2.3 Differential rate laws

Differential Rate Laws

A differential rate law tells you exactly how the concentration of each reactant affects the speed of a reaction. Understanding these laws lets you predict reaction rates, determine how concentrations change over time, and calculate useful quantities like half-lives.

Differential Rate Law Derivation

The differential rate law expresses reaction rate as a function of reactant concentrations and a rate constant. For a general reaction:

$aA + bB \rightarrow cC + dD$

the differential rate law takes the form:

$\text{Rate} = k[A]^m[B]^n$

• $k$ is the rate constant, a value specific to a given reaction at a given temperature.
• $m$ and $n$ are the reaction orders with respect to reactants A and B.
• The overall reaction order is the sum $m + n$.

One thing that trips people up: reaction orders are determined experimentally. You cannot just read them off the stoichiometric coefficients in the balanced equation. A reaction with a coefficient of 2 in front of A could still be first-order with respect to A.

Some common examples:

• First-order in A: $\text{Rate} = k[A]$ (doubling [A] doubles the rate)
• Second-order in B: $\text{Rate} = k[B]^2$ (doubling [B] quadruples the rate)

Rate Expression from Concentration Changes

The rate of a reaction can also be written in terms of how fast reactant or product concentrations change over time. For the same general reaction $aA + bB \rightarrow cC + dD$:

• For reactants: $\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt}$
• For products: $\text{Rate} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$

The negative sign appears for reactants because their concentrations decrease over time, and we want the rate to be a positive number. The stoichiometric coefficients ($a$, $b$, $c$, $d$) normalize everything so that you get the same rate value regardless of which species you track.

For example, consider $2A \rightarrow B$. If A disappears at 0.6 M/s, the rate is:

$\text{Rate} = -\frac{1}{2}\frac{d[A]}{dt} = -\frac{1}{2}(-0.6 \text{ M/s}) = 0.3 \text{ M/s}$

And B forms at 0.3 M/s, consistent with the 2:1 stoichiometry.

Rate Law Components and Relationships

The differential rate law ties together three pieces: the rate constant, the concentrations, and the reaction orders.

Reaction orders tell you the sensitivity of the rate to each reactant's concentration:

• $m = 1$ (first-order in A): rate is directly proportional to [A]
• $m = 2$ (second-order in A): rate is proportional to $[A]^2$
• $m = 0$ (zero-order in A): rate doesn't depend on [A] at all

The rate constant $k$ captures the intrinsic reactivity of the system at a particular temperature. It must be determined experimentally. Its units depend on the overall reaction order, because the rate always needs to come out in units of concentration/time (typically M/s):

Integrated Rate Law for First-Order Reactions

The differential rate law tells you the rate right now. The integrated rate law tells you the concentration at any future time. You get it by integrating the differential form.

For a first-order reaction where $\text{Rate} = k[A]$:

1. Write the rate as a derivative: $-\frac{d[A]}{dt} = k[A]$
2. Separate variables: $\frac{d[A]}{[A]} = -k \, dt$
3. Integrate both sides from the initial condition ($[A]_0$ at $t = 0$) to ($[A]_t$ at time $t$):

$\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = -k \int_0^t dt$

1. This gives the integrated rate law:

$\ln\frac{[A]_t}{[A]_0} = -kt$

which can be rearranged to $[A]_t = [A]_0 \, e^{-kt}$.

What you can do with this:

• Find concentration at any time. If $[A]_0 = 1.0$ M and $k = 0.1 \text{ s}^{-1}$, the concentration after 10 s is:

$[A]_{10} = 1.0 \text{ M} \times e^{-(0.1)(10)} = 1.0 \times e^{-1} = 0.37 \text{ M}$

• Calculate the half-life. Set $[A]_t = \frac{1}{2}[A]_0$ and solve for $t$:

$t_{1/2} = \frac{\ln 2}{k}$

For a first-order reaction with $k = 0.05 \text{ min}^{-1}$: $t_{1/2} = \frac{0.693}{0.05} = 13.9 \text{ min}$

A distinctive feature of first-order reactions is that the half-life is constant. It doesn't matter whether you start with 2.0 M or 0.001 M; the time to reach half the current concentration is always the same. This is not true for zero-order or second-order reactions.