Chemical Kinetics — Notes

The rate of a reaction is the change in concentration of a reactant or product per unit time (unit $\text{mol L}^{-1}\ \text{s}^{-1}$; for gases, $\text{atm s}^{-1}$). For $\mathrm{R} \rightarrow \mathrm{P}$: average rate $= -\dfrac{\Delta[\ce{R}]}{\Delta t} = \dfrac{\Delta[\ce{P}]}{\Delta t}$; the instantaneous rate is the slope of the tangent to the concentration–time curve, $-\dfrac{d[\ce{R}]}{dt}$.

For $\ce{aA + bB -> cC + dD}$, rate $= -\dfrac{1}{a}\dfrac{\Delta[\ce{A}]}{\Delta t} = -\dfrac{1}{b}\dfrac{\Delta[\ce{B}]}{\Delta t} = \dfrac{1}{c}\dfrac{\Delta[\ce{C}]}{\Delta t} = \dfrac{1}{d}\dfrac{\Delta[\ce{D}]}{\Delta t}$.

Factors: concentration (or pressure) of reactants, temperature, catalyst, surface area of solids, nature of reactants (and, for photochemical reactions, light).

Rate law: rate $= k[\ce{A}]^x[\ce{B}]^y$, with $x$ and $y$ found experimentally (they need not equal the coefficients). $k$ is the rate constant: the rate at unit concentrations; it depends on temperature, not on concentration.

Order $= x + y$; it may be 0, a whole number or a fraction. Unit of $k$ for order $n$: $(\text{mol L}^{-1})^{1-n}\ \text{s}^{-1}$ — zero order $\text{mol L}^{-1}\ \text{s}^{-1}$, first $\text{s}^{-1}$, second $\text{L mol}^{-1}\ \text{s}^{-1}$.

Method of initial rates: compare runs in which only one concentration changes: $\dfrac{r_2}{r_1} = \left(\dfrac{[\ce{A}]_2}{[\ce{A}]_1}\right)^x$.

Molecularity is the number of reacting species colliding in an elementary step: 1, 2 or (rarely) 3; never zero or fractional. For a complex reaction, the slowest step is rate-determining; e.g. iodide-catalysed decomposition of $\ce{H2O2}$: slow $\ce{H2O2 + I- -> H2O + IO-}$, fast $\ce{H2O2 + IO- -> H2O + I- + O2}$, so rate $= k[\ce{H2O2}][\ce{I-}]$.

Zero order: $[\ce{R}] = [\ce{R}]_0 - kt$; $[\ce{R}]$ vs $t$ is a straight line of slope $-k$; $t_{1/2} = \dfrac{[\ce{R}]_0}{2k}$. Examples: $\ce{2NH3 -> N2 + 3H2}$ on hot platinum at high pressure; decomposition of HI on gold.

First order: $\ln[\ce{R}] = \ln[\ce{R}]_0 - kt$, or $k = \dfrac{2.303}{t}\log\dfrac{[\ce{R}]_0}{[\ce{R}]}$; $\log[\ce{R}]$ vs $t$ is a straight line of slope $-k/2.303$; $t_{1/2} = \dfrac{0.693}{k}$, independent of $[\ce{R}]_0$. After $n$ half-lives, $[\ce{R}] = [\ce{R}]_0/2^n$. Examples: radioactive decay, decomposition of $\ce{N2O5}$, hydrogenation of ethene. For a gas-phase reaction $\mathrm{A(g)} \rightarrow \mathrm{B(g)} + \mathrm{C(g)}$ followed by total pressure, $k = \dfrac{2.303}{t}\log\dfrac{p_0}{2p_0 - p_t}$.

Pseudo-first-order: one reactant in large excess (e.g. water in the acid hydrolysis of ethyl ethanoate, inversion of cane sugar) so that rate $\approx k'[\text{other reactant}]$.

For many reactions a 10 K rise roughly doubles the rate. Arrhenius equation: $k = Ae^{-E_\text{a}/RT}$, where $A$ is the frequency factor and $E_\text{a}$ the activation energy; $e^{-E_\text{a}/RT}$ is the fraction of molecules with energy at least $E_\text{a}$. $\ln k = \ln A - \dfrac{E_\text{a}}{RT}$: a plot of $\ln k$ against $1/T$ is a straight line of slope $-E_\text{a}/R$.

Two temperatures: $\log\dfrac{k_2}{k_1} = \dfrac{E_\text{a}}{2.303R}\left(\dfrac{T_2 - T_1}{T_1T_2}\right)$, with $R = 8.314\ \text{J K}^{-1}\ \text{mol}^{-1}$.

Activation energy is the extra energy reactants need to form the activated complex at the top of the energy profile. At higher temperature, the Maxwell–Boltzmann distribution spreads out and the fraction of molecules beyond $E_\text{a}$ rises sharply.

Catalyst: provides an alternative path of lower $E_\text{a}$; is unchanged at the end; does not change $\Delta H$, $\Delta G$ or $K$; speeds up forward and backward reactions equally.

Reactant molecules behave as hard spheres; a reaction happens only through effective collisions: those with energy at least the threshold energy and with proper orientation. Rate $= P\,Z_\text{AB}\,e^{-E_\text{a}/RT}$, where $Z_\text{AB}$ is the collision frequency and $P$ the probability (steric) factor.

Raising the concentration increases $Z_\text{AB}$; raising the temperature increases $Z_\text{AB}$ slightly and the energetic fraction greatly; a catalyst lowers $E_\text{a}$. Limitation: molecules are treated as structureless spheres, so $P$ is an empirical correction.