# Lecture 1: Chemical Kinetics: Rate Laws
## General Chemistry II
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## Learning Objectives
By the end of this lecture, students will be able to:
1. Define reaction rate and distinguish between average and instantaneous rates
2. Write rate expressions relating the rates of disappearance of reactants and appearance of products
3. Determine the rate law for a reaction from experimental data
4. Distinguish between zero-order, first-order, and second-order reactions
5. Calculate rate constants and their units for various reaction orders
6. Use integrated rate laws to determine concentrations as a function of time
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## Lecture Content
### I. Introduction to Chemical Kinetics
Chemical kinetics is the branch of chemistry concerned with measuring and understanding reaction rates and the factors that influence them. While thermodynamics tells us whether a reaction is spontaneous, kinetics tells us how fast it actually occurs. These are independent considerations: a reaction can be thermodynamically favorable yet kinetically slow. The conversion of diamond to graphite, for example, is spontaneous under standard conditions, but it proceeds at an immeasurably slow rate at room temperature.
Several factors affect how quickly a reaction proceeds. Increasing the concentration of reactants generally speeds up a reaction by making collisions between reacting molecules more frequent. Raising the temperature increases both the frequency and energy of molecular collisions. A catalyst provides an alternative reaction pathway with a lower activation energy. For heterogeneous reactions, increasing the surface area of a solid reactant exposes more molecules to the reaction environment. Finally, the intrinsic nature of the reactants themselves plays a role, since some bonds are easier to break than others.
### II. Defining Reaction Rate
The reaction rate describes the change in concentration of a reactant or product per unit time. For a general reaction aA + bB -> cC + dD, the rate can be expressed in terms of any species involved: Rate = -(1/a)(d[A]/dt) = -(1/b)(d[B]/dt) = (1/c)(d[C]/dt) = (1/d)(d[D]/dt). The negative signs in front of the reactant terms account for the fact that reactant concentrations decrease over time, while the stoichiometric coefficients ensure that the calculated rate is the same regardless of which species is measured.
There are two ways to quantify rate. The average rate is calculated over a finite time interval as Delta[X]/Delta t, representing the slope of a secant line on a concentration-versus-time plot. The instantaneous rate, by contrast, is the rate at a specific moment and corresponds to the slope of the tangent line to the concentration curve at that point. As the time interval Delta t approaches zero, the average rate converges to the instantaneous rate.
A concentration vs. time graph for a generic reaction A -> B. Panel A: Shows the decreasing curve for [A] and the increasing curve for [B] over time. A secant line is drawn between two points on the [A] curve, labeled "average rate." Panel B: Shows the same [A] curve with a tangent line drawn at a specific time point, labeled "instantaneous rate." The slope of the tangent is highlighted. Axes are labeled with [Concentration] (mol/L) on the y-axis and Time (s) on the x-axis.
### III. The Rate Law
The rate law, also called the rate equation, expresses the reaction rate as a function of reactant concentrations: Rate = k[A]^m[B]^n. In this expression, k is the rate constant, a proportionality factor that is specific to a given reaction at a given temperature. The exponents m and n are the reaction orders with respect to A and B, respectively. A critical point is that these exponents are not necessarily equal to the stoichiometric coefficients of the balanced equation; they must be determined experimentally. The sum m + n gives the overall reaction order. The rate law is typically established from experimental data, most commonly using the method of initial rates.
### IV. Determining Reaction Order: Method of Initial Rates
The method of initial rates involves measuring the initial rate of reaction across several experiments in which only one reactant concentration changes at a time. By comparing pairs of experiments, you can isolate the effect of each reactant on the rate. If doubling the concentration of A doubles the rate, the reaction is first order in A (m = 1). If doubling [A] quadruples the rate, the reaction is second order in A (m = 2). If doubling [A] has no effect on the rate, the reaction is zero order in A (m = 0).
The general mathematical relationship is (Rate2/Rate1) = ([A]2/[A]1)^m. Taking the logarithm of both sides allows you to solve for m directly. Once all reaction orders are known, you can substitute any experiment's data into the rate law expression to determine the numerical value of k.
A table showing the method of initial rates for the reaction 2NO + O2 -> 2NO2. Three experiments are displayed with columns for Experiment Number, [NO] (mol/L), [O2] (mol/L), and Initial Rate (mol/L/s). Experiment 1: [NO] = 0.010, [O2] = 0.010, Rate = 2.5 x 10^-5. Experiment 2: [NO] = 0.020, [O2] = 0.010, Rate = 1.0 x 10^-4. Experiment 3: [NO] = 0.010, [O2] = 0.020, Rate = 5.0 x 10^-5. Arrows between experiments highlight which concentration is changed and by what factor, with the resulting rate change shown.
### V. Units of the Rate Constant
The units of the rate constant k depend on the overall order of the reaction, because k must have whatever units are needed to make the rate come out in M/s (concentration per time). For a zero-order reaction, k has units of mol/(L*s) or M/s. For a first-order reaction, k has units of s^-1. For a second-order reaction, k has units of L/(mol*s) or M^-1*s^-1. More generally, for an nth-order reaction, k carries units of M^(1-n)*s^-1.
### VI. Integrated Rate Laws
While differential rate laws relate rate to concentration, integrated rate laws relate concentration directly to time, making them especially useful for predicting how concentrations change as a reaction progresses.
For zero-order reactions, the rate is simply equal to k, and the integrated form is [A] = [A]_0 - kt. A plot of [A] versus time yields a straight line with a slope of -k, and the half-life is given by t_(1/2) = [A]_0 / (2k).
For first-order reactions, the rate equals k[A], and integration gives ln[A] = ln[A]_0 - kt, or equivalently [A] = [A]_0 * e^(-kt). Plotting ln[A] against time produces a straight line with slope -k. The half-life is t_(1/2) = 0.693/k, which is notably independent of the initial concentration.
For second-order reactions, the rate equals k[A]^2, and the integrated form is 1/[A] = 1/[A]_0 + kt. A plot of 1/[A] versus time gives a straight line with slope k. The half-life is t_(1/2) = 1/(k[A]_0).
Three side-by-side panels showing how to graphically determine reaction order. Panel A (Zero Order): Plot of [A] vs. time showing a straight line with negative slope -k, y-intercept [A]_0. Panel B (First Order): Plot of ln[A] vs. time showing a straight line with negative slope -k, y-intercept ln[A]_0. Panel C (Second Order): Plot of 1/[A] vs. time showing a straight line with positive slope k, y-intercept 1/[A]_0. Each panel includes the integrated rate law equation and the linear form highlighted.
### VII. Half-Life
The half-life, t_(1/2), is the time required for the concentration of a reactant to decrease to half its initial value. The behavior of the half-life varies with reaction order and serves as a diagnostic tool.
For a zero-order reaction, t_(1/2) = [A]_0/(2k). Because this depends on the initial concentration, successive half-lives get shorter as the reaction proceeds and [A]_0 effectively decreases. For a first-order reaction, t_(1/2) = 0.693/k, which is constant and independent of concentration. This is precisely why radioactive decay, which follows first-order kinetics, is conveniently described in terms of half-life. For a second-order reaction, t_(1/2) = 1/(k[A]_0), which depends on the initial concentration and increases as the reaction proceeds because the remaining reactant concentration drops.
These patterns provide an experimental method for determining reaction order: if successive half-lives remain constant, the reaction is first order; if they decrease, it is zero order; if they increase, it is second order.
A graph showing concentration [A] vs. time for a first-order reaction, illustrating the concept of half-life. The curve starts at [A]_0 and shows exponential decay. Horizontal dashed lines mark [A]_0, [A]_0/2, [A]_0/4, and [A]_0/8. Vertical dashed lines drop to the time axis, marking t_(1/2), 2*t_(1/2), and 3*t_(1/2). Each successive half-life interval is labeled and shown to be equal in duration, emphasizing the constant half-life characteristic of first-order kinetics.
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