Rate of a reaction
The rate of a reaction is the change in concentration of a reactant or product per unit time. For R → P:
rate = −d[R]/dt = +d[P]/dt
The minus sign makes the rate positive, because the reactant concentration falls. For a general reaction aA + bB → cC + dD, rates are divided by the coefficients so that every species gives the same value:
rate = −(1/a) d[A]/dt = −(1/b) d[B]/dt = (1/c) d[C]/dt = (1/d) d[D]/dt
The rate depends on concentration, temperature, catalysts and, for reactions involving solids, surface area.
Rate law, order and molecularity
The rate law is found by experiment:
rate = k [A]ˣ [B]ʸ
k is the rate constant.
x + y is the order of the reaction. Order can be 0, a whole number or a fraction, and it cannot be predicted from the balanced equation.
Molecularity is the number of molecules colliding in an elementary step. It is always a positive whole number (1, 2 or rarely 3).
For a complex reaction, the overall rate is decided by the slowest step, called the rate-determining step.
The units of k depend on the order: (mol L⁻¹)¹⁻ⁿ s⁻¹. For a zero-order reaction this is mol L⁻¹ s⁻¹, and for a first-order reaction it is s⁻¹.
Zero-order reactions
The rate does not depend on concentration: rate = k.
[R] = [R]₀ − k t, and t½ = [R]₀ / (2k)
Example: the decomposition of NH₃ on a hot platinum surface at high pressure. The surface is fully covered, so adding more gas does not speed it up.
First-order reactions
rate = k[R], and the integrated form is k = (2.303/t) log ([R]₀/[R])
The half-life is the time for the concentration to fall to half:
t½ = 0.693 / k
The half-life of a first-order reaction does not depend on the initial concentration. Radioactive decay is a first-order process.
Example. A first-order reaction has k = 0.0693 min⁻¹. t½ = 0.693 / 0.0693 = 10 min. After 30 min (three half-lives), 1/8 of the reactant remains, so 87.5% has reacted.
Pseudo-first-order reactions look first order because one reactant is present in large excess. The acid hydrolysis of an ester in plenty of water is an example: rate = k′[ester].
Temperature and the Arrhenius equation
For many reactions, the rate roughly doubles for every 10 °C rise in temperature. The Arrhenius equation describes this:
k = A e^(−Ea / RT)
Ea is the activation energy: the minimum energy colliding molecules need to react.
A is the frequency (pre-exponential) factor.
e^(−Ea/RT) is the fraction of molecules with energy at least Ea.
Taking logarithms at two temperatures:
log (k₂/k₁) = (Ea / 2.303R) × (T₂ − T₁)/(T₁T₂)
A plot of ln k against 1/T is a straight line with slope −Ea/R.
Raising the temperature increases the fraction of molecules with enough energy to react, so the rate rises sharply.
Catalysts
A catalyst increases the rate of a reaction without itself being used up. It provides an alternative pathway with lower activation energy.
A catalyst:
does not change ΔG or the equilibrium constant,
speeds up the forward and backward reactions equally, so equilibrium is reached faster,
cannot make a non-spontaneous reaction happen.
Collision theory
Molecules must collide in order to react. Only collisions with enough energy (at least the threshold energy) and the proper orientation lead to products. These are called effective collisions.