Rate evidence, mechanisms and activation energy

What you’ll learn

• Connect a rate equation to a plausible slow step.
• Interpret activation-energy plots.
• Explain how catalysts change reaction rates.

Understand the idea

A mechanism breaks an overall reaction into elementary steps. The slow step restricts the overall rate; its reactants, and any earlier equilibria supplying them, must be consistent with the observed rate equation. First-order hydrolysis in a tertiary halogenoalkane supports a slow ionisation step in an $\mathrm{S_N1}$ mechanism. Dependence on both a primary halogenoalkane and a nucleophile supports a one-step $\mathrm{S_N2}$ pathway.

In the Arrhenius equation, a plot of $\ln k$ against $1/T$ has gradient $-E_a/R$. Use kelvin and consistent joule units. A catalyst offers an alternative pathway with a lower activation barrier. A heterogeneous catalyst operates through adsorption, reaction at its surface and desorption; a homogeneous catalyst forms intermediates in the same phase and is regenerated. Neither changes the equilibrium constant at a fixed temperature.

Key formulae

$$k=Ae^{-E_a/(RT)}$$

$$\ln k=\ln A-\frac{E_a}{R}\frac1T$$

$$E_a=-R\times\text{gradient}$$

$$\mathrm{rate}_{S_N1}=k[\mathrm{RX}],\qquad\mathrm{rate}_{S_N2}=k[\mathrm{RX}][\mathrm{Nu^-}]$$

Worked example

A plot of $\ln k$ against $1/T$ has gradient $-7200\,\mathrm K$. Calculate $E_a$ in $\mathrm{kJ\,mol^{-1}}$ using $R=8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$.

1. Use $E_a=-R\times\text{gradient}$.
2. Calculate $-8.314(-7200)=59860.8\,\mathrm{J\,mol^{-1}}$.
3. Divide by $1000$ to convert joules to kilojoules.

Answer: $E_a=59.9\,\mathrm{kJ\,mol^{-1}}$ (3 s.f.).