8.1 Reaction stoichiometry and kinetics

Chemical reactions sit at the core of chemical engineering. Stoichiometry tells you how much reacts and forms, while kinetics tells you how fast it happens. Together, they give you the quantitative foundation for designing reactors and optimizing processes.

This section covers the math behind balanced reactions, rate laws, reaction mechanisms, and how to extract kinetic parameters from experimental data.

Chemical Reaction Stoichiometry

Quantitative Relationships in Chemical Reactions

Stoichiometry is the bookkeeping of chemical reactions. It uses the coefficients in a balanced equation to establish exact molar ratios between reactants and products.

For example, in $2H_2 + O_2 \rightarrow 2H_2O$, the stoichiometric coefficients tell you that 2 moles of $H_2$ react with 1 mole of $O_2$ to produce 2 moles of $H_2O$. Every calculation you do in reaction engineering starts from these ratios.

The limiting reactant is the one that gets completely consumed first, and it caps the maximum amount of product you can form. Any other reactant present beyond what the stoichiometry requires is in excess.

• To identify the limiting reactant, divide the moles of each reactant by its stoichiometric coefficient. The reactant with the smallest value is limiting.
• In $2Al + 3CuO \rightarrow Al_2O_3 + 3Cu$, if you have fewer moles of Al relative to its coefficient than CuO relative to its coefficient, Al is limiting and determines how much Cu you can produce.

Conversion and Yield in Chemical Reactions

Conversion measures how much of a reactant has been used up:

$X_A = \frac{\text{moles of A consumed}}{\text{initial moles of A}}$

A conversion of 0.90 (or 90%) means 90% of reactant A has reacted. High conversion is generally desirable because it means less unreacted feed to deal with downstream.

Yield compares what you actually produced to what stoichiometry says you could have produced:

$\text{Percent Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100\%$

• Theoretical yield is calculated from the limiting reactant using stoichiometric ratios.
• Actual yield is what you measure experimentally.
• If the theoretical yield is 100 g and you obtain 80 g, your percent yield is 80%.

Yield is almost always less than 100% due to side reactions, incomplete conversion, or losses during separation.

Reactor Design Considerations

Stoichiometry directly shapes how you design and operate a reactor:

• Feed ratios matter. Reactants need to be supplied in (or near) stoichiometric proportions. Off-ratio feeds lead to unconverted material or can promote unwanted side reactions.
• Conversion and yield targets drive sizing. Achieving higher conversion typically requires a larger reactor, longer residence time, or more aggressive conditions (higher temperature or pressure).
• Side reactions complicate everything. Undesired reactions consume your reactants and reduce selectivity (the fraction of converted reactant that forms the desired product). Byproducts may also require additional separation steps. For instance, in ethylene oxide production, $CO_2$ forms as a byproduct and must be removed downstream.

Fundamentals of Chemical Kinetics

Reaction Rates and Rate Laws

While stoichiometry tells you how much, kinetics tells you how fast. The reaction rate is the change in concentration of a reactant or product per unit time, typically expressed in units like mol/(L·s).

Three main factors influence how fast a reaction proceeds: temperature, reactant concentrations, and whether a catalyst is present.

The rate law is the mathematical expression that links reaction rate to reactant concentrations:

$r = k[A]^m[B]^n$

• $k$ is the rate constant (depends on temperature, not concentration)
• $[A]$ and $[B]$ are reactant concentrations
• $m$ and $n$ are the orders with respect to each reactant

The overall order is $m + n$. These exponents are determined experimentally and do not necessarily match the stoichiometric coefficients.

Rate Constants and Reaction Orders

The rate constant $k$ is what connects concentrations to rate. Its units change depending on the overall reaction order so that the rate always comes out in concentration/time:

Each order has a distinct physical meaning:

• Zero-order: Rate doesn't depend on concentration at all. This can happen when a catalyst surface is saturated.
• First-order: Rate is directly proportional to concentration. Double $[A]$, double the rate.
• Second-order: Rate depends on concentration squared (or the product of two concentrations). Double $[A]$ in a second-order-in-A reaction, and the rate quadruples.

Factors Affecting Reaction Rates

Temperature is usually the most powerful lever. Higher temperature means molecules move faster and collide with more energy, making it easier to overcome the activation energy barrier ($E_a$). The Arrhenius equation quantifies this:

$k = A \, e^{-E_a/(RT)}$

• $A$ = pre-exponential (frequency) factor
• $E_a$ = activation energy (J/mol)
• $R$ = gas constant (8.314 J/(mol·K))
• $T$ = absolute temperature (K)

Even a modest temperature increase can dramatically raise $k$ because of that exponential dependence.

Catalysts speed up reactions by providing an alternative pathway with a lower $E_a$. They participate in the reaction mechanism but are regenerated, so they're not consumed overall. Examples include platinum in automotive catalytic converters and enzymes in biological systems.

Reactant concentrations affect rate as described by the rate law. The sensitivity depends on the reaction order: a first-order reactant doubles the rate when its concentration doubles, while a zero-order reactant has no effect on rate regardless of concentration changes.

Rate-Determining Steps in Reactions

Reaction Mechanisms and Elementary Steps

Most reactions don't happen in a single molecular event. Instead, they proceed through a reaction mechanism, which is a sequence of elementary steps that, when summed, give the overall balanced equation.

Each elementary step describes one actual molecular event (a collision, a bond break, a rearrangement). Unlike overall reactions, the rate law for an elementary step can be written directly from its stoichiometry:

• Unimolecular (one molecule reacts): $r = k[A]$
• Bimolecular (two molecules collide): $r = k[A][B]$
• Termolecular (three molecules collide simultaneously): extremely rare because three-body collisions are highly improbable

Rate-Determining Steps and Overall Reaction Rates

In a multi-step mechanism, the rate-determining step (RDS) is the slowest step. It acts as a bottleneck: the overall reaction can't go faster than this step allows.

The RDS is typically the step with the highest activation energy (or equivalently, the smallest rate constant). The overall rate law for the reaction is governed by the RDS and its molecularity:

• If the RDS is unimolecular ($A \rightarrow \text{products}$), the overall rate law is first-order.
• If the RDS is bimolecular ($A + B \rightarrow \text{products}$), the overall rate law is second-order: $r = k[A][B]$.

To identify the RDS experimentally, you vary reactant concentrations and observe which ones affect the overall rate. If changing the concentration of a species has no effect on the rate, that species likely isn't involved in (or before) the RDS.

Factors Influencing the Rate-Determining Step

The identity of the RDS isn't always fixed. Changing conditions can shift which step is slowest:

• Temperature changes alter the rate constants of each elementary step by different amounts (because each step has its own $E_a$). A temperature increase could speed up the original RDS enough that a different step becomes the new bottleneck.
• Pressure changes affect steps involving gaseous species, potentially shifting the RDS in gas-phase reactions.
• Concentration changes can also matter. Increasing the concentration of a reactant involved in the RDS speeds up the overall reaction. Changing the concentration of a reactant that only appears in a fast step may have little or no effect.

Understanding which step is rate-determining helps you target the right variable when optimizing a process.

Reaction Order and Rate Constants

Integrated Rate Laws

The rate laws discussed above are differential rate laws (rate as a function of concentration). Integrated rate laws rearrange these into concentration as a function of time, which is what you actually measure in experiments.

To determine reaction order from data, plot concentration in each of these forms against time. Whichever plot gives a straight line tells you the order.

Reaction Half-Lives

The half-life ($t_{1/2}$) is the time for a reactant's concentration to drop to half its initial value.

• First-order: $t_{1/2} = \frac{\ln 2}{k}$. This is constant and independent of initial concentration, which is a signature feature of first-order kinetics.
• Second-order: $t_{1/2} = \frac{1}{k[A]_0}$. Here the half-life depends on the starting concentration, so each successive half-life is longer than the last.

Half-lives give you a quick way to estimate how far a reaction has progressed:

• After 1 half-life: 50% consumed
• After 2 half-lives: 75% consumed
• After 3 half-lives: 87.5% consumed
• After $n$ half-lives: $[A] = [A]_0 \left(\frac{1}{2}\right)^n$

Experimental Determination of Rate Laws

Method of initial rates is the most common approach for finding reaction orders:

1. Run the reaction multiple times, each time changing the initial concentration of one reactant while holding all others constant.
2. Measure the initial rate for each run (before significant concentration changes occur).
3. Compare runs: if doubling $[A]$ doubles the rate, the reaction is first-order in A. If doubling $[A]$ quadruples the rate, it's second-order in A.

Arrhenius plot for finding $E_a$ and $A$:

1. Measure the rate constant $k$ at several different temperatures.
2. Plot $\ln(k)$ vs. $1/T$.
3. The result should be a straight line with slope $= -E_a/R$ and y-intercept $= \ln(A)$.

This lets you predict $k$ at any temperature, which is essential for scaling reactions from the lab to industrial conditions.