Can C\u3csub\u3eP\u3c/sub\u3e Be Less Than C\u3csub\u3eV\u3c/sub\u3e?

Can CP be less than CV? This is a fundamental question in physics, chemistry, chemical engineering, and mechanical engineering. This question hangs in the minds of many students, instructors, and researchers. The first instinct is to answer “Yes, for water between 0 and 4 °C” if one knows that water expands as temperature decreases in this temperature range. The same question is asked in several Physical Chemistry and Physics textbooks. Students are supposed to answer that water contracts when heated at below 4 °C in an isobaric process. Because work is done to the contracting water, less heat is required to increase the water temperature in an isobaric process than in an isochoric process. Therefore, CP is less than CV. However, this answer is fundamentally flawed because it assumes, implicitly and incorrectly, that the internal energy change of water depends solely on its temperature change. Neglecting the variation of the internal energy with volume (internal pressure) will invalidate the Clausius inequality and violate the second law of thermodynamics. Once the internal pressure is properly taken into account, it becomes clear that CP cannot be less than CV for any substance at any temperature regardless of the sign of the thermal expansion coefficient of the substance

Size and Site Dependence of the Catalytic Activity of Iridium Clusters towards Ethane Dehydrogenation

This research focuses on optimizing transition metal nanocatalyst immobilization and activity to enhance ethane dehydrogenation. Ethane dehydrogenation, catalyzed by thermally stable Irn (n = 8, 12, 18) atomic clusters that exhibit a cuboid structure, was studied using the B3LYP method with triple-ζ basis sets. Relativistic effects and dispersion corrections were included in the calculations. In the dehydrogenation reaction Irn + C2H6 → H−Irn−C2H5 → (H)2−Irn−C2H4, the first H-elimination is the rate-limiting step, primarily because the reaction releases sufficient heat to facilitate the second H-elimination. The catalytic activity of the Ir clusters strongly depends on the Ir cluster size and the specific catalytic site. Cubic Ir8 is the least reactive towards H-elimination in ethane: Ir8 + C2H6 → H−Ir8−C2H5 has a large (65 kJ/mol) energy barrier, whereas Ir12 (3×2×2 cuboid) and Ir18 (3×3×2 cuboid) lower this energy barrier to 22 kJ/mol and 3 kJ/mol, respectively. The site dependence is as prominent as the size effect. For example, the energy barrier for the Ir18 + C2H6 → H−Ir18−C2H5 reaction is 3 kJ/mol, 48 kJ/mol, and 71 kJ/mol at the corner, edge, or face-center sites of the Ir18 cuboid, respectively. Energy release due to Ir cluster insertion into an ethane C–H bond facilitates hydrogen migration on the Ir cluster surface, and the second H-elimination of ethane. In an oxygen-rich environment, oxygen molecules may be absorbed on the Ir cluster surface. The oxygen atoms bonded to the Ir cluster surface may slightly increase the energy barrier for H-elimination in ethane. However, the adsorption of oxygen and its reaction with H atoms on the Ir cluster releases sufficient heat to yield an overall thermodynamically favored reaction: Irn + C2H6 + ½ O2 → Irn + C2H4 + H2O. These results will be useful towards reducing the energy cost of ethane dehydrogenation in industry

Breaking Bonds with the Left Eigenstate Completely Renormalized Coupled-Cluster Method

The recently developed [P. Piecuch and M. Wloch, J. Chem. Phys.123, 224105 (2005)] size-extensive left eigenstate completely renormalized (CR) coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as CCL, is compared with the full configuration interaction (FCI) method for all possible types of single bond-breaking reactions between C, H, Si, and Cl (except H2) and the H2SiSiH2 double bond-breaking reaction. The CCL method is in excellent agreement with FCI in the entire region R=1–3Re for all of the studied single bond-breaking reactions, whereR and Re are the bond distance and the equilibrium bond length, respectively. The CCL method recovers the FCI results to within approximately 1mhartree in the region R=1–3Reof the H–SiH3, H–Cl, H3Si–SiH3, Cl–CH3, H–CH3, and H3C–SiH3bonds. The maximum errors are −2.1, 1.6, and 1.6mhartree in the R=1–3Re region of the H3C–CH3, Cl–Cl, and H3Si–Clbonds, respectively, while the discrepancy for the H2SiSiH2 double bond-breaking reaction is 6.6 (8.5)mhartree at R=2(3)Re. CCL also predicts more accurate relative energies than the conventional CCSD and CCSD(T) approaches, and the predecessor of CR-CC(2,3) termed CR-CCSD(T)

Using a Spreadsheet To Solve the Schrödinger Equations for the Energies of the Ground Electronic State and the Two Lowest Excited States of H2

We have designed an exercise suitable for a lab or project in an undergraduate physical chemistry course that creates a Microsoft Excel spreadsheet to calculate the energy of the S0 ground electronic state and the S1 and T1 excited states of H2. The spreadsheet calculations circumvent the construction and diagonalization of the Fock matrix and thus can be accomplished by any undergraduate chemistry student with basic calculus skills. The wave functions of the S0, S1, and T1 states of H2 are constructed from the symmetry-adapted bonding and antibonding molecular orbitals (MO). All quantum mechanical integrals are estimated using the Monte Carlo integration method. Due to the stochastic nature of the spreadsheet calculations, 25 runs were carried out to obtain the mean energy of the S0, S1, and T1 electronic states of H2. The accuracy of the spreadsheet calculations is comparable to that of the HF/STO-3G calculations. The atomic and molecular orbitals and the energy components can be easily calculated and plotted for better visualization and understanding of essential quantum chemical concepts. This spreadsheet can also be adapted to tackle a wider range of quantum chemistry problems with different levels of complexity

Theoretical Study of the Pyrolysis of Methyltrichlorosilane in the Gas Phase. 1. Thermodynamics

Structures and energies of the gas-phase species produced during and after the various unimolecular decomposition reactions of methyltrichlorosilane (MTS) with the presence of H2 carrier gas were determined using second-order perturbation theory (MP2). Single point energies were obtained using singles + doubles coupled cluster theory, augmented by perturbative triples, CCSD(T). Partition functions were obtained using the harmonic oscillator-rigid rotor approximation. A 114-reaction mechanism is proposed to account for the gas-phase chemistry of MTS decompositions. Reaction enthalpies, entropies, and Gibbs free energies for these reactions were obtained at 11 temperatures ranging from 0 to 2000 K including room temperature and typical chemical vapor deposition (CVD) temperatures. Calculated and experimental thermodynamic properties such as heat capacities and entropies of various species and reaction enthalpies are compared, and theory is found to provide good agreement with experiment

Assessing density functionals for the prediction of thermochemistry of Ti-O-Cl species

Titanium dioxide (TiO2) nanoparticles are widely used in contaminant remediation, photocatalysis, and solar cell manufacturing. The low-cost production of TiO2 nanoparticles via the combustion of titanium tetrachloride (TiCl4) in oxygen is thus an important industrial process. To accurately model the flame synthesis of TiO2 nanoparticles, reliable thermodynamic data of Ti-O-Cl species are indispensable but often unavailable. We therefore carried out benchmark calculations, using the left-eigenstate completely renormalized singles, doubles, and perturbative triples (CR-CC(2,3), aka CR-CCL) method with the cc-pVTZ basis set, to obtain the equilibrium structures and vibrational frequencies of selected Ti-O-Cl species; we then performed single-point CCSD(T)/aug-cc-pVLZ (L = 3-5) calculations to extrapolate the CCSD(T)/CBS energies. After analyzing the experimental and calculated enthalpy of selected Ti-O-Cl species, the standard enthalpy of formation of the TiOCl2 molecule is determined to be -600.5 kJ/mol at 298 K. The standard enthalpy of all other Ti-O-Cl species are determined accordingly. Finally, we assessed the accuracy of 42 popular density functionals for the Ti-O-Cl species. Among these assessed functionals, the B98 functional, tightly followed by B97-1 and B3LYP, exhibits the best overall performance in the prediction of the thermochemistry of the Ti-O-Cl species

Breaking Bonds of Open-Shell Species with the Restricted Open-Shell Size Extensive Left Eigenstate Completely Renormalized Coupled-Cluster Method

The recently developed restricted open-shell, size extensive, left eigenstate, completely renormalized (CR), coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as ROCCL, is compared with the unrestricted CCSD(T) [UCCSD(T)] and multireference second-order perturbation theory (MRMP2) methods to assess the accuracy of the calculated potential energy surfaces (PESs) of eight single bond-breaking reactions of open-shell species that consist of C, H, Si, and Cl; these types of reactions are interesting because they account for part of the gas-phase chemistry in the silicon carbide chemical vapor deposition. The full configuration interaction (FCI) and multireference configuration interaction with Davidson quadruples correction [MRCI(Q)] methods are used as benchmark methods to evaluate the accuracy of the ROCCL, UCCSD(T), and MRMP2 PESs. The ROCCL PESs are found to be in reasonable agreement with the corresponding FCI or MRCI(Q) PESs in the entire region R = 1−3Re for all of the studied bond-breaking reactions. The ROCCL PESs have smaller nonparallelity error (NPE) than the UCCSD(T) ones and are comparable to those obtained with MRMP2. Both the ROCCL and UCCSD(T) PESs have significantly smaller reaction energy errors (REE) than the MRMP2 ones. Finally, an efficient strategy is proposed to estimate the ROCCL/cc-pVTZ PESs using an additivity approximation for basis set effects and correlation corrections

Tools for Prescreening the Most Active Sites on Ir and Rh Clusters toward C−H Bond Cleavage of Ethane: NBO Charges and Wiberg Bond Indexes

B3LYP calculations were carried out to study the insertion of iridium (Ir) and rhodium (Rh) clusters into a C−H bond of ethane, which is often the ratelimiting step of the catalytic cycle of oxidative dehydrogenation of ethane. Our previous research on Ir catalysis correlates the diffusivity of the lowest unoccupied molecular orbital of the Ir clusters and the relative activities of the various catalytic sites. The drawback of this research is that the molecular orbital visualization is qualitative rather than quantitative. Therefore, in this study on C−H bond activation by the Ir and Rh clusters, we conducted analyses of natural bond orbital (NBO) charges and Wiberg bond indexes (WBIs), both of which are not only quantitative but also independent of the basis sets. We found strong correlation between the NBO charges, the WBIs, and the relative activities of the various catalytic sites on the Ir and Rh clusters. Analyses of the NBO charges and the WBIs provide a fast and reliable means of prescreening the most active sites on the Ir and Rh clusters and potentially on other similar transition-metal clusters that activate the C−H bonds of ethane and other light alkanes