Order of Reaction: MDCAT Chemistry notes
Order of Reaction MDCAT notes: rate laws, zero, first, second, third and pseudo first order, fractional and negative orders, half-life and molecularity.
Rate law and order
For a reaction $a\mathrm{A} + b\mathrm{B} \rightarrow$ products, the rate law is
$$\text{rate} = k[\mathrm{A}]^x[\mathrm{B}]^y$$
- The order with respect to A is $x$, the power to which $[\mathrm{A}]$ is raised in the rate equation.
- The overall order is $x + y$.
- Orders are found by experiment. Only for an elementary (single-step) reaction do they equal the coefficients, giving rate $= k[\mathrm{A}]^a[\mathrm{B}]^b$.
- Order can be zero, whole, fractional or negative.
Examples: first order in A and second in B gives rate $= k[\mathrm{A}][\mathrm{B}]^2$ (third order overall); first order in A and third in B gives rate $= k[\mathrm{A}][\mathrm{B}]^3$.
Types of order
| Order | Rate law | Example |
|---|---|---|
| Zero | rate $= k$ | Photochemical reactions, e.g. photosynthesis and $\mathrm{H_2 + Cl_2}$ in light |
| First | rate $= k[\mathrm{A}]$ | Decomposition of $\mathrm{N_2O_5}$; radioactive decay |
| Second | rate $= k[\mathrm{A}][\mathrm{B}]$ | $\mathrm{NO + O_3 \rightarrow NO_2 + O_2}$, rate $= k[\mathrm{NO}][\mathrm{O_3}]$ |
| Third | rate $= k[\mathrm{A}]^2[\mathrm{B}]$ | $\mathrm{2NO + 2H_2 \rightarrow N_2 + 2H_2O}$, rate $= k[\mathrm{NO}]^2[\mathrm{H_2}]$ |
| Pseudo first | behaves as first order | Hydrolysis of an ester or sucrose in large excess of water |
| Negative | rate falls as concentration rises | Rate inversely proportional to a concentration |
A pseudo first order reaction has more than one reactant, but one is in such excess that its concentration hardly changes, so the rate depends on only one concentration.
Effect of doubling concentration
| Order in A | Rate when [A] is doubled |
|---|---|
| 0 | Unchanged |
| 1 | Doubled |
| 2 | Four times |
| 3 | Eight times |
In general, if concentration is multiplied by $m$ and rate by $m^n$, the order is $n$.
Half-life and order
- First order: $t_{1/2} = \dfrac{0.693}{k}$, independent of initial concentration. A constant half-life means first order.
- Second order: $t_{1/2} = \dfrac{1}{k[\mathrm{A}]_0}$, inversely proportional to initial concentration.
- General: $t_{1/2} \propto \dfrac{1}{[\mathrm{A}]_0^{\,n-1}}$
Mechanism and rate-determining step
The slowest step of a mechanism determines the rate. The rate law matches the molecules in the slow step. For rate $= k[\mathrm{NO}]^2[\mathrm{H_2}]$, the slow step must involve two NO and one $\mathrm{H_2}$, e.g. $\mathrm{2NO + H_2 \rightarrow N_2 + H_2O_2}$ (slow), then $\mathrm{H_2O_2 + H_2 \rightarrow 2H_2O}$ (fast).
Order versus molecularity
| Order | Molecularity |
|---|---|
| Sum of powers in the experimental rate law | Number of molecules colliding in an elementary step |
| Can be zero, fractional or negative | Always a whole number (1, 2 or rarely 3) |
Common MDCAT traps
- Order is not simply the coefficient in an overall equation unless the reaction is elementary.
- Rate independent of concentration means zero order.
- Constant half-life indicates first order, not zero order.
- Rate ∝ 1/concentration means negative order, not second order.
- Overall order of rate $= k[\mathrm{A}][\mathrm{B}]^2$ is 3.
Quick revision
- Order is the power of concentration in the rate equation.
- Photochemical reactions are zero order.
- Doubling [A] quadruples the rate for second order.
- First-order half-life is $0.693/k$.
- Excess water makes ester hydrolysis pseudo first order.