1.1 Rate laws and reaction orders

Chemical reactions happen at different speeds, and understanding these rates is central to chemistry. Rate laws give you a mathematical way to connect reactant concentrations to how fast a reaction proceeds. With rate laws, you can predict concentrations at any point in time, figure out how long a reaction takes, and start to uncover what's happening at the molecular level.

Reaction Rates and Rate Laws

Definition and measurement of reaction rate

Reaction rate tells you how quickly reactant concentrations drop or product concentrations rise over time. For a reactant A, the rate is expressed as:

$-\frac{\Delta[A]}{\Delta t}$

The negative sign accounts for the fact that reactant concentration decreases over time. For a product P, there's no negative sign because the concentration is increasing:

$\frac{\Delta[P]}{\Delta t}$

Concentration is measured in molarity (M), which is moles of solute per liter of solution. Time can be in seconds, minutes, or hours depending on how fast the reaction is.

One thing that trips people up: for a reaction like $2A \rightarrow B$, two moles of A are consumed for every mole of B produced. That means the rate of disappearance of A is twice the rate of appearance of B. To get a single, unambiguous rate, you divide by the stoichiometric coefficient:

$\text{Rate} = -\frac{1}{2}\frac{\Delta[A]}{\Delta t} = \frac{\Delta[B]}{\Delta t}$

More generally, for $aA + bB \rightarrow cC + dD$:

$\text{Rate} = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$

There are several ways to measure reaction rates experimentally:

• UV-Vis spectroscopy tracks absorbance changes when a reactant or product absorbs light at a specific wavelength
• Pressure monitoring works for gaseous reactions where the total number of moles of gas changes
• Conductivity measurements apply when reactions produce or consume ions in solution

Rate law and order from data

The rate law is an equation that links the reaction rate to reactant concentrations:

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

Here, $k$ is the rate constant, $[A]$ and $[B]$ are reactant concentrations, and $m$ and $n$ are the reaction orders with respect to each reactant. The overall reaction order is the sum $m + n$.

You cannot determine reaction orders from the balanced equation. They must come from experimental data. The balanced equation tells you stoichiometry, not mechanism.

Common reaction orders and what they mean:

• Zero-order ($m = 0$): Changing the concentration of that reactant has no effect on the rate
• First-order ($m = 1$): Doubling the concentration doubles the rate
• Second-order ($m = 2$): Doubling the concentration quadruples the rate
• Fractional and negative orders are possible but less common at this level

How to determine the rate law from experimental data (method of initial rates):

1. Run multiple experiments where you change the initial concentration of one reactant at a time while holding everything else constant (temperature, other concentrations).
2. Measure the initial rate for each experiment.
3. Compare two experiments to find each order. Pick two trials where only $[A]$ changes. Set up the ratio:

$\frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[A]_2}{[A]_1}\right)^m$

Solve for $m$. For example, if doubling $[A]$ causes the rate to quadruple, then $2^m = 4$, so $m = 2$.

1. Repeat for each reactant to find all orders.
2. Plug the orders, a known rate, and the corresponding concentrations back into the rate law to solve for $k$. Use any single trial's data for this step, and make sure the units work out.

Differential vs. integrated rate laws

These are two forms of the same information, but they answer different questions.

The differential rate law tells you the instantaneous rate based on current concentrations:

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

This is what you use when you have initial rate data and want to find reaction orders.

The integrated rate law tells you concentration as a function of time. You get it by integrating the differential form. This is what you use when you want to know "how much reactant is left after 30 seconds?" or "how long until half the reactant is gone?"

The integrated form depends on the reaction order:

Notice that only the first-order half-life is independent of initial concentration. This is why radioactive decay (a first-order process) has a constant half-life regardless of how much material you start with. For zero-order and second-order reactions, the half-life changes as the reaction progresses because it depends on $[A]_0$.

Each integrated rate law has the form $y = mx + b$. The "linear plot" column tells you which transformation of the data to plot on the y-axis against time on the x-axis. Whichever plot gives a straight line tells you the order:

• Straight line for $[A]_t$ vs. $t$: zero-order, slope = $-k$
• Straight line for $\ln[A]_t$ vs. $t$: first-order, slope = $-k$
• Straight line for $\frac{1}{[A]_t}$ vs. $t$: second-order, slope = $+k$

Calculations using rate laws

To find a concentration at a specific time, pick the integrated rate law that matches your reaction order, plug in your known values, and solve.

Example: First-order reaction

A first-order reaction has $k = 0.05 \text{ s}^{-1}$ and $[A]_0 = 1.0 \text{ M}$. Find $[A]$ after 10 seconds.

1. Write the first-order integrated rate law: $\ln[A]_t = -kt + \ln[A]_0$
2. Substitute known values: $\ln[A]_{10} = -(0.05 \text{ s}^{-1})(10 \text{ s}) + \ln(1.0)$
3. Simplify: $\ln[A]_{10} = -0.5 + 0 = -0.5$
4. Solve by exponentiating both sides: $[A]_{10} = e^{-0.5} = 0.61 \text{ M}$

The concentration dropped from 1.0 M to 0.61 M in 10 seconds.

You can also rearrange these equations to solve for time. For instance, "how long until the concentration reaches 0.25 M?"

$\ln(0.25) = -(0.05)t + \ln(1.0)$

$-1.386 = -0.05t$

$t = 27.7 \text{ s}$

The same logic applies to zero-order and second-order reactions. Just use the correct integrated rate law and keep your units consistent.

Always check that the units of $k$ match the reaction order: