Second-Order Reactions

Count rate-law exponents to identify overall order. Counting reactants in the balanced equation does not identify order.

Two common second-order disappearance laws are k[A]² and k[A][B]. They each have exponents summing to two. Other empirical power-law combinations can sum to two; only an explicitly elementary step has a direct molecularity-to-exponent connection.

$r = k[A][B]$

The reaction shown by that equation is a second-order reaction, but it is first-order in each reactant. You can understand why by looking at the exponent on each reactant. Because each exponent is 1, we say that this reaction is First-Order in A and First-Order in B. You then add both exponents up to determine what the overall reaction order is. Two reactants in first order add up to an overall second order reaction.

For r=k[A]^x[B]^y, overall order is x+y. The forms k[A]² and k[A][B] are common examples, not the only possible power laws. In particular, overall first order does not restrict the reaction to one physical reactant.

$r = k[A]^2$

A second order rate constant, $k$, has units of $\frac{L}{mol\cdot s}$, and $r$ remains the same as it has always been:

−d[A]/dt=k[A]² is the disappearance law used here. The minus sign makes the disappearance rate positive when [A] decreases. A normalized reaction rate may differ by a stoichiometric factor.

For a Second-Order rate equation, if you were to graph $[A] \text{ vs. }t$, you would see a scatterplot that looks something like this:

For −d[A]/dt=k[A]², plot 1/[A] in L mol⁻¹ versus t in s. It is linear with positive slope k in L mol⁻¹ s⁻¹. A general law k[A][B] does not automatically have this integrated form.

Once again, we will investigate the cause of this relationship by integrating the rate equation to find the integrated second-order rate equation.

The reciprocal law applies to −d[A]/dt=k[A]². It also applies to −d[A]/dt=k[A][B] if [A]=[B] throughout the interval.

For a fixed-volume 1:1 reaction A+B→products, equal initial concentrations remain equal if there are no other sources or sinks. Then k[A][B]=k[A]². Unequal initial concentrations or different stoichiometric consumption require a different integration.

Now it’s time to figure out the half-life equation for a second-order reaction. Starting with our integrated rate equation:

$\frac1{[A]}=\frac1{[A]_0}+kt$

Set $\frac{[A]_0}2=[A]$:

$\frac2{[A]_0}=\frac1{[A]_0}+kt$

$2=1+[A]_0\cdot kt$

$1=[A]_0\cdot kt$

$t_{\frac12}=\frac1{[A]_0\cdot k}$

There is our formula for the half-life of a second-order reaction.

Reviewed replacement, Example3.3.1: A hypothetical fixed-volume NO₂ disappearance model specifies −d[NO₂]/dt=(12 L mol⁻¹ s⁻¹)[NO₂]² and [NO₂]₀=0.0030 M. Find the half-life. This explicitly defines k on the disappearance basis and does not claim a measured nitrogen/oxygen catalytic-converter mechanism.

Reviewed replacement, Example3.3.2: In a hypothetical fixed-volume model for A+B→P, [A]₀=[B]₀=6.00 M and −d[A]/dt=k[A][B]. Successive intervals to halve the remaining concentration are35.0,70.0 and140.0 s. The cumulative times to [A]₀/2, [A]₀/4 and [A]₀/8 are35.0,105.0 and245.0 s. Determine k. These formal data replace the source’s inconsistent35,70,105 successive intervals and are not a precipitation procedure. The actual sodium carbonate formula is Na₂CO₃; calcium carbonate is CaCO₃.

If one reactant’s concentration stays effectively constant over the measured interval, absorb its concentration factor into an effective constant. This is a pseudo-order approximation; it does not change the underlying empirical law.

For an illustrative A+B→P reaction with measured disappearance law k[A][B], consider B in sufficient excess that its fractional change is negligible. This generic model is clearer than inferring methane combustion kinetics from its balanced equation.

The reviewed generic stoichiometry is A+B→P. It is separate from a real methane combustion mechanism; stoichiometry alone cannot establish that reaction’s rate law.

Excess means the concentration change is small enough for the desired accuracy; it does not mean exactly constant automatically. A controlled reservoir can also hold [B] fixed. Check that condition before using a pseudo-first-order law.

Suppose kinetic data establish −d[A]/dt=k[A][B]. We are declaring this law; coefficients alone do not establish it.

With [B]≈[B]fixed, −d[A]/dt=k′[A], where k′=k[B]fixed and k′ has inverse-time units. The model is pseudo-first order in A within that range.

Here’s a list of all the equation for the Zeroth-Order, First-Order, and Second-Order reactions:

Zeroth-Order:

The rate equation: $r =k$
The integrated rate equation: $[A] = [A]_0-kt$
The half-life equation: $t_{\frac12}=\frac{[A]_0}{2k}$

First-Order:

The rate equation: $r =k[A]$
The integrated rate equation: $[A] =[A]_0e^{-kt}$
The half-life equation: $t_\frac12=\frac{\ln2}k$

Second-Order:

Second-order summary: for −d[A]/dt=k[A]², 1/[A]=1/[A]₀+kt and t½=1/(k[A]₀). The same form applies to k[A][B] only when [A]=[B] throughout. These formulas use the explicitly declared species-disappearance convention.