Part III, Chapter 7

Rate Laws

Integrated rate laws, half-life, and determination of reaction order

Historical Context

Chemical kinetics emerged as a quantitative science in the mid-19th century. Ludwig Wilhelmy (1850) performed the first kinetic study, measuring the rate of sucrose inversion in acid. Peter Waage and Cato Guldberg formulated the law of mass action in 1864. Jacobus van't Hoff and Svante Arrhenius (Nobel Prize 1903) established the temperature dependence of reaction rates.

The systematic classification of reactions by order and the development of integrated rate laws provided experimentalists with tools to extract rate constants from concentration-vs-time data. The method of initial rates, developed in the early 20th century, remains a standard approach in physical chemistry laboratories.

Key contributors: Ludwig Wilhelmy (1850), Waage & Guldberg (1864), van't Hoff (1884), Arrhenius (1889, Nobel 1903).

Derivation 1: Zeroth-Order Rate Law

For a zeroth-order reaction A → products, the rate is independent of concentration:

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

Integrating with initial condition $[\text{A}](0) = [\text{A}]_0$:

$$[\text{A}](t) = [\text{A}]_0 - kt$$

Valid until $[\text{A}] = 0$ at $t = [\text{A}]_0/k$

Half-life (time for concentration to drop to half):

$$t_{1/2} = \frac{[\text{A}]_0}{2k} \quad \text{(depends on initial concentration)}$$

Zeroth-order kinetics occur when the reaction is limited by a saturated catalyst or enzyme. The linear plot of [A] vs t with slope -k is diagnostic.

Derivation 2: First-Order Rate Law

For A → products with first-order kinetics:

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

This is a separable ODE. Separating variables and integrating:

$$\int_{[\text{A}]_0}^{[\text{A}]} \frac{d[\text{A}]'}{[\text{A}]'} = -k \int_0^t dt' \implies \ln\frac{[\text{A}]}{[\text{A}]_0} = -kt$$
$$[\text{A}](t) = [\text{A}]_0 \, e^{-kt}$$

Half-life: $t_{1/2} = \frac{\ln 2}{k}$ (independent of $[\text{A}]_0$!)

The constant half-life is the hallmark of first-order kinetics. This is why radioactive decay (a first-order process) has a characteristic half-life independent of the amount of material. A plot of $\ln[\text{A}]$ vs t is linear with slope -k.

Derivation 3: Second-Order Rate Law

For 2A → products (or A + B → products with [A]=[B]):

$$-\frac{d[\text{A}]}{dt} = k[\text{A}]^2$$

Separating and integrating:

$$\int_{[\text{A}]_0}^{[\text{A}]} \frac{d[\text{A}]'}{[\text{A}]'^2} = -k\int_0^t dt' \implies \frac{1}{[\text{A}]} - \frac{1}{[\text{A}]_0} = kt$$
$$\frac{1}{[\text{A}](t)} = \frac{1}{[\text{A}]_0} + kt$$

Half-life: $t_{1/2} = \frac{1}{k[\text{A}]_0}$ (inversely proportional to $[\text{A}]_0$)

For $A + B \to$ products with $[\text{A}]_0 \neq [\text{B}]_0$, partial fractions give:$\frac{1}{[\text{B}]_0 - [\text{A}]_0}\ln\frac{[\text{B}][\text{A}]_0}{[\text{A}][\text{B}]_0} = kt$

Derivation 4: Method of Initial Rates

For a reaction with rate law $r = k[\text{A}]^m[\text{B}]^n$, the orders m and n can be determined by measuring initial rates at different initial concentrations.

Taking the logarithm:

$$\ln r_0 = \ln k + m \ln[\text{A}]_0 + n \ln[\text{B}]_0$$

By holding [B]_0 constant and varying [A]_0:

$$m = \frac{\ln(r_0^{(2)} / r_0^{(1)})}{\ln([\text{A}]_0^{(2)} / [\text{A}]_0^{(1)})} \bigg|_{[\text{B}]_0 \text{ const}}$$

Example: Determining Order

If doubling [A]_0 doubles the rate: m = ln(2)/ln(2) = 1 (first order in A)

If doubling [A]_0 quadruples the rate: m = ln(4)/ln(2) = 2 (second order in A)

If doubling [A]_0 has no effect: m = ln(1)/ln(2) = 0 (zeroth order in A)

Derivation 5: Pseudo-First-Order Kinetics

When one reactant is in large excess, its concentration barely changes during the reaction. For $r = k[\text{A}][\text{B}]$ with $[\text{B}]_0 \gg [\text{A}]_0$:

$$r \approx k[\text{B}]_0 \cdot [\text{A}] = k_{\text{obs}} \cdot [\text{A}]$$

where $k_{\text{obs}} = k[\text{B}]_0$ is the pseudo-first-order rate constant. The reaction appears first-order in [A]:

$$[\text{A}](t) = [\text{A}]_0 \exp(-k_{\text{obs}} \, t)$$

By varying $[\text{B}]_0$ and plotting $k_{\text{obs}}$ vs$[\text{B}]_0$, the true second-order rate constant k is obtained from the slope.

This technique is widely used in practice because it simplifies data analysis and is especially important for reactions in aqueous solution where water (the solvent) is a reactant present in vast excess.

Applications

Radioactive Decay

Nuclear decay is the archetypal first-order process. Carbon-14 dating uses$t_{1/2} = 5730$ years. The independence of half-life from concentration makes radiometric dating possible.

Pharmacokinetics

Drug elimination from the body often follows first-order kinetics. The half-life determines dosing intervals. Zeroth-order kinetics apply when metabolic enzymes are saturated (e.g., ethanol metabolism at high doses).

Enzyme Kinetics

Michaelis-Menten kinetics transitions from first-order (low [S]) to zeroth-order (high [S], enzyme saturated). The rate law is$v = V_{\max}[\text{S}]/(K_M + [\text{S}])$.

Atmospheric Chemistry

Ozone depletion, smog formation, and greenhouse gas lifetimes are all analyzed using kinetic rate laws. The atmospheric lifetime of a species is its first-order decay time constant.

Simulation: Integrated Rate Laws

This simulation compares zeroth, first, and second order kinetics side-by-side, showing both the concentration-time curves and their linearized forms. The diagnostic linear plots reveal which order best fits experimental data.

Zeroth, First, and Second Order Integrated Rate Laws

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Simulation: Half-Lives and Pseudo-First-Order

This simulation compares how concentration decay differs across reaction orders, shows how half-life depends on initial concentration for each order, and demonstrates the pseudo-first-order technique with varying excess reagent concentrations.

Half-Life Comparison and Pseudo-First-Order Kinetics

Python
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Consecutive Half-Lives and Order Determination

A practical method for determining reaction order uses consecutive half-lives. If the time for [A] to fall from [A]_0 to [A]_0/2 is $t_1$, and from [A]_0/2 to [A]_0/4 is $t_2$, then the ratio $t_2/t_1$ reveals the order:

Zeroth order: $t_2/t_1 = 1/2$. Each successive half-life is shorter because the rate is constant but there is less reactant.
First order: $t_2/t_1 = 1$. All half-lives are equal -- the defining characteristic of first-order kinetics.
Second order: $t_2/t_1 = 2$. Each successive half-life doubles because the rate decreases as [A] drops.

In general, for an nth-order reaction with respect to a single reactant:

$$t_{1/2} = \frac{2^{n-1} - 1}{(n-1) k [\text{A}]_0^{n-1}} \quad (n \neq 1)$$

The ratio of successive half-lives: $t_{1/2}^{(2)} / t_{1/2}^{(1)} = 2^{n-1}$

Complex Rate Laws and Fractional Orders

Not all reactions have integer orders. Fractional and even negative orders arise from complex mechanisms. Some important examples:

Michaelis-Menten Kinetics

$$v = \frac{V_{\max}[\text{S}]}{K_M + [\text{S}]}$$

First order at low [S] (when $[\text{S}] \ll K_M$), zeroth order at high [S] (when $[\text{S}] \gg K_M$). The Lineweaver-Burk plot ($1/v$ vs $1/[\text{S}]$) linearizes this for parameter extraction.

Surface Reactions (Langmuir-Hinshelwood)

$$r = \frac{k K_A K_B P_A P_B}{(1 + K_A P_A + K_B P_B)^2}$$

The apparent order with respect to each reactant varies from +1 at low pressure to -1 at high pressure. This is responsible for the complex kinetics observed in heterogeneous catalysis.

Parallel Reactions

When A can react by two paths (A → B with rate $k_1[\text{A}]$ and A → C with rate $k_2[\text{A}]$), the overall decay is first order with $k_{\text{obs}} = k_1 + k_2$. The product ratio is$[\text{B}]/[\text{C}] = k_1/k_2$ (kinetic selectivity).

Temperature Dependence of Rate Constants

A common rule of thumb is that reaction rates double for every 10 K increase in temperature. This can be quantified using the Arrhenius equation:

$$\frac{k(T + 10)}{k(T)} = \exp\left(\frac{E_a \cdot 10}{RT(T+10)}\right)$$

For $E_a = 50$ kJ/mol at T = 300 K, this ratio is about 1.9 -- close to the rule of thumb. But for $E_a = 100$ kJ/mol, it is about 3.7, and for$E_a = 20$ kJ/mol, only 1.3.

Preview: The full theory of temperature dependence -- the Arrhenius equation, the Eyring equation, and transition state theory -- is developed in Chapter 9. There we will see how quantum mechanics and statistical mechanics provide a microscopic understanding of the pre-exponential factor and activation energy.

Chapter Summary

• Zeroth order: $[\text{A}] = [\text{A}]_0 - kt$; rate independent of concentration; $t_{1/2} = [\text{A}]_0/(2k)$.

• First order: $[\text{A}] = [\text{A}]_0 e^{-kt}$; constant half-life $t_{1/2} = \ln 2/k$.

• Second order: $1/[\text{A}] = 1/[\text{A}]_0 + kt$; $t_{1/2} = 1/(k[\text{A}]_0)$.

• Method of initial rates: Determine order by comparing rates at different initial concentrations.

• Pseudo-first-order: Excess reactant simplifies bimolecular kinetics to apparent first order.

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