Kinetic-energy dependence of competitive spin-allowed and spin-forbidden reactions: V++CS2

Revisiting the Discrepancy between Experimental and Theoretical Predictions of the Adiabaticity of Ti+ + CH3OH

We revisit the naked transition metal cation (Ti+) and methanol reaction and go beyond the standard Landau-Zener (LZ) picture when modeling the intersystem crossing (ISC) rate between the lowest doublet and quartet states. We use both (i) unconstrained Born-Oppenheimer molecular dynamics (BOMD) calculations with an approximate two-state method to estimate population transfer between spin diabats and (ii) constrained dynamics to explore energetically accessible portions of the NDOF - 1 crossing seam, where NDOF is the total number of internal degrees of freedom. Whereas previous LZ calculations (that necessarily relied on the Condon approximation to be valid) fell short and predicted much slower crossing probabilities than shown in experiment, we show that ISC can occur rapidly because the spin-orbit coupling (SOC) between the doublet and quartet surfaces can vary by 2 orders of magnitude (depending on where in the seam the crossing occurs during dynamics) and the crossing region is revisited multiple times during a dynamics run of a few hundred femtoseconds. We further isolate the two important nuclear coordinates that tune the SOC and modulate the transition, highlighting exactly how and why organometallic ISC can occur rapidly for small systems with floppy internal nuclear vibrational modes.

A guided ion beam investigation of UO2+ thermodynamics and f orbital participation: Reactions of U+ + CO2, UO+ + O2, and UO+ + CO

A guided ion beam tandem mass spectrometer was employed to study the reactions of U+ + CO2, UO+ + O2, and the reverse of the former, UO+ + CO. Reaction cross sections as a function of kinetic energy over about a three order of magnitude range were studied for all systems. The reaction of U+ + CO2 proceeds to form UO+ + CO with an efficiency of 118% ± 24% as well as generating UO2+ + C and UCO+ + O. The reaction of UO+ + O2 forms UO2+ in an exothermic, barrierless process and also results in the collision-induced dissociation of UO+ to yield U+. In the UO+ + CO reaction, the formation of UO2+ in an endothermic process is the dominant reaction, but minor products of UCO+ + O and U+ + (O + CO) are also observed. Analysis of the kinetic energy dependences observed provides the bond energies, D0(U+-O) = 7.98 ± 0.22 and 8.05 ± 0.14 eV, D0(U+-CO) = 0.73 ± 0.13 eV, and D0(OU+-O) = 7.56 ± 0.12 eV. The values obtained for D0(U+-O) and D0(OU+-O) agree well with the previously reported literature values. To our knowledge, this is the first experimental measurement of D0(U+-CO). An analysis of the oxide bond energies shows that participation of 5f orbitals leads to a substantial increase in the thermodynamic stability of UO2+ relative to ThO2+ and especially transition metal dioxide cations.

Quantitative Aspects of Gas-Phase Metal Ion Chemistry: Conservation of Spin, Participation of f Orbitals, and C-H Activation and C-C Coupling

In this Featured Article, I reflect on over 40 years of guided ion beam tandem mass spectrometry (GIBMS) studies involving atomic metal cations and their clusters throughout the periodic table. Studies that have considered the role of spin conservation (or lack thereof) are a primary focus with a quantitative assessment of the effects examined. A need for state-specific studies of heavier elements is noted, as is a more quantitative assessment of spin-orbit interactions in reactivity. Because GIBMS experiments explicitly evaluate the kinetic energy dependence of reactions over a wide range, several interesting and unusual observations are highlighted. More detailed studies of such unusual reaction events would be welcome. Activation of C-H bonds and ensuing C-C coupling events are reviewed, with future work encouraged. Finally, studies of lanthanides and actinides are examined with an eye on understanding the role of f orbitals in the chemistry, both as participants (or not) in the bonding and as sources/sinks of electron density. This area seems to be ripe for more quantitative experiments.

Thermochemistry of the Ir+ + SO2 reaction using guided ion beam tandem mass spectrometry and theory

Kinetic energy dependences of the reactions of Ir⁺ (⁵F₅) with SO₂ were studied using a guided ion beam tandem mass spectrometer and theory. The observed cationic products are IrO⁺, IrS⁺, and IrSO⁺, formed in endothermic reactions. Bond dissociation energies (BDEs) of the products are determined by modeling the kinetic energy dependent product cross sections: D₀(Ir⁺-O) = 4.27 ± 0.11 eV, D₀(Ir⁺-S) = 4.03 ± 0.06 eV, and D₀(Ir⁺-SO) ≥ 2.95 ± 0.06 eV. The oxide BDE agrees well with literature values, whereas the two latter results are novel measurements. Quantum mechanical calculations are performed at the B3LYP level of theory using the def2-TZVPPD basis set for all product BDEs with additional calculations for IrS⁺, IrO₂ ⁺, and IrSO⁺ at the coupled cluster with single, double, and perturbative triple excitation levels with def2-QZVPPD and aug-cc-pVXZ (X = T and Q and for IrS⁺, also X = 5) basis sets and complete basis set extrapolations. These theoretical BDEs agree reasonably well with the experimental values. ¹A₁ (IrO₂ ⁺), ⁵Δ₄ (IrS⁺), and ³A″/¹A' (IrSO⁺) are found to be the ground states after including empirical spin-orbit corrections. The potential energy surfaces including intermediates and transition states for each reaction are also calculated at the B3LYP/def2-TZVPPD level. The formation of MO⁺ (M = Re, Os, and Ir) from M⁺ + SO₂ reactions is compared with those from the M⁺ + O₂ and M⁺ + CO reactions, where interesting trends in cross sections are observed. Overall, these studies suggest that the M⁺ + O₂ reactions had restrictions associated with reactions along A' and A″ surfaces.

Evaluation of the Pr + O → PrO+ + e- chemi-ionization reaction enthalpy and praseodymium oxide, carbide, dioxide, and carbonyl cation bond energies

Guided ion beam tandem mass spectrometry (GIBMS) was used to measure the kinetic energy dependent product ion cross sections for reactions of the lanthanide metal praseodymium cation (Pr+) with O2, CO2, and CO and reactions of PrO+ with CO, O2, and Xe. PrO+ is formed through barrierless exothermic processes when the atomic metal cation reacts with O2 and CO2, whereas all other reactions are observed to be endothermic. Analyses of the kinetic energy dependences of these cross sections yield 0 K bond dissociation energies (BDEs) for PrO+, PrC+, PrCO+, and PrO2+. The 0 K BDE for PrO+ is determined to be 7.62 ± 0.09 eV from the weighted average of five independent thresholds. This value is combined with the well-established ionization energy (IE) of Pr to indicate an exothermicity of the chemi-ionization reaction, Pr + O → PrO+ + e-, of 2.15 ± 0.09 eV. Additionally, BDEs of Pr+-C, OPr+-O, and Pr+-CO are determined to be 2.97 ± 0.10. 2.47 ± 0.11, and 0.31 ± 0.07 eV. Theoretical Pr+-O, Pr+-C, OPr+-O, and Pr+-CO BDEs are calculated for comparison with experimental values. The Pr+-O BDE is underestimated at the B3LYP and PBE0 level of theory but better agreement is obtained using the coupled-cluster with single, double, and perturbative triple excitations, CCSD(T), level. Density functional theory approaches yield better agreement for the BDEs of Pr+-C, OPr+-O, and Pr+-CO.

Counter-Intuitive Gas-Phase Reactivities of [V2 ]+ and [V2 O]+ towards CO2 Reduction: Insight from Electronic Structure Calculations

[V2 O]+ remains "invisible" in the thermal gas-phase reaction of bare [V2 ]+ with CO2 giving rise to [V2 O2 ]+ ; this is because the [V2 O]+ intermediate is being consumed more than 230 times faster than it is generated. However, the fleeting existence of [V2 O]+ and its involvement in the [V2 ]+ → [V2 O2 ]+ chemistry are demonstrated by a cross-over labeling experiment with a 1:1 mixture of C16 O2 /C18 O2 , generating the product ions [V2 16 O2 ]+ , [V2 16 O18 O]+ , and [V2 18 O2 ]+ in a 1:2:1 ratio. Density functional theory (DFT) calculations help to understand the remarkable and unexpected reactivity differences of [V2 ]+ versus [V2 O]+ towards CO2 .

On the Crucial Role of Isolated Electronic States in the Thermal Reaction of ReC+ with Dihydrogen

Presented here is that isolated, long‐lived electronic states of ReC⁺ serve as the root cause for distinctly different reactivities of this diatomic ion in the thermal activation of dihydrogen. Detailed high‐level quantum chemical calculations support the experimental findings obtained in the highly diluted gas phase using FT‐ICR mass spectrometry. The origin for the existence of these long‐lived excited electronic states and the resulting implications for the varying mechanisms of dihydrogen splitting are addressed.

Thermochemical studies of reactions of Re+ with SO2 using guided ion beam experiments and theory

The kinetic energy dependent reactions of Re⁺ with SO₂ were studied with guided ion beam tandem mass spectrometry. ReO⁺, ReO₂⁺, and OReS⁺ species were observed as products, all in endothermic reactions. Modeling of the kinetic energy dependent cross sections yields 0 K bond dissociation energies (BDEs, in eV) of 4.78 ± 0.06 (Re⁺-O), 5.75 ± 0.02 (Re⁺-O₂), and 4.35 ± 0.14 (Re⁺-SO). The latter two values can be combined with other information to derive the additional values 6.05 ± 0.05 (ORe⁺-O) and 4.89 ± 0.19 (ORe⁺-S). BDEs of ReO⁺ and ReO₂⁺ agree with literature values whereas the values for OReS⁺ are the first measurements. The former result is obtained even though formation of ground state ReO⁺ is spin-forbidden. Quantum mechanical calculations at the B3LYP level of theory with a def2-TZVPPD basis set yield results that agree reasonably well with experimental values. Additional calculations at the BP86 and CCSD(T) levels of theory using def2-QZVPPD and aug-cc-pVxZ (x = T, Q, and 5) basis sets were performed to compare thermochemistry with experiment to determine that ReO₂⁺ rather than the isobaric ReS⁺ is formed. Product ground states are ³Δ₃(ReO⁺), ³B₁(OReO⁺), ⁵Π_-1(ReS⁺), and ³A''(OReS⁺) after including empirical spin-orbit corrections, which means that formation of ground state products is spin-forbidden for all three product channels. The potential energy surfaces for the ReSO₂⁺ system were also explored at the B3LYP/def2-TZVPPD level and exhibited no barriers in excess of the endothermicities for all products. BDEs for rhenium oxide and sulfide diatomics and triatomics are compared and discussed. The present result for formation of ReO⁺ is compared to that for formation of ReO⁺ in the reactions of Re⁺ + O₂ and CO, where the former system exhibited interesting dual cross section features. Results are consistent with the hypothesis that the distinction of in-plane and out-of-plane C_S symmetry in the triatomic systems might be the explanation for the two endothermic features observed in the Re⁺ + O₂ reaction.

Evaluation of the exothermicity of the chemi-ionization reaction Nd + O → NdO+ + e- and neodymium oxide, carbide, dioxide, and carbonyl cation bond energies

The exothermicity of the chemi-ionization reaction, Nd + O → NdO⁺ + e^-, has been indirectly determined by measuring the thermochemistry for reactions of the lanthanide metal neodymium cation (Nd⁺) with O₂, CO₂, and CO and reactions of NdO⁺ with CO, O₂, and Xe. Guided ion beam tandem mass spectrometry was used to measure the kinetic energy dependent product ion cross sections for these reactions. NdO⁺ is formed through a barrierless exothermic process when the atomic metal cation reacts with O₂ and CO₂. All other reactions are observed to be endothermic. Analyses of the kinetic energy dependences of these cross sections yield 0 K bond dissociation energies (BDEs) for several species. The 0 K BDE for Nd⁺-O is determined to be 7.28 ± 0.10 eV from the average of four independent thresholds. This value is combined with the well-established Nd ionization energy to indicate an exothermicity of the title reaction of 1.76 ± 0.10 eV, which is lower and more precise than literature values. In addition, the Nd⁺-C, ONd⁺-O, and Nd⁺-CO BDEs are determined to be 2.61 ± 0.30, 2.12 ± 0.30, and 0.30 ± 0.21 eV. Additionally, theoretical BDEs of Nd⁺-O, Nd⁺-C, ONd⁺-O, and Nd⁺-CO are calculated at several levels for comparison with the experimental values. B3LYP calculations seriously underestimate the Nd⁺-O BDE, whereas MP2 and coupled-cluster with single, double-and perturbative triple excitations values are in reasonable agreement. Good agreement is generally obtained for Nd⁺-C, ONd⁺-O, and Nd⁺-CO BDEs as well.

Thermodynamic and Kinetic Aspects of the Reactions of Titanium(I) ion with the Sulfurtransfer Reagent SCO via the C22O Bond Activation Pathway

Theoretical studies on the thermodynamic and kinetic aspects of the reactions of Ti+ ion with the sulfur-transfer reagent SCO via the C22O bond activation pathway have been carried out over the temperature range 200 — 1200K using the DFT=B3LYP method, general statistical thermodynamics and Eyring transition state theory with the Wigner correction. The relevant reactions comprise: reaction 1 [Formula: see text], and reaction 2 [Formula: see text] in which the spin multiplicity changes from the quartet state to the doublet state in the crossing region. It is concluded that the order of the equilibrium constants (K) and the reaction rate constants (k) are consistent with that of their corresponding exoergic energies DE and reaction barriers, respectively, and their differences at low temperature are larger than those at high temperature. Step 2 of reaction 1 is both thermodynamically and kinetically favoured over the entire temperature range. Moreover, both reactions 1 and 2 are exothermic and proceed spontaneously in which their entropy increases, and the magnitudes of their thermodynamic constants all decrease with rising temperature. The calculated DH8, DG8, and DS8 values of the two reactions are relatively similar. However, because reactants 4Ti+ and SCO have a lower energy than that of reactants 2Ti+ and SCO, it is reaction 2, in which the quartet state is changed to the doublet state through intersystem crossing, which should be seen as dominant.

Methane Activation by Gas Phase Atomic Clusters

The increasing supply of natural gas has created a strong demand for developing efficient catalytic processes to upgrade methane, the most stable alkane molecule, into value-added chemicals. Currently, methane conversion in laboratory and industry is mostly performed under high-temperature conditions. A lot of effort has been devoted to exploring chemical entities that are able to activate the C-H bond of methane at lower temperatures, preferably room temperature. Gas phase atomic clusters with limited numbers of atoms are ideal models of active sites on heterogeneous catalysts. The cluster systems are being actively studied to activate methane under room-temperature conditions. State-of-the-art mass spectrometry, photoelectron imaging spectroscopy, and quantum chemistry calculations have been combined in our laboratory to reveal the molecular-level mechanisms of methane activation by atomic clusters. In this Account, we summarize our recent progress on thermal methane activation by metal oxide clusters doped with noble-metal atoms (Au, Pt, and Rh) as well as by oxygen-free species including carbides and borides of base metals (V, Ta, Mo, and Fe). In contrast to the generations of CH₃^• free radicals in many of the previously reported cluster reactions with methane, the generations of stable products such as formaldehyde, acetylene, and syngas as well as closed-shell species AuCH₃ and B₃CH₃ have been identified for the cluster reaction systems herein. Besides the well recognized mechanisms of methane activation by the O^-• radicals through hydrogen atom abstraction and by metal atoms through oxidative addition, the new mechanisms of synergistic methane activation by Lewis acid-base pairs (such as Au^δ+-O^δ- and B^δ+-B^δ-) and by dinuclear metal centers (such as Ta-Ta) have been recently revealed. In the reactions between methane and oxide clusters doped with noble-metal atoms, the oxide cluster "supports" can accept the H atoms and the CH _x species delivered through the noble-metal atoms and then transform methane into stable oxygenated compounds. The product selectivity (such as formaldehyde versus syngas) can be controlled by different noble-metal atoms (such as Pt versus Rh). The electronic structures of base metal centers can be engineered through carburization so that the low-spin states can be accessible to reduce the C-H bond of methane. Such active base metal centers in low-spin states resemble related noble-metal atoms in methane activation. The boron clusters (such as B₃ in VB₃⁺) can be polarized by the metal cations to form the Lewis acid-base pair B^δ+-B^δ- to cleave the C-H bond of methane very easily. These molecular-level mechanisms may well be operative in related heterogeneous catalysis and can be a fundamental basis to design efficient catalysts for activation and conversion of methane under mild conditions.

Temperature and Isotope Dependent Kinetics of Nickel-Catalyzed Oxidation of Methane by Ozone

The temperature dependent kinetics of Ni⁺ + O₃ and of NiO⁺ + CH₄/CD₄ are measured from 300 to 600 K using a selected-ion flow tube apparatus. Together, these reactions comprise a catalytic cycle converting CH₄ to CH₃OH. The reaction of Ni⁺ + O₃ proceeds at the collisional limit, faster than previously reported at 300 K. The NiO⁺ product reacts further with O₃, also at the collisional limit, yielding both higher oxides (up to NiO₅⁺ is observed) as well as undergoing an apparent reduction back to Ni⁺. This apparent reduction channel is due to the oxidation channel yielding NiO₂⁺* with sufficient energy to dissociate. ⁴NiO⁺ + CH₄ (CD₄) (whereas ⁴NiO⁺ refers to the quartet state of NiO⁺) proceeds with a rate constant of (2.6 ± 0.4) × 10^-10 cm³ s^-1 [(1.8 ± 0.5) × 10^-10 cm³ s^-1] at 300 K and a temperature dependence of ∼ T^-0.7±0.3 (∼ T^-1.1±0.4), producing only the ²Ni⁺ + ¹CH₃OH channel up to 600 K. Statistical modeling of the reaction based on calculated stationary points along the reaction coordinate reproduces the experimental rate constant as a function of temperature but underpredicts the kinetic isotope shift. The modeling was found to better represent the data when the crossing from quartet to doublet surface was incomplete, suggesting a possible kinetic effect in crossing from quartet to doublet surfaces. Additionally, the modeling predicts a competing ³NiOH⁺ + ²CH₃ channel to become increasingly important at higher temperatures.

Reactivity of 4Fe+(CO)n=0-2 + O2: oxidation of CO by O2 at an isolated metal atom

The kinetics of ⁴Fe⁺(CO)_n=0-2 + O₂ are measured under thermal conditions from 300-600 K using a selected-ion flow tube apparatus. Both the bare metal and n = 2 cations are inert to reaction over this temperature range, but ⁴Fe⁺(CO) reacts rapidly (k = 3.2 ± 0.8 × 10^-10 cm³ s^-1 at 300 K, 52% of the collisional rate coefficient) to form FeO⁺ + CO₂. This is an example of the oxidation of CO by O₂ occurring entirely on a single non-noble metal atom. The reaction of the bare metal reaction is known to be endothermic, such that this result is expected; however, the n = 2 reaction has highly exothermic product channels available, such that the lack of reaction is surprising in light of the n = 1 reactivity. Stationary points along all three reaction coordinates are calculated using the TPSSh hybrid functional. These surfaces show that the n = 1 reaction is an example of two-state reactivity; the reaction proceeds initially on the sextet surface over a submerged barrier to a structure with an O-O bond distance longer than that in O₂, but must cross to the quartet surface in order to proceed over a second submerged barrier to rearrange to form CO₂. The n = 2 reaction does not proceed because, on all spin surfaces, the transition state corresponding to O-O separation is at higher energy than the separated reactants. The difference between the n = 1 and n = 2 reactions is not a result of steric effects, but rather because the O₂ is more strongly bound to Fe in the entrance well of the n = 1 case, and that energy is available to overcome the rate-limiting barrier to O-O cleavage. Experimental verification of some of these details are provided by guided ion beam tandem mass spectrometry results. The kinetic energy dependence of the n = 1 reaction shows evidence for a curve crossing and yields relevant thermochemistry for competing reaction channels.

Methane Activation by 5 d Transition Metals: Energetics, Mechanisms, and Periodic Trends

Although it has been known for almost three decades that several 5d transition-metal cations will activate methane at room temperature, a more detailed examination of these reactions across the periodic table has only recently been completed. In this Minireview, we compare and contrast studies of the kinetic energy dependence of these reactions as studied using guided-ion-beam tandem mass spectrometry. Thermochemistry for the various products observed (MH⁺ , MH₂⁺ , MC⁺ , MCH⁺ , MCH₂⁺ , and MCH₃⁺ ) are collected and periodic trends evaluated and discussed. The mechanisms for the reactions as elucidated by synergistic quantum chemical calculations are also reviewed. Recent spectroscopic evidence for the structures of the MCH₂⁺ dehydrogenation products are discussed as well.

Cationic Noncovalent Interactions: Energetics and Periodic Trends

In this review, noncovalent interactions of ions with neutral molecules are discussed. After defining the scope of the article, which excludes anionic and most protonated systems, methods associated with measuring thermodynamic information for such systems are briefly recounted. An extensive set of tables detailing available thermodynamic information for the noncovalent interactions of metal cations with a host of ligands is provided. Ligands include small molecules (H2, NH3, CO, CS, H2O, CH3CN, and others), organic ligands (O- and N-donors, crown ethers and related molecules, MALDI matrix molecules), π-ligands (alkenes, alkynes, benzene, and substituted benzenes), miscellaneous inorganic ligands, and biological systems (amino acids, peptides, sugars, nucleobases, nucleosides, and nucleotides). Hydration of metalated biological systems is also included along with selected proton-based systems: 18-crown-6 polyether with protonated peptides and base-pairing energies of nucleobases. In all cases, the literature thermochemistry is evaluated and, in many cases, reanchored or adjusted to 0 K bond dissociation energies. Trends in these values are discussed and related to a variety of simple molecular concepts.

Mechanistic exploration of the catalytic cycles for the CO oxidation by O2 over FeO(1-3) application of the energetic span model

Carbon monoxide (CO) and oxygen (O2) catalyzed by small neutral iron oxide clusters (FeO(1-3)) was investigated at the density functional level of theory using the Becke-Perdew-Wang functional (BPW91). Three reaction pathways along with singlet, triplet and quintet states were calculated for ascertaining the presence of some spin inversion during the catalytic cycle. The catalytic cycle was found to be "two state reactivity" resulting from the crossing among the multistate energetic profiles. The Landau-Zener equation was used to calculate the thermally-averaged spin transition probabilities for the non-adiabatic surface crossing reaction. In order to predict the efficiency of catalyst the energetic span model developed by Kozuch was implemented, whereas this model is not suitable for handling the diabatic reaction, this feature we must take into consideration. To this end, a kinetic assessment is carried out with an expansion of the energetic span model, including the spin-crossing effects. This approximation enables one to measure the efficiency of catalytic cycle including spin-crossing effects by quantum mechanical computation.

The competitive O-H versus C-H bond activation of ethanol and methanol by VO2(+) in gas phase: a DFT study

The activation of ethanol and methanol by VO2(+) in gas phase has been theoretically investigated by using density functional theory (DFT). For the VO2(+)/ethanol system, the activation energy (ΔE) is found to follow the order of ΔE(C(β)-H) < ΔE(C(α)-H) ≈ ΔE(O-H). Loss of methyl and glycol occurs respectively via O-H and C(β)-H activation, while acetaldehyde elimination proceeds through two comparable O-H and C(α)-H activations yielding both VO(H2O)(+) and V(OH)2(+). Loss of water not only gives rise to VO(CH3CHO)(+) via both O-H and C(α)-H activation but also forms VO2(C2H4)(+) via C(β)-H activation. The major product of ethylene is formed via both O-H and C(β)-H activation for yielding VO(OH)2(+) and VO2(H2O)(+). In the methanol reaction, both initial O-H and C(α)-H activation accounts for formaldehyde and water elimination, but the former pathway is preferred.

Thermodynamic and kinetic properties of scandium (I) ion reacting with SCO in gas phase

Theoretical studies on the thermodynamic and kinetic properties of the reactions of scandium (I) ion with the sulfur-transfer reagent SCO via the C-O bond activation pathway have been carried out over the temperature range of 200-1200 K using the DFT/B3LYP method, general statistical thermodynamics, and Eyring transition state theory with Wigner correction. The relevant reactions include reaction 1 1Sc+ + SCO → 1IM1 → 1TS1 → 1IM2 (Step 1) → 1TS2 → 1IM3 → 1ScO+ + 1CS (Step 2), and reaction 2 3Sc+ + SCO → 3IM1 → CP → 1IM2 → 1TS2 → 1IM3 → 1ScO+ +1CS in which the spin multiplicity changes from the triplet state to the singlet state in the crossing region. It was concluded that the order of the equilibrium constants (K) and the reaction rate constants (k) are consistent with that of their corresponding exoergic energies, ΔE, and reaction barriers, respectively. Step 2 of reaction 1 is both thermodynamically and kinetically favored over the whole temperature range. Moreover, both Reaction 1 and reaction 2 are exothermic and spontaneous processes in which their entropy increases, and the magnitudes of their thermodynamic values all decrease with increasing temperature.