Chemistry · Analytical and physical chemistry · Year 2

Chemical kinetics and Arrhenius

When does a medicine become unusable, and how can a test at 40 °C tell us?

The situation

In the quality-control laboratory of a pharmaceutical factory a new batch of medicine is being tested: samples are kept in thermostat chambers at 25, 40 and 60 °C, while a spectrophotometer and a conductivity meter record the progress of the reactions minute by minute. To say that a medicine is “good for two years” you do not have to wait two years — the shelf life at room temperature is calculated from the decomposition accelerated by heat. In the next room, behind a protective screen, a mixture of hydrogen and chlorine explodes from a single flash. How does the rate of a reaction depend on concentration and temperature, how are the order and the activation energy found, and why do some reactions “run away” as chains?

Topic

Chemical kinetics: order, Arrhenius, complex reactions. Quality-control laboratory of a pharmaceutical company: when does a medicine tablet become unfit for use? Samples in thermostats (25, 40, 60 °C), a spectrophotometer and…

Key ideas

Order of reaction
Arrhenius equation
Complex reactions

Formulas

v = - \frac{d c}{d t} = k c^{n}

The kinetic equation (differential form): $n$ is the order of reaction, $k$ the rate constant; $n$ is found from experiment.

\ln c = \ln c_{0} - k t, \frac{1}{c} = \frac{1}{c_{0}} + k t

Integral forms: first and second order (equal initial concentrations). Half-life $t_{1 / 2} = \ln 2 / k$ and $1 / (k c_{0})$ .

k = A e^{- E_{a} / R T}, \ln \frac{k_{2}}{k_{1}} = \frac{E_{a}}{R} (\frac{1}{T_{1}} - \frac{1}{T_{2}})

The Arrhenius equation: the graph of $\ln k$ against $1 / T$ is a straight line with slope $- E_{a} / R$ .

\lg k = \lg k_{0} + 1.02 z_{A} z_{B} \frac{\sqrt{I}}{1 + \sqrt{I}}

The Brønsted–Bjerrum equation (water, 25 °C): the dependence of the rate of a reaction between ions on the ionic strength $I$ of the solution.

k = \frac{k_{B} T}{h} e^{Δ S^{‡} / R} e^{- Δ H^{‡} / R T}

The Eyring equation (transition state theory): $k_{B} T / h$ ≈ 6.2·10¹² s⁻¹ (298 K).

Theory

Rate, order and molecularity

The rate of a reaction is the change of concentration in unit time: $v = - d c / d t$ . According to the law of mass action (Guldberg and Waage), $v = k {[A]}^{a} {[B]}^{b}$ ; the sum of the exponents, $n = a + b$ , is the order of reaction. The order is found only from experiment: it can be an integer, a fraction or zero. Molecularity, on the other hand, is the number of particles colliding in a single elementary step (uni-, bi- or termolecular) — it is always a whole number and has meaning only for the elementary steps of a reaction mechanism. Integral equations: zero order $c = c_{0} - k t$ , $t_{1 / 2} = c_{0} / 2 k$ ; first order $\ln c = \ln c_{0} - k t$ , $t_{1 / 2} = \ln 2 / k$ — independent of concentration; second order $1 / c = 1 / c_{0} + k t$ , $t_{1 / 2} = 1 / (k c_{0})$ . The unit of $k$ depends on the order: mol/(L·s), s⁻¹, L/(mol·s). If one of the reactants is in large excess, its concentration hardly changes and the reaction proceeds with a pseudo-order: crystal violet (CV⁺) with 0.02 mol/L NaOH — pseudo-first order, $k^{'} = k [O H^{-}]$ . The simple effect of concentration and temperature on rate is in the “Rate of reaction and catalyst” experiment.

Methods of determining the order (laboratory)

Integral method: plot the measured $c (t)$ in the coordinates $c$ , $\ln c$ , $1 / c$ — whichever gives a straight line (the coefficient of determination $R^{2}$ closest to 1) shows the order; $k$ comes from the slope. Differential method (van 't Hoff): $\lg v = \lg k + n \lg c$ — the slope is $n$ . Half-life method: $t_{1 / 2}$ is measured at different $c_{0}$ , $n = 1 + \lg (t_{1 / 2}^{'} / t_{1 / 2}^{''}) / \lg (c_{0}^{''} / c_{0}^{'})$ . The Ostwald isolation method: all reactants but one are taken in excess and the order with respect to that one is found separately; then the concentration of the excess reactant is changed ( $k^{'} \propto {[O H^{-}]}^{1}$ → overall order 2). Instruments: the spectrophotometer — the Bouguer–Lambert–Beer law $A = ε l c$ (CV⁺: 590 nm, ε ≈ 8.7·10⁴ L/(mol·cm)); the conductivity meter — in saponification the mobile OH⁻ (198 S·cm²/mol) is replaced by the slower CH₃COO⁻ (41) and $κ$ falls; in the hydrolysis of methyl ethanoate a 5 mL sample is titrated with 0.5 mol/L NaOH every 10 minutes, $k t = \ln \frac{V_{\infty} - V_{0}}{V_{\infty} - V_{t}}$ . The effect of a catalyst: hydrolysis is accelerated by H⁺ ions, $k = k_{H} [H^{+}]$ — if you double the HCl concentration, $k$ doubles too (catalysis itself is in the “Catalysis and enzymes” experiment). Safety: NaOH and HCl burn skin and eyes — wear goggles and gloves; crystal violet stains; esters are highly flammable.

Temperature and ionic strength: van 't Hoff, Arrhenius, Brønsted

When the temperature rises by 10 °C, the rate increases 2–4 times (van 't Hoff's rule). The exact law is the Arrhenius equation $k = A e^{- E_{a} / R T}$ : the slope of the graph of $\ln k$ against $1 / T$ is $- E_{a} / R$ ; $E_{a}$ can also be found from $k$ at two temperatures. For example, for the acid inversion of sucrose $E_{a}$ ≈ 108 kJ/mol. In an accelerated stability test (ICH Q1A) a medicine is stored for 6 months at 40 ± 2 °C and 75 % humidity; from the values of $k$ at different temperatures the Arrhenius straight line is extended to 25 °C and the shelf life $t_{90} = \ln (10 / 9) / k = 0.105 / k$ (10 % decomposition) is calculated. When assessing the storage conditions of a medicine, 83 kJ/mol is often taken for $E_{a}$ (the mean kinetic temperature formula). Ionic strength $I = \frac{1}{2} \sum c_{i} z_{i}^{2}$ : in a reaction between ions the charge of the activated complex is $z_{A} + z_{B}$ ; the ionic atmosphere stabilises it or the reverse — the primary salt effect: $\lg k = \lg k_{0} + 1.02 z_{A} z_{B} \sqrt{I} / (1 + \sqrt{I})$ . $S_{2} O_{8}^{2 -} + 2 I^{-} \overset{}{\to} 2 SO_{4}^{2 -} + I_{2}$ ( $z_{A} z_{B} = + 2$ ) — speeds up if KNO₃ is added; CV⁺ + OH⁻ ( $- 1$ ) — slows down; neutral ester + OH⁻ (0) — unchanged. The iodine clock: thiosulfate immediately reduces the iodine as it forms, $I_{2} + 2 S_{2} O_{3}^{2 -} \overset{}{\to} 2 I^{-} + S_{4} O_{6}^{2 -}$ ; the moment the thiosulfate is used up, starch turns blue, so $1 / t$ is proportional to the initial rate.

Complex reactions

In a reversible reaction $A \overset{}{⇌} B$ equilibrium is approached at a rate $k_{1} + k_{- 1}$ , $K = k_{1} / k_{- 1}$ (the α ⇌ β mutarotation of glucose in water stops at a ratio of ≈ 36 : 64). In parallel reactions $A \overset{}{\to} B$ , $A \overset{}{\to} C$ A disappears at $k_{1} + k_{2}$ , and the ratio of products is always $[B] / [C] = k_{1} / k_{2}$ . In consecutive reactions $A \overset{}{\to} B \overset{}{\to} C$ the intermediate B first accumulates and then is used up; the maximum is at $t_{m a x} = \ln (k_{1} / k_{2}) / (k_{1} - k_{2})$ . The overall rate is set by the slowest step — the rate-determining step. The steady-state method (Bodenstein): if B is very reactive ( $k_{2} ≫ k_{1}$ ), $d [B] / d t \approx 0$ and $[B] \approx k_{1} [A] / k_{2}$ — the main way of simplifying complex mechanisms. In life: the substance of a swallowed tablet passes from the stomach into the blood and is removed from the blood — the concentration of the drug in the blood changes exactly along the curve of a consecutive reaction. The vessels in the scene are a hydraulic model: the flow through a capillary is proportional to the liquid height, so the volumes obey the equations of first-order kinetics exactly.

Active collisions and the transition state

The collision theory (Arrhenius, Trautz, Lewis): $k = P Z_{0} e^{- E_{a} / R T}$ — $Z_{0}$ is the number of collisions, $P$ the steric factor, and $e^{- E_{a} / R T}$ the fraction of collisions whose energy along the line of centres exceeds $E_{a}$ . The “tail” of the Maxwell–Boltzmann distribution grows sharply as the temperature rises: for $E_{a}$ = 50 kJ/mol the fraction at 300 K is ≈ 2·10⁻⁹, and at 310 K it is ≈ 1.9 times larger — the reason for van 't Hoff's rule. Transition state theory (Eyring, Evans, Polanyi, 1935): a reaction follows the lowest path (the minimum-energy path) across the potential energy surface; its highest point is the saddle point, the activated complex. For $H + H_{2} \overset{}{\to} H_{2} + H$ on the LEPS surface the saddle point is at $r_{1} = r_{2}$ ≈ 0.93 Å, and the barrier is ≈ 0.42 eV (≈ 40 kJ/mol). In $k = (k_{B} T / h) e^{Δ S^{‡} / R} e^{- Δ H^{‡} / R T}$ both the enthalpy and the entropy of activation (how “ordered” the complex is) matter.

Chain and photochemical reactions; safety and the model

$H_{2} + Cl_{2} \overset{}{\to} 2 HCl$ hardly proceeds in the dark, but in light it goes as a chain (Bodenstein, 1913; mechanism — Nernst, 1918): initiation $C l_{2} \overset{\overset{}{h ν}}{\to} 2 C l^{∙}$ , propagation $C l^{∙} + H_{2} \to H C l + H^{∙}$ and $H^{∙} + C l_{2} \to H C l + C l^{∙}$ , termination — radicals combine on the wall or with O₂ (an inhibitor). The quantum yield $Φ$ is the number of molecules formed per absorbed photon: in this reaction it is 10⁴–10⁶ (by the Stark–Einstein law, in the primary act one photon excites one molecule; the rest is the chain). If heat is released faster than it is dissipated, a thermal explosion occurs (Semenov); in branched chains ( $H_{2} + O_{2}$ ) the number of radicals itself multiplies. Safety: the H₂ + Cl₂ mixture is handled only by the teacher, behind a protective screen, in a fume cupboard. Model: solutions are ideal (activities = concentrations, with Debye–Hückel only in the salt effect), 25 °C; the values of $k$ and $E_{a}$ for CV⁺, the esters, persulfate and the drug are model numbers chosen to be of the order found in the literature; a small random error is added to the measurements; time is sped up (as stated on the panel); LEPS is an approximate surface; the balls on the hill are an energy analogy; the explosion criterion is schematic.

Did you know?

The first scientist to measure the rate of a reaction quantitatively was Ludwig Wilhelmy (1850): observing the acid inversion of sucrose with a polarimeter, he showed that the rate is proportional to the amount of sucrose remaining.
The concept of activation energy was introduced by Svante Arrhenius in 1889 to explain the dependence of the rate of sucrose inversion on temperature; in 1903 he received the Nobel Prize for the theory of electrolytic dissociation.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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