About 3 107chemical compounds are known, and it is likely that fH°, S°m, C°P,m, and fG° for most known compounds will never be measured. Several methods have been proposed for estimating thermodynamic properties of a compound for which data do not exist. Chemical engineers often use estimation methods. It’s a lot cheaper and faster to estimate needed unknown thermodynamic quantities than to measure them, and quantities obtained by estimation methods are sufficiently reliable to be useful for many purposes. An outstanding compilation of reliable estimation methods for thermodynamic and transport properties (Chapter 15) of liquids and gases is Prausnitz, Poling, and O’Connell.
Bond Additivity
Many properties can be estimated as the sum of contributions from the chemical bonds. One uses experimental data on compounds for which data exist to arrive at typical values for the bond contributions to the property in question. These bond con-tributions are then used to estimate the property in compounds for which data are un-available. It should be emphasized that this approach is only an approximation.
Bond additivity methods work best for ideal-gas thermodynamic properties and usually cannot be applied to liquids or solids because of the unpredictable effects of intermolecular forces. For a compound that is a liquid or solid at 25°C and 1 bar, the ideal-gas state (like a supercooled liquid state) is not stable. Let Pvpbe the liquid’s vapor pressure at 25°C. To relate observable thermodynamic properties of the liquid at 25°C and 1 bar to ideal-gas properties at 25°C and 1 bar, we use the following isothermal process at 25°C (Fig. 5.13): (a) change the liquid’s pressure from 1 bar to Pvp; (b) reversibly vaporize the liquid at 25°C and Pvp; (c) reduce the gas pressure to zero; (d ) wave a magic wand that transforms the real gas to an ideal gas; (e) compress the ideal gas to P 1 bar. Since the differences between real-gas and ideal-gas prop-erties at 1 bar are quite small, one usually replaces steps (c) , (d ), and (e) with a com-pression of the gas (assumed to behave ideally) from pressure Pvpto 1 bar. Also, step (a) usually has a negligible effect on the liquid’s properties. Thus, knowledge of Hm
of vaporization enables estimates of enthalpies and entropies of the liquid to be found from estimated ideal-gas enthalpies and entropies. Methods for estimation of vapHm are discussed in Prausnitz, Poling, and O’Connell, chap. 7.
Benson and Buss constructed a table of bond contributions to C°P,m,298, S°m,298, and
fH°298for compounds in the ideal-gas state [S. W. Benson and J. H. Buss, J. Chem.
Phys., 29, 546 (1958)]. Addition of these contributions enables one to estimate ideal-gas S°m,298and C°P,m,298values with typical errors of 1 to 2 cal/(mol K) and fH°298 val-ues with typical errors of 3 to 6 kcal/mol. It should be noted that a contribution to
(a)
(b)
(c)
(d)
(e) Liquid at Pvp
Ideal vapor at 0 bar Liquid at 1 bar
Vapor at 0 bar
Ideal vapor at 1 bar Vapor at Pvp
Figure 5.13
Conversion of a liquid at 25°C and 1 bar to an ideal gas at 25°C and 1 bar.
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S°m,298that arises from the symmetry of the molecule must be included to obtain valid results (see the discussion of the symmetry number in Chapter 21).
For example, some bond additivity contributions to fH°298/(kcal/mol) are
COC COH COO OOH
2.73 3.83 12.0 27.0
fH°298/(kcal/mol) of C2H6(g) and C4H10(g) are then predicted to be 2.73 6(3.83)
20.2 and 3(2.73) 10(3.83) 30.1, as compared with the experimental values
20.0 for ethane, 30.4 for butane, and 32.1 for isobutane. Since the fH°298 values are for formation from graphite and H2, the bond-contribution values have built-in allowances for the enthalpy changes of the processes C(graphite) → C(g) and H2(g) → 2H(g).
Bond Energies
Closely related to the concept of bond contributions to fH° is the concept of average bond energy. Suppose we want to estimate H°298 of a gas-phase reaction using molecular properties. We have H°298 U°298 (PV)°298. As noted in Sec. 5.4, the
(PV)° term is generally substantially smaller than the U° term, and H° generally varies slowly with T. Therefore, H°298will usually be pretty close to U°0, the reac-tion’s change in ideal-gas internal energy in the limit of absolute zero. Intermolecular forces don’t contribute to ideal-gas internal energies, and at absolute zero, molecular translational and rotational energies are zero. Therefore U°0 is due to changes in molecular electronic energy and in molecular zero-point vibrational energy (Sec. 2.11).
We shall see in Chapter 20 that electronic energies are much larger than vibrational energies, so it is a good approximation to neglect the change in zero-point vibrational energy. Therefore U°0and H°298are largely due to changes in molecular electronic energy. To estimate this change, we imagine the reaction occurring by the following path:
(5.44) In step (a), we break all bonds in the molecule and form separated atoms. It seems plausible that the change in electronic energy for step (a) can be estimated as the sum of the energies associated with each bond in the reacting molecules. In step (b), we form products from the atoms and we estimate the energy change as minus the sum of the bond energies in the products.
To show how bond energies are found from experimental data, consider the gas-phase atomization process
(5.45) (Atomization is the dissociation of a substance into gas-phase atoms.) We define the average COH bond energy in methane as one-fourth of H°298for the reaction (5.45).
From the Appendix, fH°298 of CH4 is 74.8 kJ/mol. H°298 for sublimation of graphite to C(g) is 716.7 kJ/mol. Hence fH°298of C(g) is 716.7 kJ/mol, as listed in the Appendix. (Recall that fH° is zero for the stable form of an element. At 25°C, the stable form of carbon is graphite and not gaseous carbon atoms.) fH°298of H(g) is listed as 218.0 kJ/mol. [This is H°298for H2(g) → H(g).] For (5.45) we thus have
Hence the average COH bond energy in CH4is 416 kJ/mol.
To arrive at a carbon–carbon single-bond energy, consider the process C2H6(g) → 2C(g) 6H(g). Appendix fH°298values give H°298 2826 kJ/mol for this reaction.
¢H°298 3716.7 41218.02 174.82 4 kJ>mol 1663.5 kJ>mol
1 2
CH41g2 S C1g2 4H1g2
Gaseous reactants S1a2gaseous atoms S1b2 gaseous products
Chapter 5
Standard Thermodynamic Functions of Reaction
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Section 5.10 Estimation of Thermodynamic Properties 167
C–(C)(H)3 C–(C)2(H)2 C–(C)3H C–(C)4 O–(C)(H) O–(C)2 C–(C)(H)2O C–(H)3(O)
41.8 20.9 10.0 0.4 158.6 99.6 33.9 41.8
This H°298is taken as the sum of contributions from six COH bonds and one COC bond. Use of the CH4value 416 kJ mol1for the COH bond, gives the COC bond energy as [2826 6(416)] kJ/mol 330 kJ/mol.
The average-bond-energy method would then estimate the heat of atomization of propane CH3CH2CH3(g) at 25°C as [8(416) 2(330)] kJ/mol 3988 kJ/mol. We break the formation of propane into two steps:
The Appendix fH° data give H°298for the first step as 3894 kJ/mol. We have esti-mated H°298for the second step as 3988 kJ/mol. Hence the average-bond-energy es-timate of fH°298of propane is 94 kJ/mol. The experimental value is 104 kJ/mol, so we are off by 10 kJ/mol.
Some values for average bond energies are listed in Table 19.1 in Sec. 19.1. The COH and COC values listed differ somewhat from the ones calculated above, so as to give better overall agreement with experiment.
The bond-additivity-contribution method and the average-bond-energy method of finding fH°298are equivalent to each other. Each bond contribution to fH°298of a hy-drocarbon is a combination of bond energies and the enthalpy changes of the processes C(graphite) → C(g) and H2(g) → 2H(g) (see Prob. 5.55).
To estimate H°298 for a gas-phase reaction, one uses (5.44) to write H°298
atH°298,re atH°298,pr, where atH°reand atH°pr, the heats of atomization of the reac-tants and products, can be found by adding up the bond energies. Corrections for strain energies in small-ring compounds, resonance energies in conjugated compounds, and steric energies in bulky compounds are often included.
Thus, the main contribution to H° of a gas-phase reaction comes from the change in electronic energy that occurs when bonds are broken and new bonds formed. Changes in translational, rotational, and vibrational energies make much smaller contributions.
Group Additivity
Bond additivity and bond-energy calculations usually give reasonable estimates of gas-phase enthalpy changes, but can be significantly in error. An improvement on bond additivity is the method of group contributions. Here, one estimates thermody-namic quantities as the sum of contributions from groups in the molecule. Corrections for ring strain and for certain nonbonded interactions (such as the repulsion between two methyl groups that are bonded to adjacent carbons and that are in a gauche con-formation) are included. A group consists of an atom in the molecule together with the atoms bonded to it. However, an atom bonded to only one atom is not considered to produce a group. The molecule (CH3)3CCH2CH2Cl contains three C–(H)3(C) groups, one C–(C)4 group, one C–(C)2(H)2 group, and one C–(C)(H)2(Cl) group, where the central atom of each group is listed first.
The group-contribution method requires tables with many more entries than the bond-contribution method. Tables of gas-phase group contributions to fH°, C°P,m, and S°mfor 300 to 1500 K are given in S. W. Benson et al., Chem. Rev., 69, 279 (1969), and S. W. Benson, Thermochemical Kinetics, 2d ed., Wiley-Interscience, 1976. See also N. Cohen and S. W. Benson, Chem. Rev., 93, 2419 (1993). These tables give C°P,m and S°mideal-gas values with typical errors of 1 cal/(mol K) and fH° ideal-gas values with typical errors of 1 or 2 kcal/mol. Some gas-phase group additivity values for
fH°298/(kJ/mol) are
3C1graphite2 4H21g2 S 3C1g2 8H1g2 S C3H81g2
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Group-additivity values for fH°298have been tabulated for solid, for liquid, and for gaseous C-H-O compounds in N. Cohen, J. Phys. Chem. Ref. Data, 25, 1411 (1996). The average absolute errors are 1.3 kcal/mol for gases, 1.3 kcal/mol for liq-uids, and 2.2 kcal/mol for solids. (A few compounds with large errors were omitted in calculating these errors.)
The computer programs CHETAH (www.chetah.usouthal.edu/), NIST Therm/Est (www.esm-software.com/nist-thermest), and NIST Organic Structures and Properties (www.esm-software.com/nist-struct-prop) use Benson’s group-additivity method to estimate thermodynamic properties of organic compounds.
Sign of S°
Now consider S°. Entropies of gases are substantially higher than those of liquids or solids, and substances with molecules of similar size have similar entropies. Therefore, for reactions involving only gases, pure liquids, and pure solids, the sign of S° will usu-ally be determined by the change in total number of moles of gases. If the change in moles of gases is positive, S° will be positive; if this change is negative, S° will be negative;
if this change is zero, S° will be small. For example, for 2H2(g) O2(g) → 2H2O(l), the change in moles of gases is 3, and this reaction has S°298 327 J/(mol K).
Other Estimation Methods
Thermodynamic properties of gas-phase compounds can often be rather accurately calculated by combining statistical-mechanics formulas with quantum-mechanical calculations (Secs. 21.6, 21.7, 21.8) or molecular-mechanics calculations (Sec. 19.13).