 332.2 (for P(g)), and 196.6 (for Te(g)) in place of the current values4 of 278.5, 316.4, and 209.5, respectively (all in kJ mol−1), but other values are unchanged. Formation enthalpies, if unavailable, may be roughly estimated using Yoder and Flora’s SSA39,40 (simple salt approximation) as the sum of the component materials. Thus, for example Δf H(CaNb2O6 , s)/kJ mol−1 ≅ Δf H(CaO, s) + Δf H(Nb2 O5 , s) = ( −634.9) + (− 1898.3) = − 2533.2

This is 95% of the reported value of −2675.2 kJ mol−1, and it is not unreasonable to add a standard 5% to the estimated value to account for the extra stability of the combined material.

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CONCLUSIONS We have brought a large number of reported “experimental” cohesive energies to the same basis of “kJ per mole of formula units”, and have compared the values obtained with the thermodynamic enthalpy sum for the same materials. We have thereby identified a number of lattice energies incorrectly reported as cohesive energies. We suggest that authors present their cohesive energies on this basis for ease of comparison of the properties of materials. We have also shown that the thermodynamic enthalpy sum is a valid measure of the cohesive energy of ternary ionic solids.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b01056. Tables S1 and S2, which are versions of Tables 1 and 2, sorted alphabetically by chemical formula; Table S3 listing

a

0.91 1100.0 1

1353.1 1 1353.1 632.8 111.8

P(g) Si(g)

Cd(g) CdSiP2 kcal

323.4

−29.9

455.6

632.8 130.4

P(g) Si(g)

Zn(g) ZnSiP2 kcal

336.0

−41.0c

455.6

418.9 243.3 In(g)

AgInTe2 kcal

249.5

−29.4

Ag(g)

284.9

Te(g)

1230.1

1259.8

976.5

1405.8

1043.9

1

1

1405.8

0.91

0.90

0.94 1043.9

0.92 1092.0 418.9 284.9 Ag(g)

Ga(g)

AgGaTe2 kcal

261.0

−33.5

272.0

Te(g)

1040.2 454.2 Se(g) 284.9

243.3 In(g)

Ag(g) −57.8 AgInSe2 kcal

292.5

Ga(g)

AgGaSe2 kcal

348.7

(−106.6)

272.0

454.2 Ag(g)

284.9

Se(g)

1009.3

1223.8

1

1223.8 1

1092.0

0.85

(0.77) (1459.0) 1 1459.0 1117.7

cohesive E enthalpy sum ΔfH(X, g) X(g) ΔfH(M, g) M(g) ΔfH/kJ mol‑1 material cohesive E ref and energy unit

Table 3. continued

b


value/kJ mol‑1

n

n·cohesive E/kJ mol‑1


Inorganic Chemistry

G


Article

Inorganic Chemistry

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(23) Yadav, D. S.; Singh, D. V. Phys. Scr. 2012, 85, 015701. (24) Hilsum, C.; Rose-Innes, A. C. Semiconducting III-V Compounds; Pergamon Press: New York, 1961. (25) Li, C.; Lim Hoe, J.; Wu, P. J. Phys. Chem. Solids 2003, 64, 201− 212. (26) Marcus, Y. Ion Properties; Marcel Dekker: New York, 1997. (27) Harding, J. H.; Pyper, N. C. Philos. Mag. Lett. 1995, 71, 113−121. (28) Buckingham Potential. Wikipedia; https://en.wikipedia.org/ wiki/Buckingham_potential (April 2016). (29) Ladd, M. F. C. J. Chem. Phys. 1974, 60, 1954−1957. (30) Jenkins, H. D. B.; Roobottom, H. K. Lattice Energies. In CRC Handbook of Chemistry and Physics; Haynes, W. M., Ed.; CRC Press: Boca Raton, FL, 2015. (31) Schlosser, H. Phys. Status Solidi B 1993, 179, K1−K3. (32) Verma, A. S. Phys. Status Solidi B 2009, 246, 345−353. (33) Koh, A. K. J. Phys. Chem. Solids 1997, 58, 467−473. (34) Gupta, V. P.; Sipani, S. K. Phys. Status Solidi B 1982, 111, 295− 301. (35) Kavanoz, H. B. High Temp. Mater. Processes 2015, 34, 81−86. (36) Aresti, A.; Garbato, L.; Rucci, A. J. Phys. Chem. Solids 1984, 45, 361−365. (37) Velikokhatnyi, O. I.; Kumta, P. N. J. Power Sources 2012, 202, 190−199. (38) Jacob, K. T.; Rajput, A. J. Alloys Compd. 2015, 620, 256−262. (39) Yoder, C. H.; Flora, N. J. Am. Mineral. 2005, 90, 488−496. (40) Yoder, C. H.; Rowand, J. P. Am. Mineral. 2006, 91, 747−752.

cohesive enthalpies of the alkaline-earth halides (Von Szentpaly7 has a more extensive list for these and related materials); Figure S1, a plot of apparent “cohesive energy” versus thermodynamic enthalpy sum demonstrating that they are not correlated (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: + 61 8 9848-3334. Fax: + 61 8 9266-4699. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS L.G. acknowledges the support of Curtin University in the provision of research facilities and excellent library services. D.A.S. acknowledges the financial support of Curtin University through the provision of an Early Career Research Fellowship.

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REFERENCES

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