Unit 4: Empirical Chemical Kinetics & Integrated Rate Laws

Chemical kinetics investigates the time evolution of reacting systems and the microscopic pathways through which reactants transform into products.

Definition of Reaction Rate and Extent of Reaction

Consider a general closed homogeneous chemical reaction with stoichiometric coefficients $\nu_i$:

where $\nu_i < 0$ for reactants and $\nu_i > 0$ for products.

The extent of reaction $\xi$ (SI unit: moles) is defined by de Donder as:

The intensive rate of reaction $r$ (or $v$) per unit volume $V$ is defined as:

For a constant-volume system, introducing concentration $c_i = [A_i] = n_i / V$:

For a specific reaction such as $a A + b B \longrightarrow c C + d D$:

This definition guarantees that the reaction rate $r$ is a uniquely defined, positive quantity independent of which participant species is monitored.

Master Classification Table: Analytical Solutions to Empirical Rate Laws

| Reaction Order | Differential Rate Law | Integrated Rate Law | Linear Coordinate Plot | Slope ($m$) & Intercept ($b$) | Rate Constant Units | Half-Life Expression ($t_{1/2}$) | Three-Quarter Life ($t_{3/4}$) | |---|---|---|---|---|---|---|---| | Zero Order ($n=0$) | $-\frac{d[A]}{dt} = k$ | $[A]_t = [A]_0 - k t$ | $[A]_t \text{ vs } t$ | $m = -k$, $b = [A]_0$ | $\text{M}\cdot\text{s}^{-1}$ | $t_{1/2} = \frac{[A]_0}{2 k}$ | $t_{3/4} = \frac{3 [A]_0}{4 k}$ | | First Order ($n=1$) | $-\frac{d[A]}{dt} = k [A]$ | $\ln[A]_t = \ln[A]_0 - k t$ | $\ln[A]_t \text{ vs } t$ | $m = -k$, $b = \ln[A]_0$ | $\text{s}^{-1}$ | $t_{1/2} = \frac{\ln 2}{k}$ | $t_{3/4} = \frac{\ln 4}{k} = 2 t_{1/2}$ | | Second Order (symmetric) | $-\frac{d[A]}{dt} = k [A]^2$ | $\frac{1}{[A]_t} = \frac{1}{[A]_0} + k t$ | $\frac{1}{[A]_t} \text{ vs } t$ | $m = +k$, $b = \frac{1}{[A]_0}$ | $\text{M}^{-1}\cdot\text{s}^{-1}$ | $t_{1/2} = \frac{1}{k [A]_0}$ | $t_{3/4} = \frac{3}{k [A]_0} = 3 t_{1/2}$ | | Second Order (asymmetric)| $-\frac{d[A]}{dt} = k [A][B]$ | $\ln\left(\frac{[B]_t [A]_0}{[A]_t [B]_0}\right) = ([B]_0 - [A]_0) k t$ | $\ln\left(\frac{[B]_t}{[A]_t}\right) \text{ vs } t$ | $m = ([B]_0 - [A]_0) k$ | $\text{M}^{-1}\cdot\text{s}^{-1}$ | Dependent on $[B]_0 / [A]_0$ | Dependent on $[B]_0 / [A]_0$ | | Third Order (symmetric) | $-\frac{d[A]}{dt} = k [A]^3$ | $\frac{1}{[A]_t^2} = \frac{1}{[A]_0^2} + 2 k t$ | $\frac{1}{[A]_t^2} \text{ vs } t$ | $m = +2k$, $b = \frac{1}{[A]_0^2}$ | $\text{M}^{-2}\cdot\text{s}^{-1}$ | $t_{1/2} = \frac{3}{2 k [A]_0^2}$ | $t_{3/4} = \frac{15}{2 k [A]_0^2} = 5 t_{1/2}$ | | $n$-th Order ($n \ne 1$) | $-\frac{d[A]}{dt} = k [A]^n$ | $\frac{1}{[A]_t^{n-1}} = \frac{1}{[A]_0^{n-1}} + (n-1) k t$ | $\frac{1}{[A]_t^{n-1}} \text{ vs } t$ | $m = (n-1)k$ | $\text{M}^{1-n}\cdot\text{s}^{-1}$ | $t_{1/2} = \frac{2^{n-1} - 1}{(n-1) k [A]_0^{n-1}}$ | $t_{3/4} = \frac{4^{n-1} - 1}{(n-1) k [A]_0^{n-1}}$ | | Reversible First-Order | $-\frac{d[A]}{dt} = k_1 [A] - k_{-1} [B]$ | $\ln\left(\frac{[A]_0 - [A]_{\text{eq}}}{[A]_t - [A]_{\text{eq}}}\right) = (k_1 + k_{-1}) t$| $\ln([A]_t - [A]_{\text{eq}}) \text{ vs } t$| $m = -(k_1 + k_{-1})$ | $\text{s}^{-1}$ | $t_{1/2} = \frac{\ln 2}{k_1 + k_{-1}}$ | Relaxation time $\tau = \frac{1}{k_1 + k_{-1}}$ |

A critical distinction in chemical kinetics lies between the purely empirical concept of reaction order and the theoretical concept of molecularity.

Empirical Differential Rate Law

For many reactions far from equilibrium, the reaction rate depends on instantaneous reactant concentrations via an empirical power law:

where:

• $k$ is the rate constant (or specific reaction rate), independent of concentration but a strong function of temperature and ionic strength.
• $\alpha, \beta, \gamma$ are the partial orders of reaction with respect to species $A, B, C$.
• The overall order of reaction is $n = \alpha + \beta + \gamma + \cdots$.

Reaction Order:

• An experimentally determined, empirical quantity.
• Can be an integer ($0, 1, 2, 3$), a fraction ($1/2, 3/2$), or even negative.
• Does NOT necessarily bear any relation to stoichiometric coefficients $a, b, c$ unless the reaction is elementary.

Molecularity:

• A theoretical concept applied strictly to an elementary reaction step.
• Represents the exact number of reactant particles that must collide simultaneously to form the transition state.
• Strictly a positive integer: unimolecular ($1$), bimolecular ($2$), or termolecular ($3$). Quadrimolecular elementary steps have zero probability of occurrence.

In a zero-order reaction, the rate of reaction is entirely independent of the reactant concentration.

Differential and Integrated Equations

where $k_0$ has SI units of $\text{mol}/(\text{m}^3\cdot\text{s})$ or $\text{M}\cdot\text{s}^{-1}$.

Separating variables and integrating from $t = 0$ ($[A] = [A]_0$) to time $t$:

A plot of $[A]$ versus $t$ is a straight line with slope $-k_0$ and intercept $[A]_0$.

Half-Life ($t_{1/2}$)

The half-life $t_{1/2}$ is the time required for concentration to decrease to half its initial value ($[A] = [A]_0 / 2$):

In zero-order kinetics, the half-life is directly proportional to initial concentration $[A]_0$.

Total Reaction Lifetime ($t_{\text{end}}$)

The reaction terminates completely when $[A] = 0$:

Physical Occurrence in Heterogeneous Surface Catalysis

Zero-order kinetics commonly occur when a reaction takes place on a saturated solid catalyst surface or enzyme active site (e.g., decomposition of ammonia on hot tungsten, $2 NH_3 \xrightarrow{W} N_2 + 3 H_2$). When all catalytic active sites are fully covered by adsorbed molecules (Langmuir coverage $\theta \approx 1$), increasing gas concentration cannot increase the reaction rate.

First-order kinetics govern processes where the rate is directly proportional to the concentration of a single reactant.

Differential and Integrated Equations

where $k_1$ has SI units of $\text{s}^{-1}$ (time$^{-1}$).

Separating variables:

Linearized form:

A plot of $\ln[A]$ versus $t$ yields a straight line with slope $-k_1$.

Concentration of Product ($P$)

For $A \longrightarrow P$, with $[P]_0 = 0$:

Half-Life ($t_{1/2}$)

At $t = t_{1/2}$, $[A] = [A]_0 / 2$:

Cardinal Feature: The half-life of a first-order process is strictly independent of the initial concentration $[A]_0$. Successive half-lives remain constant: after $n$ half-lives, $[A] = [A]_0 / 2^n$.

Mean Lifetime ($\tau$)

The average lifetime of a reacting particle is the reciprocal of the first-order rate constant:

Second-order reactions involve the collision of two molecules, classified into two distinct cases.

Case 1: Single Reactant or Equal Initial Concentrations ($2 A \to P$ or $A + B \to P$ with $[A]_0 = [B]_0$)

where $k_2$ has SI units of $\text{m}^3/(\text{mol}\cdot\text{s})$ or $\text{M}^{-1}\text{s}^{-1}$.

Separating variables:

A plot of $1/[A]$ versus $t$ is linear with slope $+k_2$ and intercept $1/[A]_0$.

Half-Life:

In second-order kinetics, the half-life is inversely proportional to initial concentration. Each successive half-life doubles ($t_{1/2}, 2 t_{1/2}, 4 t_{1/2}$).

Case 2: Unequal Initial Concentrations ($A + B \to P$ with $[A]_0 \neq [B]_0$)

Let $x$ be the extent of concentration reacted at time $t$: $[A] = [A]_0 - x$, $[B] = [B]_0 - x$.

Using partial fractions:

Integrating from $x = 0$ at $t = 0$:

Plotting $\ln([B]/[A])$ versus $t$ yields a straight line with slope $([B]_0 - [A]_0) k_2$.

University Honors Research Monograph: Microfluidic Flow Kinetics & Automated Machine-Learning Profiling

Continuous-flow microchemical reactors have transformed empirical reaction kinetics from laborious manual batch aliquot sampling into continuous, autonomous real-time multidimensional mapping:

• Peclet and Reynolds Number Hydrodynamics: In microchannels of hydraulic diameter $D_h \approx 200 - 500\;\mu\text{m}$, fluid flow is strictly laminar:

Turbulent back-mixing is eliminated, while molecular diffusion distances are miniature ($< 100\;\mu\text{m}$), ensuring complete radial mixing in milliseconds.

• Plug-Flow Reaction Profiling: Under plug-flow conditions, axial distance $z$ downstream from the mixing junction translates linearly to reaction residence time:

Varying total volumetric pumping rate $Q_{\text{total}}$ rapidly sweeps reaction times from $10\text{ ms}$ to $10\text{ minutes}$.

• Closed-Loop Bayesian Optimization: Coupling microfluidic flow reactors to in-line high-pressure NMR, HPLC-MS, and ATR-FTIR flow cells enables machine learning algorithms (Bayesian optimization with Gaussian process regression) to automatically map multi-variable kinetic response surfaces ($T$, $P$, catalyst loading, residence time), resolving full multi-step reaction mechanisms and rate constants in under 48 hours.

Higher-order and non-integer kinetics emerge in complex termolecular gas reactions and catalytic systems.

Third-Order Reactions ($3 A \to P$)

where $k_3$ has SI units of $\text{M}^{-2}\text{s}^{-1}$.

Integrating:

Half-life for third-order kinetics:

True gas-phase termolecular elementary reactions are rare due to the vanishing probability of simultaneous three-body collisions; prominent examples include gas-phase oxidation of nitric oxide ($2 NO + O_2 \longrightarrow 2 NO_2$) and radical recombination ($H + H + M \longrightarrow H_2 + M$).

Fractional-Life Method for General Order $n$

For an $n$-th order reaction ($-\frac{d[A]}{dt} = k [A]^n$, $n \neq 1$):

The half-life scaling is:

Pseudo-Unimolecular Reactions (Isolation Method)

When a bimolecular reaction $A + B \to P$ is conducted with species $B$ in overwhelming stoichiometric excess ($[B]_0 \gg [A]_0$): $[B](t) \approx [B]_0 = \text{constant}$ throughout the course of reaction.

where $k_{\text{obs}} = k_2 [B]_0$ is the pseudo-first-order rate constant (units: $\text{s}^{-1}$). This reduces second-order mathematics to simple first-order exponential decays, forming the basis of the isolation method.

When the forward and reverse reactions proceed at comparable rates, the system approaches a dynamic chemical equilibrium.

First-Order Opposing Reaction ($A \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} B$)

Let initial concentrations be $[A]_0$ and $[B]_0 = 0$. At time $t$, $[A] = [A]_0 - x$ and $[B] = x$.

The net rate of consumption of $A$ is:

At dynamic equilibrium ($t \to \infty$), net rate vanishes ($dx/dt = 0$):

where $K_c$ is the thermodynamic equilibrium constant. This satisfies the Principle of Microscopic Reversibility.

Expressing $k_1 [A]_0$ in terms of equilibrium extent:

Substituting back into the rate differential equation:

Separating variables and integrating:

In terms of reactant concentration:

Key Insight: The approach to equilibrium is a first-order exponential relaxation whose apparent rate constant is the sum of forward and backward rate constants ($k_{\text{obs}} = k_1 + k_{-1}$). Measuring $k_{\text{obs}}$ along with the equilibrium ratio $K_c = k_1 / k_{-1}$ allows exact evaluation of both individual rate constants.

Determining multi-step kinetic rate laws in modern organic synthesis and industrial process development relies on continuous, in situ physical property tracking using operando reaction calorimetry and continuous-flow spectroscopy.

1. In Situ Heat-Flow Reaction Calorimetry

Because virtually all chemical transformations exhibit a non-zero enthalpy of reaction ($\Delta_r H \ne 0$), the instantaneous rate of heat generation $q_r(t)$ in a closed or flow reactor is directly proportional to the instantaneous chemical reaction rate $r(t) = -d[A]/dt$:

Dynamic Heat Balance in a Controlled Reactor

In an automated heat-flow reaction calorimeter (such as an RC1 or Omnical system), a thermostat jacket maintains reactor temperature $T_r$ under isothermal control. The heat balance equation is:

where:

• $U$ is overall heat transfer coefficient ($\text{W}/(\text{m}^2\cdot\text{K})$).
• $A$ is wetted heat exchange area ($\text{m}^2$).
• $T_j$ is cooling/heating jacket temperature.
• $q_{\text{stir}}$ is mechanical agitator power dissipation.
• $C_p$ is total heat capacity of the reactor assembly and liquid contents.

Under true isothermal steady conditions ($dT_r/dt = 0$), calibration pulses determine the instantaneous heat transfer product $U A$. Subtracting baseline power yields direct, millisecond-resolved chemical heat release rates $q_r(t)$ without withdrawing samples or perturbing chemical equilibria.

Reaction Progress Kinetic Analysis (RPKA) via Calorimetry

Integrating heat release from time $t = 0$ to completion ($t = \infty$) yields total chemical enthalpy release:

The instantaneous fractional conversion $\chi(t)$ and reactant concentration $[A](t)$ are given by:

Plotting instantaneous reaction rate $r(t) = q_r(t) / (V (-\Delta_r H))$ directly against concentration $[A](t)$ generates "graphical rate equations" that reveal reaction orders, catalyst deactivation, and product inhibition in a single experiment.

2. Online UV-Vis / FTIR Attenuated Total Reflectance (ATR) Spectroscopy

In situ ATR-FTIR utilizes a zinc selenide ($\text{ZnSe}$) or diamond internal reflection element placed inside the reaction medium. An infrared beam undergoes multiple total internal reflections at the crystal-solution interface, generating an evanescent wave that penetrates $1 - 3\;\mu\text{m}$ into the solution.

By Beer-Lambert's law for evanescent absorption:

where $d_p = \frac{\lambda}{2 \pi \sqrt{n_{\text{crystal}}^2 \sin^2 \theta - n_{\text{sol}}^2}}$ is penetration depth. Fourier transform deconvolution generates complete concentration profiles $[C_i](t)$ for reactants, transient reactive intermediates, and products with sub-second temporal resolution.