Chapter 1 Introduction
1.4 Analytic, Testing and Calculation Methods
1.4.20 Calculation of Heat of formation
Due to the highly energetic character of new HEDMs, bomb calorimetric measurements oftentimes can only be performed with very small amounts, consequently doubtful combustion energies are obtained. Therefore extended computational studies were accomplished, which are presented in the following. In earlier times ab initio calculations based on the ideal gas phase reaction enthalpy (method 1) were used. However, it has been shown that by using the atomization energy method (method 2) more precise results are obtained. All calculations were carried out using the Gaussian G03W (revision B.03) program package.[115]
Method 1
The method is explained in the following by computing the heat of formation of triaminoguanidinium dinitramide (TAG+DN–, Chapter 3). The structure, energy and
frequency calculations were performed at the Hartree-Fock level followed by a Møller- Plesset correlation energy correction, truncated at second order.[116] For all atoms H, C,
N and O an augmented correlation consistent polarized double-zeta basis set was used (aug-cc-pVDZ).[117] The ab initio computational results are summarized in Table 1.6.
The molecular volumes of TAG and DN were taken or back-calculated from literature- known X-ray structural data (Table 1.7).The lattice energies (UL) and lattice enthalpies
(ΔHL) of TAG-DN were calculated according to the equations provided by Jenkins et
Table 1.6 MP2/aug-cc-pVDZ computational results.
Compound Formula Symbol Point group
–E / a.u. zpe [a] /
kcal mol–1 triamino- guanidinium CH9N6+ TAG C3 370.773187 100.8 dinitramide N3O4– DN C2 463.977161 17.9 water H2O C2v 76.260910 13.4 dinitrogen N2 D∞h 109.280650 3.1 carbon dioxide CO2 D∞h 188.169700 7.1 dioxygen O2 D∞h 150.004290 2.0
[a] The zero point energies (zpe) were calculated at the B3LYP/aug-cc-pVDZ level of theory
Table 1.7 Molecular volumes.
Symbol VM / Å3 VM / nm3
NH4+[118] A 21.0 0.021
C(NHNH2)3+[119] TAG 112 0.112
N(NO2)2–[120] DN 89.1 0.089
Table 1.8 Lattice energies (UL) and lattice enthalpies (ΔHL) of TAG-DN.
VM / nm3 UL / kJ mol–1 ΔHL / kJ mol–1
TAG-DN 0.201 504.3 509.3
C(NHNH2)3+(g) + N(NO2)2–(g) + 1.25 O2(g) CO2(g) + 4.5 H2O(g) + 4.5 N2 (g)
With the values given in Table 1.9 the ∆Eel. was calculated (all species in the gas phase):
∆Eel. (TAG-DN) =–2232.8kJ mol–1
The ∆Eel. value was converted into the gas phase reaction enthalpy (Δrxn.H) after
Table 1.9 Gas phase reaction enthalpy
equations
P∆V = Σ νi RT 6.75 RT
∆vibU = Σ νi (zpe)i –166.7 kJ mol–1
∆transU = Σ νi (1.5) RT 10.125 RT
∆rotU = Σ νi (Frot / 2) RT 8 RT
∆rxnH298 / kJ mol–1 –2337.9
Using the lattice enthalpies and the enthalpy of vaporization for water (ΔvapH = 44.0 kJ
mol–1) [122] the enthalpy of combustion according to the following equation was
calculated.
[C(NHNH2)3]+[N(NO2)2]–(s) + 1.25 O2(g) CO2(g) + 4.5 H2O(l) + 4.5 N2 (g)
∆comb.H298(TAG-DN) =– 1828.6 kJ mol–1 (for H2O(g)) = – 2026.6 kJ mol–1 (for H2O(l))
With the known enthalpies of formation of carbon dioxide (ΔfH°298(CO2(g)) = –393.5 kJ
mol–1[122]) and water (ΔfH°298(H2O(g)) = –241.8 kJ mol–1[122]) the enthalpy of formation of
TAG-DN can now be calculated to:
ΔfH°298(TAG-DN(s)) = +347.0 kJ mol–1.
The energy of formation (ΔfUo298) can be obtained from the above calculated enthalpy of
formation according to the following equation with Δn being the change of moles of the gaseous components (Δn(TAG-DN) = –11):
ΔfU°298 =ΔfH°298–Δn RT
ΔfU°(TAG-DN(s)) = +374.3 kJ mol–1 = +1772.7 kJ kg–1
Method 2
This method is explained by computing the heat of formation of 5-aminotetrazolium dinitramide (HAT+DN–, Chapter 3). The enthalpies (H) and free energies (G) were
order to obtain very accurate energies. The CBS models use the known asymptotic convergence of pair natural orbital expressions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. CBS-4 begins with a HF/3- 21G(d) structure optimization; the zero point energy is computed at the same level. It then uses a large basis set SCF calculation as a base energy, and a MP2/6-31+G calculation with a CBS extrapolation to correct the energy through second order. A MP4(SDQ)/6-31+(d,p) calculation is used to approximate higher order contributions. In this study we applied the modified CBS-4M method (M referring to the use of minimal population localization) which is a re-parametrized version of the original CBS-4 method and also includes some additional empirical corrections.[123,124] The enthalpies of the
gas-phase species M were computed according to the atomization energy method (eq. 1) (Tables 1.10–1.12).[125,126]
fH°(g, M, 298) = H(Molecule, 298) – ∑H°(Atoms, 298) + ∑fH°(Atoms, 298) (1)
Table 1.10 CBS-4M results.
p.g. –H298 / a.u. –G298 / a.u. NIMAG
HAT+ Cs 313.534215 313.567694 0 DN– C2 464.499549 464.536783 0 AF Cs 313.533549 313.570115 0 2-MeHAT C1 352.783676 352.821342 0 HTZ+ C2v 258.229407 258.259436 0 H 0.500991 0.514005 0 C 37.786156 37.803062 0 N 54.522462 54.539858 0 O 74.991202 75.008515 0
Table 1.11 Literature values for atomic ΔfH° / kcal mol–1.
Ref. [125] NIST [122]
Table 1.12 Enthalpies of the gas-phase species M. M M ΔfH°(g,M) / kcal mol–1 HAT+ CH4N5 +235.0 DN– N(NO2)2–, N3O4– –29.6 AF+ (H2N)2CN3+, CH4N5+ +235.4 HTZ+ +246.3 2-MeHAT+ C2H6N5+ +221.2
The lattice energies (UL) and lattice enthalpies (ΔHL) were calculated from the
corresponding molecular volumes (Table 1.13) according to the equations provided by Jenkins et al. [118] and are summarized in Table 1.14.
Table 1.13 Molecular volumes.
VM / Å3 VM / pm3
DN– 89 [a] 0.089
[HAT]+ 69 [b] 0.069
[NH4][DN] 110 [120] 0.110
[a]this work, back-calculated from V(ADN) using the molecular volume for NH4+ from the literature; [b] The
molecular volume of [HAT]+ was calculated from the molecular volume of [HAT][NO3]– VM(NO3–).
Table 1.14 Lattice energies and lattice enthalpies.
VM / nm3 UL / kJ mol–1 ΔHL / kJ mol–1 ΔHL / kcal mol–1
[HAT][DN] 0.172 525.6 530.6 126.8
With the calculated lattice enthalpies the gas-phase enthalpies of formation (Table 1.12) were converted into the solid state (standard conditions) enthalpies of formation (Table 1.15). These molar standard enthalpies of formation (ΔHm) were used to
calculate the molar solid state energies of formation (ΔUm) according to eq. (2).
The enthalpy of formation of the solid species M (ΔfH°(s,M)) was calculated to be
+78.6 kcal mol–1.
Table 1.15 Solid state energy of formation (ΔfU°)
ΔfH°(s) / kcal mol–1 Δn ΔfU°(s) / kcal mol–1 M / g mol–1 ΔfU°(s) / kJ kg–1 [HAT][DN] +78.6 –8 83.3 192.1 +1813.3