🧂Physical Chemistry II Unit 1 Review

Chemical reactions proceed at vastly different speeds, and quantifying those speeds is central to physical chemistry. Rate laws connect macroscopic observables (concentrations, time) to molecular-level events, giving you the tools to predict, control, and mechanistically interpret chemical change.

Reaction rate and significance

Definition and units

The reaction rate is the time derivative of concentration for a reactant or product, scaled by the stoichiometric coefficient so that every species gives the same numerical rate. For a general reaction $aA + bB \rightarrow cC + dD$ , the rate is:

$r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$

The negative signs account for the fact that reactant concentrations decrease over time. Units are typically $\text{M s}^{-1}$ (mol L $^{-1}$ s $^{-1}$ ), though $\text{M min}^{-1}$ or other time units appear depending on the timescale of the reaction.

Importance in chemical kinetics

Rates tell you how fast a reaction progresses and how that speed changes as concentrations evolve.
Controlling reaction rates is essential in industrial synthesis, pharmaceutical manufacturing, and biological systems.
Chemical kinetics as a field is built on measuring rates and connecting them to molecular mechanisms.

Experimental determination

You measure rates experimentally by tracking concentration as a function of time. Common techniques include UV-Vis spectroscopy, pressure measurements (for gas-phase reactions), and conductivity measurements.

Because the rate generally changes as the reaction proceeds, the initial rate (the rate at $t = 0$ ) is especially useful. At $t = 0$ you know the exact concentrations, and reverse reactions or product inhibition haven't yet complicated things. This makes the initial rate the cleanest data point for extracting rate law parameters.

Determining rate law expressions

Rate law definition

The rate law is an experimentally determined equation relating the rate to reactant concentrations and a rate constant:

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

$k$ is the rate constant, which depends on temperature but not on concentration.
$m$ and $n$ are the reaction orders with respect to $A$ and $B$ .
These exponents are not necessarily equal to the stoichiometric coefficients. They must be found experimentally.

Method of initial rates (step-by-step)

This is the most common way to determine a rate law in a Physical Chemistry course:

Run multiple experiments. In each trial, vary the initial concentration of one reactant while holding all others constant.
Measure the initial rate for each trial.
Compare pairs of experiments to isolate the effect of one reactant. If you double $[A]$ and the rate doubles, $m = 1$ . If the rate quadruples, $m = 2$ . If the rate doesn't change, $m = 0$ .
Mathematically, take the ratio of rates from two experiments where only $[A]$ changes:

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

Solve for $m$ by taking the logarithm of both sides:

$m = \frac{\ln(\text{Rate}_2 / \text{Rate}_1)}{\ln([A]_2 / [A]_1)}$

Repeat for each reactant to find all orders.
Calculate $k$ by substituting the determined orders and any one set of experimental data back into the rate law and solving.

Watch the units of $k$ . They depend on the overall reaction order. For an overall order $n$ , the units of $k$ are $\text{M}^{1-n}\text{s}^{-1}$ . A first-order reaction gives $k$ in $\text{s}^{-1}$ ; a second-order reaction gives $\text{M}^{-1}\text{s}^{-1}$ .

Differential vs integrated rate laws

Definition and units, The Rate Law | Introduction to Chemistry

Differential rate laws

A differential rate law expresses the instantaneous rate as a function of concentration:

$-\frac{d[A]}{dt} = k[A]^n$

This form is what you get directly from the method of initial rates. It tells you the rate right now, given the current concentrations.

Integrated rate laws

An integrated rate law is obtained by integrating the differential form. It gives concentration as an explicit function of time, which is what you actually need to predict how much reactant remains after a given period.

The results for the three most common orders (single-reactant case) are:

Order	Differential Form	Integrated Form	Linear Plot	Half-life
Zero	$-\frac{d[A]}{dt} = k$	$[A] = [A]_0 - kt$	$[A]$ vs $t$	$t_{1/2} = \frac{[A]_0}{2k}$
First	$-\frac{d[A]}{dt} = k[A]$	$\ln[A] = \ln[A]_0 - kt$	$\ln[A]$ vs $t$	$t_{1/2} = \frac{\ln 2}{k}$
Second	$-\frac{d[A]}{dt} = k[A]^2$	$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$	$\frac{1}{[A]}$ vs $t$	$t_{1/2} = \frac{1}{k[A]_0}$

How to use these to determine reaction order

Plot your experimental concentration-vs-time data in each of the three linearized forms. Whichever plot gives a straight line tells you the order. The slope of that line gives you $k$ (or $-k$ , depending on the form).

Notice that the first-order half-life is independent of initial concentration, while zero- and second-order half-lives are not. This is a common exam question and a useful conceptual check.

Reaction order and rate law relationship

Definition of reaction order

The reaction order with respect to a given reactant is the exponent on that reactant's concentration in the rate law. The overall reaction order is the sum of all individual orders:

$\text{Overall order} = m + n + \cdots$

Reaction orders are determined experimentally. They reflect the kinetics of the rate-determining step, not the overall stoichiometry.

Types of reaction orders

Zero-order ( $m = 0$ ): The rate is independent of $[A]$ . This often occurs when a catalyst surface is saturated or when a reactant is in large excess. The concentration drops linearly with time.
First-order ( $m = 1$ ): The rate is directly proportional to $[A]$ . Radioactive decay is the classic example. Concentration decays exponentially.
Second-order ( $m = 2$ ): The rate depends on $[A]^2$ (one reactant) or on $[A][B]$ (two reactants, each first-order). Many bimolecular elementary reactions fall here.
Fractional and negative orders do occur. A fractional order often signals a complex, multi-step mechanism. A negative order means that increasing that species' concentration actually slows the reaction (common with product inhibition).

Connection to reaction mechanism

The rate law constrains the mechanism but doesn't uniquely determine it. The orders tell you what the rate-determining step looks like at the molecular level:

If the rate is first-order in $A$ , one molecule of $A$ is involved in (or before) the rate-determining step.
If a species appears in the balanced equation but not in the rate law, it enters the mechanism after the rate-determining step.

Be careful with this interpretation. A rate law consistent with a proposed mechanism is necessary but not sufficient proof that the mechanism is correct. Multiple mechanisms can produce the same rate law.

🧂Physical Chemistry II Unit 1 Review