Reaction Mechanisms and Physical Organic Chemistry

Arrhenius equation and activation energy

The Arrhenius equation expresses the temperature dependence of a rate constant as k = A\exp(-E_a/RT), where A is the pre-exponential factor, E_a is the activation energy, R is the gas constant and T is the absolute temperature. The pre-exponential factor has the same units as the rate constant, so its unit depends on the overall order of the reaction. For a second-order reaction, k and A have units of concentration^{-1} time^{-1}, which in SI units is \text{m}^3\,\text{mol}^{-1}\,\text{s}^{-1}.

When rate constants at two temperatures are known, the two-point form \ln(k_2/k_1) = (E_a/R)(1/T_1 - 1/T_2) gives the activation energy directly. Both temperatures must be converted to kelvin, and E_a comes out in J mol^{-1} when R is in J mol^{-1} K^{-1}.

• activation energy — the minimum energy barrier that colliding molecules must overcome; it is obtained from the slope of \ln k against 1/T.
• enthalpy of activation — \Delta^\ddagger H^\circ, the enthalpy term of the transition-state theory expression, which differs from E_a by a term that depends on molecularity.

For a bimolecular elementary step in the gas phase, the two quantities are related by E_a = \Delta^\ddagger H^\circ + 2RT, because the rate constant of a bimolecular gas-phase step carries an extra RT factor compared with a unimolecular one. In solution, the usual relation is E_a = \Delta^\ddagger H^\circ + RT.

The standard enthalpy and internal energy of activation are linked by \Delta^\ddagger H^\circ = \Delta^\ddagger U^\circ + \Delta n^\ddagger RT, where \Delta n^\ddagger is the change in the number of molecules on going from the reactants to the activated complex. When two reactant molecules form one activated complex, \Delta n^\ddagger = 1 - 2 = -1, so the enthalpy of activation is smaller than the internal energy of activation by RT.

Gibbs energy and equilibrium constant

The standard Gibbs energy change of a reaction is related to its equilibrium constant by \Delta G^\circ = -RT\ln K, so K = \exp(-\Delta G^\circ/RT). A negative \Delta G^\circ gives K > 1, and a positive value gives K < 1. Because K depends exponentially on \Delta G^\circ/RT, a change of about 5.7 kJ mol^{-1} at 298 K alters K by a factor of ten, and a reaction with \Delta G^\circ near -10 kJ mol^{-1} at 298 K has K of about 57.

For gas-phase reactions, the equilibrium constant is written in partial pressures, K_p = \prod_i p_i^{\nu_i}, with each pressure expressed in bar. The unit of K_p is bar^{\Delta n}, so a reaction whose gas moles decrease gives a K_p with a negative power of bar. The equilibrium composition is found from an ICE table: the stoichiometric coefficients fix how many moles of each species are consumed or formed, and the partial pressures at equilibrium are then substituted into the expression for K_p.

For a reversible isothermal process, the maximum work that can be extracted equals the decrease in Helmholtz energy, -W_{\max} = \Delta A = \Delta U - T\Delta S. For gaseous species, \Delta U = \Delta H - \Delta n_{\text{gas}}RT, so when gas moles decrease the internal energy change is less negative than the enthalpy change. When the question defines the work in terms of constant-pressure conditions, the corresponding quantity is \Delta G, so the definition in the question must be followed.

Faraday's laws of electrolysis

Faraday's first law states that the amount of substance liberated at an electrode is proportional to the quantity of electricity passed through the cell. The charge is Q = It, with I in amperes and t in seconds, so any time given in hours or minutes must be converted first. The applied cell voltage does not appear in this calculation, because the amount of electrolysis depends only on the charge passed.

Faraday's second law states that the masses of different substances deposited or dissolved by the same charge are proportional to their equivalent masses. The number of moles of electrons is n_e = Q/F, with F = 96485 C mol^{-1}, and the moles of product are n_e/z, where z is the number of electrons per ion or atom. For example, Ag^+ + \text{e}^- \rightarrow \text{Ag} has z = 1, while Cu \rightarrow Cu^{2+} + 2\text{e}^- has z = 2, so equal charges dissolve half as many moles of copper as they deposit moles of silver.

• equivalent mass — molar mass divided by the number of electrons transferred per ion.
• faraday — the charge of one mole of electrons, approximately 96485 C mol^{-1}.

Raoult's law for ideal solutions

For an ideal solution, Raoult's law states that the partial vapour pressure of each component is p_i = x_i p_i^\circ, where x_i is the mole fraction of the component in the liquid and p_i^\circ is the vapour pressure of the pure component. The total vapour pressure is the sum of the partial pressures. Dalton's law then gives the vapour-phase mole fraction as y_i = p_i/P, which is not the same as the liquid-phase composition.

Because the more volatile component has the larger p_i^\circ, it is enriched in the vapour relative to the liquid. Mole fractions must be calculated from amounts in moles, not from masses or molarities, before the vapour composition is determined.

For a solution containing a non-volatile solute, Raoult's law gives the relative lowering of vapour pressure as \Delta P/P^\circ = x_{\text{solute}}. Writing x_{\text{solute}} = n/(n+N), where n is the moles of solute and N is the moles of solvent, the amount of solute is found and the molar mass is obtained as mass divided by n.

First-order radioactive decay

Radioactive decay follows first-order kinetics, so N = N_0\,e^{-kt} and the decay constant is k = \ln 2/t_{1/2}. The half-life does not depend on the initial amount of the sample. The fraction remaining after time t is e^{-kt}, and the fraction that has disintegrated is 1 - e^{-kt}. When kt is small, the disintegrated fraction can be approximated as kt, but the exponential must be kept when kt is not small.

For radiocarbon dating, the age follows from t = (t_{1/2}/\ln 2)\ln(A_0/A), where A_0 is the initial activity of the reference material and A is the activity of the sample. The count rate measured by a counter always includes a background contribution, which must be subtracted from both the sample and the reference before the ratio is taken. The rates must also be expressed in the same time units, such as counts per minute per gram of carbon.

• half-life — the time needed for half of the nuclei to decay, t_{1/2} = \ln 2/k.
• background count — counts recorded without any radioactive sample, subtracted from every measurement.

Langmuir adsorption isotherm

The Langmuir isotherm assumes monolayer adsorption on identical, independent surface sites. The fractional coverage is \theta = KP/(1+KP), where K = k_a/k_d is the equilibrium constant for adsorption, equal to the ratio of the rate constants for adsorption and desorption. At low pressure, KP \ll 1 and \theta \approx KP, so the coverage is linear in pressure. At high pressure, KP \gg 1 and \theta \rightarrow 1, corresponding to a saturated surface.

Because \theta depends only on the product KP, a fixed coverage requires that KP is constant at a fixed temperature. K must therefore vary inversely with P, so K \propto 1/P.

For a unimolecular surface reaction that follows the Langmuir mechanism, the rate is r = k\theta = kKP/(1+KP). At low pressure the rate is proportional to P, which is first order. At high pressure the surface is saturated, the rate becomes independent of P, and the reaction is zero order. Since the rate is directly proportional to the fraction of the surface covered by the reactant, the two limits follow from the coverage expression. The Freundlich isotherm is purely empirical and does not describe the saturation of a monolayer, so it is not the relevant model for this mechanism.

Solvolysis and carbocation rearrangement

Solvolysis is a substitution or elimination reaction in which the solvent acts as the nucleophile or base. Ionisation of a substrate with a good leaving group gives a carbocation, which can rearrange through 1,2-hydride or 1,2-alkyl or aryl shifts to a more stable cation before the solvent captures it.

Neighbouring-group participation is a special case in which a nearby group with lone pairs or \pi electrons displaces the leaving group intramolecularly. An aryl group forms a bridged phenonium ion, and the solvent then opens this ion. Two successive inversions, one at the participation step and one at the solvent attack, give a net retention of configuration, and the skeleton of the product follows from which carbon of the bridged ion is attacked.

Not every rearrangement is ionic, and the mechanism must be identified from the conditions. The di-\pi-methane rearrangement is a photochemical process in which two alkenes attached to one sp^3 carbon form a 1,3-diradical intermediate that closes to a cyclopropane. The barrelene-to-semibullvalene conversion is the classic example. A divinylcyclopropane rearrangement is a thermal [3,3] sigmatropic change, and a [2+2+2] cycloaddition requires three separate \pi components, so neither describes a photochemical intramolecular reorganisation of a single triene.

Adsorption isotherm models

The Freundlich isotherm is empirical, \theta = k p^{1/n}, where k and n are constants obtained by fitting. It does not predict saturation, but its two limits can be compared with the Langmuir isotherm. At low coverage, the Langmuir isotherm becomes linear, \theta \approx Kp, which corresponds to the Freundlich form with n = 1. At high coverage, \theta \rightarrow 1 becomes constant, which corresponds to 1/n \rightarrow 0, that is, n \rightarrow \infty.

The BET isotherm describes multilayer adsorption and is written as \dfrac{p}{p^*-p} on the left side with a term (c-1)p/p^* on the right side, where p^* is the saturation vapour pressure and c is a constant. At very low pressure, p \ll p^*, the BET isotherm reduces to the Langmuir monolayer isotherm. As p \rightarrow p^*, the left-hand side diverges, which corresponds to unlimited layering and not to the saturation of one monolayer.

The BET equation is usually linearised as \dfrac{z}{(1-z)V} = \dfrac{1}{V_mC} + \dfrac{C-1}{V_mC}z, where z = p/p^\circ and V_m is the volume of a monolayer. The sum of the slope and the intercept equals 1/V_m, so V_m = 1/(\text{slope} + \text{intercept}). The number of molecules in the monolayer is obtained by converting V_m with the molar volume of the gas, and the surface area follows by multiplying the number of molecules by the area each molecule occupies.

Michaelis–Menten enzyme kinetics

The Michaelis–Menten rate law is v = V_{\max}[S]/(K_M + [S]), where V_{\max} = k_{\text{cat}}[E]_0 and K_M is the substrate concentration at half the maximum rate. The Lineweaver–Burk form is \dfrac{1}{v} = \dfrac{K_M}{V_{\max}}\cdot\dfrac{1}{[S]} + \dfrac{1}{V_{\max}}, so the intercept on the 1/v axis is 1/V_{\max} and the slope is K_M/V_{\max}. Therefore V_{\max} = 1/\text{intercept} and K_M = \text{slope} \times V_{\max}.

The turnover number is k_{\text{cat}} = V_{\max}/[E]_0. Two enzymes with equal intercepts have equal V_{\max}, and if their enzyme concentrations are also equal, they have equal turnover numbers. A different slope at the same intercept means a different K_M. Since K_M = (k_{-1} + k_2)/k_1, the difference in K_M reflects different rate constants of the elementary steps, so the individual step rates are not identical.

• turnover number — the number of substrate molecules converted per enzyme molecule per unit time at saturation.

Substituent effects on ester hydrolysis

Alkaline hydrolysis, or saponification, begins with the addition of hydroxide to the carbonyl carbon, so the rate depends on how electrophilic the carbonyl carbon is and how easily the nucleophile can reach it. Electron-withdrawing groups near the carbonyl increase the rate, while electron-donating groups decrease it. Amides are the least reactive acyl derivatives because the nitrogen lone pair donates strongly into the C=O group, which lowers the electrophilicity of the carbonyl carbon.

Steric hindrance also controls the rate. On a cyclohexane ring, an equatorial ester is less hindered toward an approaching nucleophile than an axial ester, which is shielded by the axial hydrogens on the same face of the ring. The rate ranking of a set of esters and amides therefore combines electronic effects, which rank the acyl derivatives, with conformational effects, which rank the ester positions.

• saponification — base-promoted hydrolysis of an ester to a carboxylate and an alcohol.
• Bürgi–Dunitz angle — the approach angle of the nucleophile to the carbonyl, about 107°, which the axial substituent may block.

Acidity trends of organic compounds

Acidity depends on the stability of the conjugate base. Sulfonic acids have negative pKa values and are the strongest common acids, carboxylic acids have pKa values near 4 to 5, and phenols have pKa values near 10. Water has a pKa near 16, and alcohols are slightly weaker acids still.

For C–H acids, acidity increases with the s character of the carbon. A terminal alkyne C–H, with sp hybridisation, has a pKa of roughly 25, which is far less acidic than any oxygen acid. An electron-withdrawing ester group on the alkyne lowers this value somewhat, but it does not bring the alkyne close to the acidity of a carboxylic acid or a phenol.

• pKa — the negative logarithm of the acid dissociation constant; a smaller pKa means a stronger acid.
• conjugate base stability — increases with resonance delocalisation, electronegativity of the charged atom and the s character of the orbital holding the charge.

Carnot heat engine efficiency

A reversible heat engine that absorbs heat Q_h from a hot reservoir at T_h and rejects heat Q_c to a cold reservoir at T_c has the maximum efficiency \eta = W/Q_h = 1 - T_c/T_h. Both temperatures must be absolute, in kelvin. The work output is W = \eta Q_h, and the heat rejected to the cold reservoir is Q_c = (1-\eta)Q_h. The efficiency is expressed as a percentage by multiplying by 100.

No real engine working between the same two reservoirs can exceed this efficiency, and the efficiency increases as the temperature ratio T_c/T_h decreases.

• reversible engine — an engine operating without irreversible losses, so its efficiency depends only on the reservoir temperatures.

Enthalpy change stoichiometry

The enthalpy change written with a reaction equation refers to the reaction exactly as written, that is, to the stoichiometric amounts shown in the balanced equation. Enthalpy is an extensive property, so the enthalpy change for a different amount of reaction is obtained by scaling it in proportion to the extent of reaction, \xi = n_{\text{consumed}}/\nu, where \nu is the stoichiometric coefficient of the species.

The sign of the enthalpy change is retained when the amount is scaled, and the physical states of the reactants and products must match those given in the equation, because the enthalpy of a liquid differs from that of a gas. Converting a given mass into moles with the molar mass is the first step before scaling.

• extent of reaction — the amount of reaction that has proceeded, measured in moles per stoichiometric unit.

E2 elimination regio- and stereoselectivity

E2 elimination is a concerted, bimolecular reaction in which a base removes a \beta-hydrogen while the leaving group departs. It requires an anti-periplanar arrangement of the \beta C–H bond and the C–leaving group bond. With a strong base such as methoxide, E2 competes with SN2 for a secondary alkyl halide, and the more substituted alkene is formed as the major product under the Zaitsev rule.

For an acyclic substrate, the anti-periplanar geometry fixes the relative configuration of the new double bond. The E or Z descriptor is assigned by Cahn–Ingold–Prelog priorities: on each alkene carbon the higher-priority substituent is identified, and the descriptor is E when the two higher-priority groups lie on opposite sides of the double bond.

• anti-periplanar — the geometry in which the departing groups lie in the same plane and on opposite sides of the C–C bond.
• Zaitsev product — the more highly substituted alkene formed by elimination.

Reversible work of gas expansion

For expansion against a constant external pressure, the work is w = -P\Delta V, and for a reversible isothermal expansion of an ideal gas it is w = -nRT\ln(V_f/V_i). The sign convention takes work done on the system as positive, so an expansion gives negative w for the system. The magnitude of the work is what is required when the question asks for the work done in absolute value.

Units must be consistent when the pressure is in atmospheres and the volume in litres. The conversion 1\ \text{L atm} = 101.325\ \text{J} is used to express the work in joules.

• reversible process — a process carried out through a series of equilibrium states, so it gives the maximum work.

Differential thermal analysis principle

Differential thermal analysis, DTA, records the temperature difference between a sample and an inert reference as both are heated or cooled at the same rate. Physical or chemical changes in the sample, such as melting, crystallisation or decomposition, make the temperature difference deviate from the baseline, and these deviations are plotted as a function of the furnace temperature. An endothermic event gives a peak in one direction and an exothermic event gives a peak in the other.

DTA is distinct from related methods. Differential scanning calorimetry measures the difference in heat flow to the sample and reference. Thermogravimetric analysis measures the change in mass. A glass transition in DTA appears as a step change in the baseline rather than as a sharp peak, because it is a change in heat capacity and not a latent heat effect.

• reference material — an inert substance with no thermal event in the temperature range of the experiment.

Carbon dioxide phase diagram

A pressure–temperature phase diagram shows the regions where each phase is stable, separated by the sublimation, vaporisation and fusion curves that meet at the triple point. The slope of the solid–liquid curve follows from the Clausius–Clapeyron relation, dP/dT = \Delta H_{\text{fus}}/(T\Delta V_{\text{fus}}). When the solid is denser than the liquid, \Delta V_{\text{fus}} > 0 and the melting point rises with pressure. When the solid is less dense than the liquid, as for water, the slope is negative and the melting point falls with pressure.

The position of the triple point decides whether a substance has a normal boiling point. If the triple point lies below 1 atm, the liquid can exist at normal pressure. If the triple point lies above 1 atm, the substance sublimes at 1 atm and has no normal boiling point. Carbon dioxide is the textbook example of the second case, which is why solid carbon dioxide sublimes at atmospheric pressure. An isothermal compression at a temperature below the triple point temperature crosses the sublimation curve, so the solid forms before any liquid.

• triple point — the unique pressure and temperature at which solid, liquid and gas coexist in equilibrium.

Canonical ensemble constraints

An ensemble is defined by the variables that are held fixed. The canonical ensemble describes a closed system of fixed number of particles N, fixed volume V and fixed temperature T. Fixed temperature is achieved by thermal contact with a large heat reservoir, and fixed volume follows from the rigid walls of the system. The number of particles of each chemical species is also fixed, so the composition is constant.

Pressure and energy are not fixed in the canonical ensemble. They fluctuate among the microstates, and their average values follow from the equation of state and the internal energy. Other ensembles fix different variables: the microcanonical ensemble fixes E, the isothermal–isobaric ensemble fixes P and T, and the grand canonical ensemble fixes the chemical potential.

• canonical ensemble — the ensemble of constant N, V and T.

Entropy change of surroundings in phase change

For a process at constant temperature and pressure, the entropy change of the surroundings is \Delta S_{\text{surr}} = -\Delta H_{\text{sys}}/T, because heat exchanged with the surroundings at temperature T is equal and opposite to the enthalpy change of the system. This applies even when the process is irreversible, such as the freezing of a supercooled liquid, since the surroundings are at a fixed temperature.

When the process occurs at a temperature different from the transition temperature, the enthalpy change is obtained by Kirchhoff's law, \Delta H(T_2) = \Delta H(T_1) + \int_{T_1}^{T_2}\Delta C_p\,dT, where \Delta C_p = C_p(\text{products}) - C_p(\text{reactants}). The enthalpy change of the system must be computed along a path that first brings the reactant to the transition temperature, performs the transition and then changes the temperature of the product. The sign of \Delta S_{\text{surr}} is positive when the surroundings gain heat, which occurs when the system releases heat during freezing.

• supercooled liquid — a liquid cooled below its normal freezing point without solidifying, a metastable state.

Norrish type I and II photoreactions

Carbonyl compounds absorbing near-UV light, typically above 300 nm, are excited into an n\pi^* state, from which two characteristic photoreactions occur. Norrish type I is the \alpha-cleavage of the C–C bond next to the carbonyl group, which gives an acyl radical and an alkyl radical. It is favoured when the alkyl radical formed is stabilised, for example as a tertiary or benzylic radical.