Condensed phases internal energy

The vast majority of the reactions carried out in industrial scale batch reactors involve reactants in condensed phases. Since the specific volumes of both liquids and solids are very small, the difference between internal energy and enthalpy for these materials is usually negligible. Thus one often sees the statement that for batch reactions taking place at constant volume ... 

However, for condensed phases, the difference between internal energy and enthalpy is usually negligible ... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

The quasi-equilibrium theory (QET) of mass spectra is a theoretical approach to describe the unimolecular decompositions of ions and hence their mass spectra. [12-14,14] QET has been developed as an adaptation of Rice-Ramsperger-Marcus-Kassel (RRKM) theory to fit the conditions of mass spectrometry and it represents a landmark in the theory of mass spectra. [11] In the mass spectrometer almost all processes occur under high vacuum conditions, i.e., in the highly diluted gas phase, and one has to become aware of the differences to chemical reactions in the condensed phase as they are usually carried out in the laboratory. [15,16] Consequently, bimolecular reactions are rare and the chemistry in a mass spectrometer is rather the chemistry of isolated ions in the gas phase. Isolated ions are not in thermal equilibrium with their surroundings as assumed by RRKM theory. Instead, to be isolated in the gas phase means for an ion that it may only internally redistribute energy and that it may only undergo unimolecular reactions such as isomerization or dissociation. This is why the theory of unimolecular reactions plays an important role in mass spectrometry. 

Now, in a series of closely similar protonation reactions, AS° may be assumed to be invariant, and then variations in AG° will reflect primarily variations in AH°. In condensed phases and for cations of closely similar structure, these will be practically equal to the changes in internal energy, i.e. 

Interestingly, two of the other species in Table 3 are nitrolates, i.e. ethers of a-nitrooximes, an otherwise thermochemically unprecedented class of compounds. We already have briefly discussed one, 3-nitroisoxazoline, and the second is 1-nitroacetaldehyde 0-(l,l-dinitroethyl)oxime (ONo-ld-dinitroethyl acetonitronate), MeC (NOala—O—N=C(N02)Me. The latter acyclic species is a derivative of 1,1-dinitroethanol—we know of the enthalpy of formation of no other a-nitroalcohol or derivative. Nonetheless, we may ask if the two calorimetric data are internally consistent. Consider the condensed phase reaction 47, which involves formal cleavage of the O — bond in the nitroisoxazoline by the C—H bond of the dinitromethane. It is assumed that the isoxazoline has the same strain energy as the archetypal 5-atom ring species cyclopentane and cyclopentene, ca 30 kJ mol . ... 

Similarly, it can be shown that the enthalpy of mixing can be approximated by the internal energy of mixing [see Eq. (2.4)] for condensed phases, which in turn can be related to the mole fractions of the two components, and an interaction energy, a, which also has units of joules ... 

The highly excited states of molecules produced by high-energy radiation that arc chemically important are mainly the ionic states because of the rapidity of internal conversion processes. Primary excitation is relatively unimportant while secondary excitation is quite common. In the condensed phases energy dissipation is very rapid because of colli-sional deactivation, the cage effect, and excitation energy transfer processes all of which act to negate the chemical effects of secondary excitation,... 

Owing to the electron-vibrational interaction in molecules, there is one more possible decay channel for SES. This is the nonradiative relaxation (internal conversion), in which the electron energy is transferred into vibrational energy of molecules (in the condensed phase, into thermal energy of the medium). If the molecule fluoresces, there may also occur fluorescence from the lowest excited state. (According to the empirical rule of Kasha,64 the molecular fluorescence occurs from the lowest excitation level irrespective of the wavelength of the exciting radiation.)... 

Another reactive force field that is dependent on bond-order was developed by van Duin, Dasgupta, Loran, and Goddard [183] for hydrocarbons. The configurational energy is described as the sum of energy contributions from internal modes as well as non-bonding van der Waals and Coulombic interactions, but the parameters of the functions that describe each contribution is dependent upon the bond order of atoms involved in each description. It is assumed that the bond order between an atom pair is dependent on the interatomic separation. While this model has been used to predict bond dissociation energies, heats of formation and structures of simple hydrocarbons, it was not applied to predict condensed phase properties. However, the form of the potential should allow for condensed phase studies. 

In kinetically limited models, the pyrolysis rate is no longer calculated solely from a heat balance at the pyrolysis front. Instead, the rate at which the condensed-phase is volatilized depends on its temperature. This gives a local volumetric reaction rate (kg/m3-s) by assuming that all volatiles escape instantaneously to the exterior gas-phase with no internal resistance, the fuel mass flux is obtained by integrating this volumetric reaction rate in depth. One consequence is that the pyrolysis reaction is distributed spatially rather than confined to a thin front as with heat transfer limited models and the thickness of the pyrolysis front is controlled by decomposition kinetics and heat transfer rates. For a pyrolysis reaction with high activation energy or for very high heat transfer rates, the pyrolysis zone becomes thin, and kinetically limited models tend toward heat transfer limited models. 

Joule s experiments on the free expansion of an ideal gas showed that the internal energy of such a system is a function of temperature alone. For a real gas, this is only approximately true. For condensed phases, which are effectively incompressible, the volume dependence on the change in internal energy is negligible. As a result, the internal energies of liquids and solids are also considered a function of temperature alone. For this reason, the internal energy of a system may loosely be referred to as the thermal energy . 

In the ion sources, the analysed samples are ionized prior to analysis in the mass spectrometer. A variety of ionization techniques are used for mass spectrometry. The most important considerations are the internal energy transferred during the ionization process and the physico-chemical properties of the analyte that can be ionized. Some ionization techniques are very energetic and cause extensive fragmentation. Other techniques are softer and only produce ions of the molecular species. Electron ionization, chemical ionization and field ionization are only suitable for gas-phase ionization and thus their use is limited to compounds sufficiently volatile and thermally stable. However, a large number of compounds are thermally labile or do not have sufficient vapour pressure. Molecules of these compounds must be directly extracted from the condensed to the gas phase. 

Most of the chemical reactions occur in the condensed phase or in the gas phase under conditions such that the number of intermolecular collisions during the reaction time is enormous. Internal energy is quickly distributed by these collisions over all the molecules according to the Maxwell-Boltzmann distribution curve. 

We recall that for condensed phases (either liquid or adsorbed) we are generally able to equate the molar enthalpy and the molar internal energy. Here we shall use enthalpies, simply because they are more common in the literature, but in the following definitions h could be replaced by u. 

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