§9.1 §9.1 Principles of Homogeneous Catalysis: Cycles, TON, TOF & Catalyst Deactivation
A homogeneous catalyst operates in the same phase (typically liquid solution) as the reactants, offering atomic dispersion, molecularly well-defined active sites, tunable coordination spheres, and mild operating temperatures and pressures.
Fundamental Catalytic Metrics:
1. Turnover Number (TON):
The total number of moles of substrate converted into product per mole of catalyst before the catalyst completely loses its activity:
2. Turnover Frequency (TOF):
The turnover number achieved per unit time, reflecting the intrinsic catalytic rate:
3. Catalytic Cycle Dynamics:
- Catalyst Resting State (CRS): The thermodynamic ground state intermediate that accumulates in the largest concentration in solution (detectable spectroscopically).
- Turnover-Limiting Step (TLS): The elementary step possessing the highest transition state energy relative to the resting state (governs the net reaction rate).
- Catalyst Deactivation: Pathways that permanently siphon active metal species out of the catalytic loop, including bimolecular cluster dimerization, ligand degradation, or metal precipitation.
§9.2 §9.2 Homogeneous Hydrogenation by Wilkinson's Catalyst $\text{RhCl}(\text{PPh}_3)_3$
Discovered in 1965 by Sir Geoffrey Wilkinson, chlorotris(triphenylphosphine)rhodium(I) $\text{RhCl}(\text{PPh}_3)_3$ is the prototypical homogeneous hydrogenation catalyst for unhindered alkenes and alkynes.
The Catalytic Cycle (The Dihydride Pathway):
1. Initiation (Phosphine Dissociation):
In solution, the 16-electron square planar precursor undergoes reversible dissociation of one bulky triphenylphosphine ligand ($\theta = 145^\circ$):
2. Oxidative Addition of $\text{H}_2$:
Rapid, concerted oxidative addition of molecular dihydrogen yields a 16-electron cis-dihydride:
3. Alkene Coordination:
The alkene coordinates to the open site, generating an 18-electron dihydride-olefin complex:
4. Migratory Insertion (Rate-Determining Step):
Intramolecular 1,2-migratory insertion of the alkene into a mutually cis rhodium-hydride bond generates a 16-electron alkyl-hydride intermediate:
5. Reductive Elimination (Product Release):
Concerted reductive elimination of alkane regenerates the active 14-electron $[\text{RhCl}(\text{PPh}_3)_2]$ catalyst:
Substrate Selectivity:
Because the transition states are sterically crowded, hydrogenation rates follow:
§9.3 §9.3 Asymmetric Homogeneous Hydrogenation: Knowles, Noyori & Chiral Diphosphines
Asymmetric hydrogenation revolutionized pharmaceutical synthesis, converting prochiral alkenes into single enantiomers with $>99\%$ enantiomeric excess (Nobel Prize in Chemistry, 2001 to William S. Knowles and Ryoji Noyori).
Milestone Catalytic Systems:
1. Knowles' DIPAMP Catalyst:
Utilized chiral-at-phosphorus bidentate ligands for the industrial synthesis of L-DOPA (treatment for Parkinson's disease):
2. Noyori's BINAP-Ruthenium Catalysts:
Utilized axially chiral, atropisomeric $2,2'$-bis(diphenylphosphino)-$1,1'$-binaphthyl (BINAP):
- $[\text{Ru}(\text{BINAP})(\text{OAc})_2]$ hydrogenates $\alpha,\beta$-unsaturated carboxylic acids (e.g., $(S)$-naproxen at $>97\%\ ee$).
- Noyori's bifunctional ruthenium-diamine catalysts $[\text{RuCl}_2(\text{BINAP})(\text{DAIPEN})]$ hydrogenate simple ketones via a non-classical metal-ligand bifunctional outer-sphere mechanism without substrate coordination to the metal!
The Halpern 'Minor-Isomer' Mechanism:
Jack Halpern demonstrated by low-temperature NMR that the catalyst binds a prochiral enamide to form two diastereomeric complexes in a rapid pre-equilibrium:
Counter-intuitively, oxidative addition of dihydrogen into the minor diastereomer is $10^3$ to $10^4$ times faster than into the major diastereomer:
Therefore, the minor, less stable diastereomer delivers $>99\%$ of the final enantiomeric product!
§9.4 §9.4 Hydroformylation (The Oxo Process): Cobalt vs. Rhodium Catalysis
Discovered in 1938 by Otto Roelen, hydroformylation converts alkenes, carbon monoxide, and dihydrogen (syngas) into aldehydes:
It is the largest-volume homogeneous catalytic process in the global chemical industry ($>15$ million metric tons annually).
Comparison of Industrial Catalyst Systems:
| Metric | Unmodified Cobalt | Phosphine-Modified Cobalt | Rhodium-Phosphine (Low-Pressure Oxo) | | :--- | :--- | :--- | :--- | | Active Catalyst | $\text{HCo}(\text{CO})_4$ | $\text{HCo}(\text{CO})_3(\text{PBu}_3)$ | $\text{HRh}(\text{CO})(\text{PPh}_3)_2$ | | Temperature | $140 - 180^\circ\text{C}$ | $160 - 200^\circ\text{C}$ | $85 - 110^\circ\text{C}$ | | Pressure | $200 - 300\text{ bar}$ | $50 - 100\text{ bar}$ | $15 - 30\text{ bar}$ | | Activity | Moderate | Low | Extremely High ($10^3 \times \text{Co}$) | | Linear:Branched ($l:b$) | $3:1 - 4:1$ | $7:1 - 9:1$ | $15:1 - 50:1$ | | Byproduct Hydrogenation | Minimal | High (alcohols formed) | Negligible |
The modern Low-Pressure Oxo (LPO) process developed by Union Carbide / Davy Powergas employs rhodium with excess triphenylphosphine, operating under exceptionally mild conditions and delivering premium linear aldehydes.
§9.5 §9.5 Regioselectivity Control in Hydroformylation: Linear vs. Branched Aldehydes
In industrial hydroformylation of terminal alkenes, the linear aldehyde ($n$-aldehyde) is preferred for plasticizer alcohols (e.g., 2-ethylhexanol) and biodegradable detergents.
Origin of Regioselectivity:
Regioselectivity is established during the 1,2-migratory insertion of the alkene into the metal-hydride bond:
1. Anti-Markovnikov Insertion:
- The hydride transfers to the internal secondary carbon (C2), while the metal attaches to the terminal primary carbon (C1):
- Subsequent CO insertion and hydrogenolysis delivers the linear aldehyde.
2. Markovnikov Insertion:
- The hydride transfers to the terminal carbon (C1), while the metal attaches to C2:
- Delivers the branched aldehyde.
Steric Engineering of the Ligand Sphere:
In the rhodium-catalyzed cycle, the active intermediate is the trigonal bipyramidal complex $\text{HRh}(\text{CO})_2 L_2$:
- When bulky phosphines (e.g., triphenylphosphine or wide bite-angle diphosphines like Xantphos) coordinate:
- Steric repulsion between the bulky phosphine ligands and the alkyl substituent $R$ destabilizes the transition state for Markovnikov insertion.
- The alkene is forced to direct its $R$ group away from the coordination sphere, locking the system into anti-Markovnikov insertion.
- Using Xantphos ($\beta_n = 111^\circ$) raises the linear-to-branched ratio to $l:b > 50:1$ with $>98\%$ selectivity.
§9.6 §9.6 Olefin Metathesis: Historical Evolution & The Chauvin Mechanism
Olefin metathesis (from Greek metathesis, meaning 'transposition') is a chemical transformation in which carbon-carbon double bonds are cleaved and reformed through the redistribution of alkylidene fragments:
The Chauvin Mechanism (1971):
Yves Chauvin proposed that the active catalyst is a transition metal alkylidene (carbene) $M=\text{CHR}$, and that the reaction proceeds through alternating $[2+2]$ cycloadditions and cycloreversions:
1. $[2+2]$ Cycloaddition:
The metal alkylidene coordinates an alkene and undergoes a concerted, symmetry-allowed $[2+2]$ cycloaddition to form a four-membered metallacyclobutane intermediate:
2. $[2+2]$ Cycloreversion:
The metallacyclobutane cleaves across the perpendicular coordinate, regenerating a new metal alkylidene and releasing a new alkene:
3. Equilibrium and Driving Force:
Because every elementary step in the Chauvin cycle is reversible, metathesis of unstrained acyclic alkenes is an equilibrium under thermoneutral enthalpy control (driven entropically by the volatilization of ethylene gas $\text{H}_2\text{C}=\text{CH}_2 \uparrow$).
§9.7 §9.7 Evolution of Metathesis Catalysts: Schrock vs. Grubbs Systems
The development of well-defined metathesis catalysts transformed the field (Nobel Prize in Chemistry, 2005 to Yves Chauvin, Richard R. Schrock, and Robert H. Grubbs):
1. Schrock Molybdenum and Tungsten Alkylidenes:
- Structure: High-valent $d^0$ complexes $[\text{Mo}(=\text{CH}R)(=\text{NAr})(\text{OR}')_2]$ featuring an imido ligand ($=\text{NAr}$) and electron-withdrawing alkoxides (e.g., $-\text{OCMe}(\text{CF}_3)_2$).
- Properties: Exceptional catalytic activity; capable of metathesizing sterically hindered and electron-deficient alkenes.
- Drawback: Extremely sensitive to air, water, and protic functional groups (alcohols, acids).
2. Grubbs Ruthenium Alkylidenes:
- Grubbs 1st Generation (1995):
- Structure: $[(\text{PCy}_3)_2\text{Cl}_2\text{Ru}=\text{CHPh}]$.
- Low-valent $d^6$ ruthenium(II) center.
- Remarkable air- and moisture-tolerance, compatible with alcohols, water, and carboxylic acids. Moderate activity.
- Grubbs 2nd Generation (1999):
- Structure: Replace one $\text{PCy}_3$ with an $N$-heterocyclic carbene (NHC, $\text{H}_2\text{IMes}$ or $\text{IMes}$).
- NHCs are superior $\sigma$-donors that do not dissociate, accelerating the rate of phosphine dissociation and stabilizing the 14-electron ruthenacyclobutane intermediate.
- Activity matches Schrock catalysts while maintaining full functional group tolerance.
- Hoveyda-Grubbs 2nd Generation (2000):
- Replaces the phosphine entirely with a chelating ortho-isopropoxybenzylidene ligand, yielding exceptional bench-stability and recyclability.
§9.8 §9.8 Synthetic Variations of Metathesis: RCM, ROMP, CM and Stereocontrol
Olefin metathesis encompasses several major synthetic variations:
1. Ring-Closing Metathesis (RCM):
- Converts an $\alpha,\omega$-diene into a cyclic alkene with extrusion of ethylene gas:
- Powerful methodology for synthesizing 5- to 8-membered rings as well as macrocyclic lactones and natural products (e.g., epothilones).
2. Ring-Opening Metathesis Polymerization (ROMP):
- Driven by the release of ring strain from cyclic alkenes (e.g., norbornene $\Delta H_\text{strain} \approx 110\text{ kJ/mol}$, dicyclopentadiene):
- Produces living polymers with controlled molecular weights and narrow polydispersity ($PDI < 1.1$).
3. Cross-Metathesis (CM):
- Intermolecular coupling of two different acyclic alkenes. Regulated by Grubbs' classification of olefins into Type I through Type IV based on their rates of homodimerization and homocoupling.
4. $Z$-Selective Metathesis:
- Modern cyclometallated ruthenium and Schrock molybdenum catalysts enforce formation of thermodynamically less stable $(Z)$-alkenes with $>95\%\ Z$-selectivity, critical for pheromone and drug manufacturing.
Worked Practice Problems (9 Challenge Exercises)
Multi-step solved problems covering neutral vs ionic electron counting, d-electron configuration determination, 16-electron square planar stabilization, metal-metal single and multiple bond orders, bridging ligand electron partitioning, and 3c-2e bridge thermodynamic equilibria with line-by-line mathematical proofs.
In a homogeneous hydrogenation reaction, $2.5\text{ mg}$ of Wilkinson's catalyst $\text{RhCl}(\text{PPh}_3)_3$ (molar mass $925.2\text{ g/mol}$) is dissolved with $5.0\text{ g}$ of cyclohexene (molar mass $82.14\text{ g/mol}$) in $50\text{ mL}$ of benzene under $1.0\text{ bar}$ of $\text{H}_2$. After $45\text{ minutes}$, GC analysis shows $94\%$ conversion to cyclohexane. (a) Calculate the moles of catalyst and substrate. (b) Calculate the turnover number (TON) and turnover frequency (TOF) in $\text{h}^{-1}$ and $\text{s}^{-1}$.
Line-by-Line Solution:
(a) Calculation of Moles of Catalyst and Substrate:
1. Moles of Wilkinson's Catalyst:
2. Moles of Cyclohexene Substrate:
3. Moles of Product Formed at $94\%$ Conversion:
(b) Calculation of TON and TOF:
1. Turnover Number (TON):
2. Turnover Frequency (TOF):
Reaction time: $t = 45\text{ minutes} = 0.75\text{ hours} = 2700\text{ seconds}$.
- In units of $\text{h}^{-1}$:
- In units of $\text{s}^{-1}$:
- Conclusion: The catalyst achieves a TON of $\approx 2.12 \times 10^4$ and turns over at a frequency of $\approx 7.8\text{ catalytic cycles per second}$.
Predict the major organic product when each of the following polyunsaturated substrates is hydrogenated with 1 equivalent of dihydrogen in the presence of Wilkinson's catalyst: (a) Limonene (1-methyl-4-(prop-1-en-2-yl)cyclohex-1-ene), (b) 2-Methylbuta-1,3-diene (isoprene), (c) Methyl cinnamate (methyl 3-phenylprop-2-enoate) vs cinnamaldehyde.
Line-by-Line Solution:
(a) Limonene Hydrogenation:
- Structure of limonene:
- Contains an endocyclic trisubstituted double bond in the cyclohexene ring.
- Contains an exocyclic disubstituted terminal isopropenyl double bond ($-\text{C}(\text{CH}_3)=\text{CH}_2$).
- In Wilkinson's hydrogenation, the rate-determining migratory insertion occurs within a sterically congested coordination sphere. Steric congestion dictates the rate order:
- The catalyst coordinates and hydrogenates the less sterically hindered exocyclic isopropenyl double bond selectively:
leaving the endocyclic trisubstituted double bond intact.
(b) Isoprene (2-Methylbuta-1,3-diene):
- Isoprene contains two double bonds: a monosubstituted terminal double bond (C3=C4) and a 1,1-disubstituted double bond (C1=C2).
- Coordination occurs preferentially at the less hindered monosubstituted C3=C4 bond.
- Hydrogenation of C3=C4 with 1 equivalent of $\text{H}_2$ yields:
(c) Methyl Cinnamate vs. Cinnamaldehyde:
- Wilkinson's catalyst hydrogenates carbon-carbon double bonds rapidly, while aldehydes and esters are completely inert under standard conditions (carbonyl groups do not coordinate strongly to $\text{Rh}(\text{I})$).
- For cinnamaldehyde ($\text{PhCH}=\text{CH}-\text{CHO}$):
- Chemoselective reduction of the $C=C$ double bond occurs, leaving the aldehyde group intact:
Predict the initial metathesis products (including the volatile byproduct that drives the reaction to completion) for: (a) Ring-closing metathesis of diethyl diallylmalonate catalyzed by Grubbs 1st generation catalyst. (b) Cross-metathesis between allylbenzene and excess cis-1,4-diacetoxybut-2-ene. (c) Ring-opening metathesis polymerization (ROMP) of cyclopentene.
Line-by-Line Solution:
(a) Ring-Closing Metathesis (RCM) of Diethyl Diallylmalonate:
- Substrate structure: $(\text{EtO}_2\text{C})_2\text{C}(\text{CH}_2-\text{CH}=\text{CH}_2)_2$ (a 1,6-diene).
- The ruthenium carbene coordinates one terminal alkene, forms a ruthenacyclobutane, and transfers the alkylidene onto the substrate.
- Intramolecular $[2+2]$ cycloaddition with the second terminal alkene closes a cyclopentene ring:
- Byproduct: The two terminal methylene ($=\text{CH}_2$) groups combine to release ethene gas $\mathbf{\text{H}_2\text{C}=\text{CH}_2 \uparrow}$, which bubbles out of solution, shifting the equilibrium quantitatively to $100\%$ conversion.
(b) Cross-Metathesis (CM) of Allylbenzene:
- Reactants: Allylbenzene $\text{PhCH}_2-\text{CH}=\text{CH}_2$ and symmetric cis-1,4-diacetoxybut-2-ene $\text{AcOCH}_2-\text{CH}=\text{CH}-\text{CH}_2\text{OAc}$.
- Transposition of the alkylidene fragments cleaves the terminal alkene and exchanges fragments:
- Volatile byproduct: Ethene gas $\text{H}_2\text{C}=\text{CH}_2 \uparrow$.
(c) Ring-Opening Metathesis Polymerization (ROMP) of Cyclopentene:
- Cyclopentene is a cyclic alkene possessing low-to-moderate ring strain ($pprox 28\text{ kJ/mol}$).
- Coordination to the ruthenium carbene and $[2+2]$ cycloaddition forms a bicyclic ruthenacyclobutane.
- Cycloreversion opens the five-membered ring, regenerating an active propagating alkylidene chain end:
containing repeating pentamethylene units with alternating double bonds.
The rate of homogeneous hydrogenation of cyclohexene catalyzed by Wilkinson's catalyst follows the empirical equation: $\text{Rate} = \frac{k K_1 K_2 [\text{H}_2][\text{olefin}][\text{Rh}]_0}{1 + K_1 [\text{H}_2] + K_2 [\text{olefin}] + K_3 [\text{PPh}_3]}$. Derive this rate equation from the dihydride catalytic cycle using the steady-state approximation and mass balance on rhodium.
Line-by-Line Solution:
1. Catalytic Reaction Sequence (The Dihydride Route):
- Let $P = \text{PPh}_3$. The precursor $[\text{RhCl}P_3]$ undergoes dissociation:
- The 14e species $[\text{RhCl}P_2]$ undergoes oxidative addition of $\text{H}_2$:
- Coordination of olefin ($O$):
- Alternatively, direct coordination of olefin to $[\text{RhCl}P_2]$:
- The turnover-limiting step is the migratory insertion and subsequent fast reductive elimination:
2. Rate of Hydrogenation:
3. Total Rhodium Mass Balance: The total rhodium catalyst concentration $[\text{Rh}]_0$ is distributed among all rhodium-containing species in solution:
Expressing each species in terms of $[\text{RhCl}P_2]$:
- $[\text{RhCl}P_3] = \frac{[P]}{K_d} [\text{RhCl}P_2] = K_3' [P] [\text{RhCl}P_2]$
- $[\text{RhCl}(\text{H})_2 P_2] = K_1 [\text{H}_2] [\text{RhCl}P_2]$
- $[\text{RhCl}(O)P_2] = K_O [O] [\text{RhCl}P_2]$
- $[\text{RhCl}(\text{H})_2(O)P_2] = K_1 K_2 [\text{H}_2][O] [\text{RhCl}P_2]$ (typically negligible in the resting state balance under low olefin concentration)
Factoring $[\text{RhCl}P_2]$:
4. Final Rate Law: Substitute $[\text{RhCl}P_2]$ into the rate equation:
- Order Analysis:
- At low $[\text{H}_2]$, the rate is first-order in $[\text{H}_2]$; at high $[\text{H}_2]$, it approaches zero-order.
- Adding excess triphenylphosphine $[P]$ increases the denominator, inhibiting the reaction rate ($-\text{order}$ in $[\text{PPh}_3]$).
In the asymmetric hydrogenation of methyl 2-acetamidoacrylate catalyzed by $[\text{Rh}((R,R)-\text{DIPAMP})]^+$, the major catalyst-substrate complex $C_\text{maj}$ constitutes $95\%$ of the resting state, while the minor complex $C_\text{min}$ constitutes $5\%$ ($K_\text{eq} = [C_\text{maj}]/[C_\text{min}] = 19$). Oxidative addition of $\text{H}_2$ occurs with rate constants $k_\text{maj} = 0.15\text{ M}^{-1}\text{s}^{-1}$ and $k_\text{min} = 950\text{ M}^{-1}\text{s}^{-1}$. (a) Calculate the ratio of rates of product formation via the minor pathway versus the major pathway. (b) Calculate the resulting enantiomeric excess ($ee$). (c) Explain why decreasing $\text{H}_2$ pressure increases the enantiomeric excess.
Line-by-Line Solution:
(a) Ratio of Product Formation Rates:
- The rate of product formation from each diastereomeric pathway is:
- The ratio of rates is:
- Substitute the given kinetic and equilibrium values:
- $\frac{k_\text{min}}{k_\text{maj}} = \frac{950}{0.15} \approx 6333.3$
- $\frac{[C_\text{min}]}{[C_\text{maj}]} = \frac{1}{19} \approx 0.05263$
- Conclusion: Product formation via the minor diastereomer is 333 times faster than via the major diastereomer!
(b) Calculation of Enantiomeric Excess ($ee$):
- The minor diastereomer produces the $(S)$-enantiomer, and the major diastereomer produces the $(R)$-enantiomer:
- Compute $ee$:
- Result: The reaction delivers $(S)$-product with $99.4\%$ enantiomeric excess!
(c) Pressure Dependence of Enantiomeric Excess:
- The Halpern mechanism relies on rapid pre-equilibrium between $C_\text{maj}$ and $C_\text{min}$ compared to the rate of oxidative addition of dihydrogen:
- If the partial pressure of dihydrogen $P(\text{H}_2)$ is raised to high levels, the rate of oxidative addition $k_\text{min} [\text{H}_2]$ increases linearly.
- At very high $\text{H}_2$ pressure, oxidative addition begins to compete with the interconversion rate between $C_\text{maj}$ and $C_\text{min}$ (Curtin-Hammett breakdown).
- As interconversion becomes non-equilibrating, more product is forced to form through the slower but predominantly present major complex $C_\text{maj}$, which produces the undesired $(R)$-enantiomer.
- Therefore, low $\text{H}_2$ pressure preserves the rapid pre-equilibrium, maximizing the Curtin-Hammett kinetic partitioning through the fast minor pathway and increasing the enantiomeric excess.
In the rhodium-catalyzed hydroformylation of 1-hexene to heptanal (linear) and 2-methylhexanal (branched): (a) The standard enthalpies of reaction are $\Delta H_\text{lin}^\circ = -118\text{ kJ/mol}$ and $\Delta H_\text{br}^\circ = -115\text{ kJ/mol}$, with standard entropies $\Delta S_\text{lin}^\circ = -195\text{ J/(mol}\cdot\text{K)}$ and $\Delta S_\text{br}^\circ = -188\text{ J/(mol}\cdot\text{K)}$. Calculate $\Delta G^\circ$ for both pathways at $373\text{ K}$. (b) Explain why industrial regioselectivity ($l:b = 30:1$) is kinetically controlled rather than thermodynamically controlled.
Line-by-Line Solution:
(a) Calculation of $\Delta G^\circ$ at $373\text{ K}$:
1. For the Linear Product (Heptanal):
- $\Delta H_\text{lin}^\circ = -118,000\text{ J/mol}$
- $\Delta S_\text{lin}^\circ = -195\text{ J/(mol}\cdot\text{K)}$
2. For the Branched Product (2-Methylhexanal):
- $\Delta H_\text{br}^\circ = -115,000\text{ J/mol}$
- $\Delta S_\text{br}^\circ = -188\text{ J/(mol}\cdot\text{K)}$
3. Thermodynamic Free Energy Difference:
The thermodynamic equilibrium ratio at $373\text{ K}$ would be:
(b) Kinetic Origin of Industrial $l:b$ Selectivity ($30:1$):
- Thermodynamic control would predict an almost equimolar $l:b$ ratio of $1.13 : 1$ (only $53\%$ linear).
- However, the industrial process routinely achieves $l:b = 30:1$ to $50:1$ ($>97\%$ linear).
- This proves that hydroformylation is strictly kinetically controlled:
- Once the alkene undergoes irreversible 1,2-migratory insertion and CO insertion, the resulting acyl intermediates do not equilibrate back to alkene under low-pressure rhodium conditions.
- The activation energy barrier for anti-Markovnikov insertion is lower than for Markovnikov insertion by $\Delta\Delta G^\ddagger \approx 10-12\text{ kJ/mol}$ due to steric repulsion between the alkene's alkyl tail and the bulky equatorial phosphine ligands ($ ext{PPh}_3$ or diphosphines).
- This difference in activation energy dictates the high observed linear regioselectivity.
The four-membered metallacyclobutane intermediate $[L_n M(\text{C}_3\text{H}_6)]$ in olefin metathesis can adopt planar or puckered geometries. (a) Construct the orbital interaction diagram between a $d^2$ metal alkylidene $[L_n M=\text{CH}_2]$ and ethylene. (b) Explain why electron-donating $N$-heterocyclic carbenes (NHCs) in Grubbs 2nd generation catalysts stabilize the 14-electron ruthenacyclobutane transition state. (c) Explain why early transition metal metallacyclobutanes (titanium, tantalum) are isolable ground states (e.g., Tebbe's reagent), while ruthenium analogs are short-lived reactive intermediates.
Line-by-Line Solution:
(a) Orbital Interaction Diagram of $[2+2]$ Cycloaddition:
1. Metal Alkylidene $L_n M=\text{CH}_2$:
- The $M=C$ double bond consists of a $\sigma$-bonding orbital (HOMO$-1$) and a localized $\pi(M=C)$ bonding orbital (HOMO).
- The LUMO is the low-lying $\pi^*(M=C)$ antibonding orbital, polarized heavily toward the metal atom (significant $d_\pi$ character).
2. Alkene $\text{H}_2\text{C}=\text{CH}_2$:
- The HOMO is the bonding $\pi_{CC}$ orbital.
- The LUMO is the antibonding $\pi_{CC}^*$ orbital.
3. Concerted $[2+2]$ Orbital Mixing:
- Primary interaction 1: Alkene $\pi_{CC}$ (HOMO) donates into the empty metal alkylidene $\pi^*(M=C)$ (LUMO).
- Primary interaction 2: Filled alkylidene $\pi(M=C)$ (HOMO) backdonates into the empty alkene $\pi_{CC}^*$ (LUMO).
- Because the transition metal provides an accessible $d$-orbital that changes oxidation state ($M^n \rightleftharpoons M^{n+2}$), the orbital symmetry restrictions that forbid organic $[\pi 2_s + \pi 2_s]$ cycloadditions are completely lifted!
(b) Role of $N$-Heterocyclic Carbenes (NHCs) in Grubbs 2nd Generation Catalysts:
- In Grubbs 1st generation $[(\text{PCy}_3)_2\text{Cl}_2\text{Ru}=\text{CHPh}]$, the catalyst must dissociate one $\text{PCy}_3$ phosphine to generate the active 14-electron intermediate:
- Phosphine dissociation is slow ($k_1 \approx 10^{-2}\text{ s}^{-1}$), and the empty coordination site is readily recaptured by free $\text{PCy}_3$ ($k_{-1} \gg k_\text{olefin}$).
- In Grubbs 2nd generation catalysts $[(\text{NHC})(\text{PCy}_3)\text{Cl}_2\text{Ru}=\text{CHPh}]$:
- The NHC ligand is an extraordinarily powerful $\sigma$-donor with negligible $\pi$-acceptor ability.
- Its massive electron donation exerts a strong trans-effect that accelerates phosphine dissociation.
- More crucially, the electron-rich NHC ligand stabilizes the resulting electron-deficient 14-electron ruthenacyclobutane intermediate by $\sigma$-electron donation, lowering the activation barrier for the $[2+2]$ cycloaddition step by over $25\text{ kJ/mol}$ and boosting overall metathesis activity by $>10^4$.
(c) Isolability of Titanium (Tebbe) vs. Lability of Ruthenium Metallacyclobutanes:
1. Titanium Metallacyclobutanes (e.g., Grubbs' Titanacyclobutanes from Tebbe's Reagent):
- Titanium is in a high formal oxidation state $\text{Ti}(\text{IV})$ ($d^0$).
- It possesses strong, covalent, localized $\text{Ti}-\text{C}$ $\sigma$-bonds with high bond enthalpies ($D_0 \approx 330\text{ kJ/mol}$).
- Because $\text{Ti}(\text{IV})$ is $d^0$, there are no filled $d$-electrons to initiate reductive cycloreversion back to a low-valent titanium(II) species.
- Consequently, titanacyclobutanes sit in a deep thermodynamic energy well and can be isolated as bench-stable crystalline solids.
2. Ruthenium Metallacyclobutanes:
- Ruthenium resides in the $\text{Ru}(\text{IV})$ oxidation state ($d^4$).
- The metal center has accessible $d$-electrons that readily participate in orbital-assisted retro-$[2+2]$ cycloreversion back to the thermodynamically favored $\text{Ru}(\text{II})$ ($d^6$) alkylidene.
- The ruthenacyclobutane represents a shallow, transient intermediate on the potential energy surface, turning over millions of times per second.
Norbornene undergoes Ring-Opening Metathesis Polymerization (ROMP) with a 2nd generation Grubbs catalyst to yield polynorbornene. (a) Given ring strain enthalpy $\Delta H_\text{strain} = -110\text{ kJ/mol}$ and standard polymerization entropy $\Delta S^\circ = -85\text{ J/(mol}\cdot\text{K)}$, calculate the ceiling temperature $T_c$ of norbornene polymerization at $[M]_0 = 1.0\text{ M}$. (b) If polymerization follows living kinetics where $k_p = 140\text{ M}^{-1}\text{s}^{-1}$ and $[\text{Ru}]_0 = 1.0 \times 10^{-4}\text{ M}$, calculate the time required for $99\%$ monomer conversion. (c) Derive the theoretical Polydispersity Index (PDI) for a Poisson distribution with degree of polymerization $\overline{X}_n = 500$.
Line-by-Line Solution:
(a) Ceiling Temperature Calculation: At the thermodynamic ceiling temperature $T_c$:
Given:
- $\Delta H_p^\circ \approx \Delta H_\text{strain} = -110\text{ kJ/mol} = -110,000\text{ J/mol}$
- $\Delta S_p^\circ = -85\text{ J/(mol}\cdot\text{K)}$ (standard state $[M] = 1.0\text{ M}$)
- Conclusion: Because of the colossal ring strain of the bicyclo[2.2.1]heptene skeleton ($110\text{ kJ/mol}$), the ceiling temperature is over $1000^\circ\text{C}$. At all normal processing temperatures ($-20^\circ\text{C}$ to $100^\circ\text{C}$), ROMP of norbornene is completely irreversible and driven to $100\%$ conversion.
(b) Reaction Time for $99\%$ Monomer Conversion: For an ideal living polymerization with instantaneous initiation ($k_i \ge k_p$):
Integrating from $t = 0$ to $t$:
For $99\%$ conversion ($[M] / [M]_0 = 0.01$):
Given:
- $k_p = 140\text{ M}^{-1}\text{s}^{-1}$
- $[\text{Ru}]_0 = 1.0 \times 10^{-4}\text{ M}$
- Apparent rate constant $k_\text{app} = k_p [\text{Ru}]_0 = (140)(1.0 \times 10^{-4}) = 0.014\text{ s}^{-1}$
- $\ln(100) \approx 4.6052$
Solve for time $t$:
- The polymerization reaches $99\%$ conversion in under $5.5\text{ minutes}$.
(c) Polydispersity Index (PDI) for Living Poisson Distribution: In a living polymerization free of chain transfer and termination, the molecular weight distribution obeys a Poisson distribution:
- Number-average degree of polymerization: $\overline{X}_n = 500$.
- Weight-average degree of polymerization:
- The Polydispersity Index (PDI, $\text{Đ}$) is:
For $\overline{X}_n = 500$:
- Result: The polymer possesses near-monodisperse architecture with a theoretical PDI of 1.002.
Conventional Grubbs catalysts produce thermodynamically favored $(E)$-alkenes during cross-metathesis. Modern cyclometallated $Z$-selective catalysts (Grubbs-Hoveyda $Z$-catalysts) invert this preference to deliver $(Z)$-alkenes with $>95\%$ selectivity. (a) Draw the ruthenacyclobutane transition state for conventional $(E)$-selective metathesis versus $Z$-selective metathesis. (b) Explain the steric shielding mechanism of the bulky, bidentate $N$-arylamido or cyclometallated NHC ligand that forces both alkylidene substituents into a cis orientation. (c) Derive why $(Z)$-selectivity drops at high conversion if the catalyst is not completely stereoretentive.
Line-by-Line Solution:
(a) Ruthenacyclobutane Transition States:
1. Conventional $(E)$-Selective Metathesis (All-Trans TS):
- In unconstrained ruthenacyclobutanes, the four-membered ring adopts a puckered or planar geometry where the two substituents $R_1$ and $R_2$ orient themselves in a trans-diequatorial arrangement (pointing away from each other on opposite faces of the ring).
- This minimizes 1,2-steric repulsion between the substituents, cycloreverting to release the thermodynamically favored $(E)$-alkene.
2. $Z$-Selective Metathesis (All-Cis TS):
- The ruthenacyclobutane forces both substituents $R_1$ and $R_2$ to reside on the same face of the four-membered ring (cis-conformation).
- Cycloreversion of this cis-metallacyclobutane delivers the $(Z)$-alkene.
(b) Steric Shielding Mechanism of Cyclometallated $Z$-Catalysts:
- In modern $Z$-selective catalysts, one of the $N$-aryl groups of the NHC ligand is replaced by a sterically massive, cyclometallated adamantly, mesityl, or nitrato chelate that reaches directly over the ruthenium center.
- This creates an asymmetric, deep steric pocket:
- One quadrant of the metal coordination sphere is completely blocked by the bulky, rigid ligand architecture.
- The other quadrant remains open.
- When the two alkene fragments coordinate and form the ruthenacyclobutane:
- Orienting one substituent trans would force it to point directly into the heavily congested, blocked quadrant, incurring catastrophic steric clash ($>80\text{ kJ/mol}$).
- To avoid this clash, both substituents are forced to point together out of the open quadrant.
- Consequently, the only accessible transition state is the one where both $R_1$ and $R_2$ are cis to each other.
- Cycloreversion delivers the kinetically controlled $(Z)$-alkene with $>95\%$ stereocontrol.
(c) Secondary Metathesis and Degradation of $(Z)$-Selectivity at High Conversion:
- The desired $(Z)$-alkene is the kinetically controlled product, but it is thermodynamically less stable than the $(E)$-alkene by $\Delta G^\circ \approx 4 - 8\text{ kJ/mol}$ due to steric clash between the cis alkyl groups.
- Once the starting terminal alkene is depleted at high conversion ($>95\%$):
- The active catalyst can coordinate the newly formed $(Z)$-alkene product.
- This initiates secondary metathesis (cross-metathesis with itself or ethylene).
- If the catalyst undergoes minor decomposition or if non-stereospecific cycloreversion occurs even $1\%$ of the time, the $(Z)$-alkene will be isomerized irreversibly into the thermodynamically downhill $(E)$-alkene:
- Therefore, to preserve $>95\%\ Z$-selectivity, reactions must be halted before complete monomer depletion, or run with ultra-active catalysts under strict kinetic quenching.