Reaction Order and Rate Constant Units

In some of our examples, the reaction orders in the rate law happen to be the same as the coefficients in the chemical equation for the reaction. This is merely a coincidence and very often not the case.

Rate laws may exhibit fractional orders for some reactants, and negative reaction orders are sometimes observed when an increase in the concentration of one reactant causes a decrease in reaction rate. A few examples illustrating these points are provided:

$$NO_2+CO⟶NO+CO_2\qquad \text{rate}=k[NO_2]^2$$ $$CH_3CHO⟶CH_4+CO\qquad \text{rate}=k[CH_3CHO]^2$$ $$2N_2O_5⟶NO_2+O_2\qquad \text{rate}=k[N_2O_5]$$ $$NO_2+F_2⟶2NO_2F\qquad \text{rate}=k[NO_2][F_2]$$ $$NO_2Cl⟶2NO_2+Cl_2\qquad \text{rate}=k[NO_2Cl]$$

It is important to note that rate laws are determined by experiment only and are not reliably predicted by reaction stoichiometry.

The units for a rate constant will vary as appropriate to accommodate the overall order of the reaction – remembering that rate is always reported as $\frac{\Delta\:concentration}{\Delta\:time}$, and the units of concentration and time must be consistent across all terms in the rate law.

Dimensional analysis of the rate laws for reaction orders 0-3 shows that each reaction order will have a characteristic unit for its rate constant -shown in the table below, summarizing the rate constant units for common reaction orders.

Rate Constant Units for Common Reaction Orders
Overall Reaction Order (x) Rate Constant Unit (Lx−1 mol1−x s−1)
0 (zero) mol L−1 s−1
1 (first) s−1
2 (second) L mol−1 s−1
3 (third) L2 mol−2 s−1

The units in this table were derived using specific units for concentration (mol/L) and time (s), though any valid units for these two properties may be used – for example: M/hour, M/min or g L-1s-1 are all valid units for rate.

Generically: the rate constant unit for a reaction whose overall order is x to be $L^{x-1}\;mol^{1-x}\;s^{-1}$. (Assuming concentration is measured in M and time in s.)

An example of the derivation of the units of the rate constant for a 3rd order reaction: $$ rate = k [A][B][C] \\ \text{Substituting in the units:} \qquad \qquad \qquad \\ \left( \frac{M}{s} \right) = k [M][M][M] = k [M]^3 \\ \left( \frac{M}{s \cdot M^3} \right) = k \\ k = \frac{1}{s \cdot M^2} \\ k = \frac{1}{s \cdot \left( \frac{mol}{L} \right)^2} = \frac{L^2}{s \cdot mol^2} = L^{2}mol^{-2}s^{-1} $$

Check Your Learning: