12.1 Reaction Kinetics and Performance Metrics

Reaction kinetics lies at the heart of chemical reaction engineering, describing the rate at which chemical species are consumed or generated during a reaction. In the FE Chemical exam, mastering these concepts requires a solid grasp of rate laws, stoichiometry, Arrhenius temperature dependence, and key performance indicators like conversion, yield, and selectivity. The FE Reference Handbook v10.6 provides the primary equations in the 'Chemical Engineering' section under 'Chemical Reaction Engineering'.

Homogeneous Reaction Rates and Stoichiometry

A homogeneous reaction occurs in a single phase (typically liquid or gas). The rate of reaction of a species $i$, denoted as $r_i$, is defined as the change in moles of $i$ per unit time per unit volume of reacting mixture:

$$r_i = \frac{1}{V} \frac{dN_i}{dt}$$

where $V$ is the volume of the reaction mixture, $N_i$ is the number of moles of species $i$, and $t$ is time. The rate of reaction $r_i$ is negative for reactants (since they are consumed) and positive for products (since they are formed). To define a rate of reaction that is independent of the species chosen, we use the stoichiometry of the reaction. For a general reaction:

$$a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}$$

where $a, b, c,$ and $d$ are stoichiometric coefficients, the relative rates of reaction are related by:

$$\frac{-r_{\text{A}}}{a} = \frac{-r_{\text{B}}}{b} = \frac{r_{\text{C}}}{c} = \frac{r_{\text{D}}}{d}$$

Here, $-r_{\text{A}}$ and $-r_{\text{B}}$ represent the rates of disappearance of reactants A and B, respectively, while $r_{\text{C}}$ and $r_{\text{D}}$ are the rates of formation of products C and D. A classic FE exam question will provide the rate of disappearance of one reactant and ask for the rate of appearance of a product. In such cases, direct substitution into these ratios is the key.

Rate Laws and Reaction Order

The rate of reaction is generally a function of species concentrations and temperature. Under isothermal conditions, the rate of consumption of reactant A is typically represented by a power-law rate equation:

$$-r_{\text{A}} = k C_{\text{A}}^{\alpha} C_{\text{B}}^{\beta}$$

where $C_{\text{A}}$ and $C_{\text{B}}$ are the molar concentrations of A and B, respectively. The parameters $\alpha$ and $\beta$ are the reaction orders with respect to reactants A and B. The sum of these individual orders, $n = \alpha + \beta$, is the overall reaction order.

• Elementary Reactions: For elementary reactions, the reaction orders match the stoichiometric coefficients (e.g., $\alpha = a$ and $\beta = b$).
• Non-Elementary Reactions: For non-elementary reactions, there is no direct correspondence, and the orders must be determined experimentally.

The proportionality constant $k$ is the reaction rate constant. The units of $k$ depend on the overall reaction order $n$ and are derived from the rate law:

$$[k] = \left(\frac{\text{volume}}{\text{mole}}\right)^{n-1} \cdot \frac{1}{\text{time}}$$

In common SI units:

• For a zero-order reaction ($n=0$): $k$ has units of $\text{mol}/(\text{L}\cdot\text{s})$.
• For a first-order reaction ($n=1$): $k$ has units of $\text{s}^{-1}$.
• For a second-order reaction ($n=2$): $k$ has units of $\text{L}/(\text{mol}\cdot\text{s})$.
• For a third-order reaction ($n=3$): $k$ has units of $\text{L}^2/(\text{mol}^2\cdot\text{s})$.

Temperature Dependence: The Arrhenius Equation

The rate constant $k$ is strongly dependent on temperature. This behavior is modeled by the Arrhenius equation:

$$k = A \exp\left(-\frac{E_a}{R T}\right)$$

where:

• $A$ is the pre-exponential factor (same units as $k$), representing the frequency of collisions.
• $E_a$ is the activation energy ($\text{J/mol}$ or $\text{cal/mol}$), representing the minimum energy barrier that must be overcome for a reaction to occur.
• $R$ is the universal gas constant ($8.314\text{ J/(mol}\cdot\text{K)}$ or $1.987\text{ cal/(mol}\cdot\text{K)}$).
• $T$ is the absolute temperature in Kelvin ($\text{K}$).

To compare rate constants at two different temperatures ($T_1$ and $T_2$), the Arrhenius equation is integrated to yield the two-point form:

$$\ln\left(\frac{k_2}{k_1}\right) = \frac{-E_a}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)$$

This can be rewritten as:

$$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

On the FE Chemical exam, you are frequently asked to find $E_a$ given rate constants at two temperatures, or to calculate a rate constant at a new temperature. A common error is entering temperatures in Celsius rather than Kelvin, which will lead to incorrect answers that often match distractor options.

Performance Metrics: Conversion, Yield, and Selectivity

To evaluate and design chemical reactors, engineers use quantitative performance metrics.

1. Conversion ($X_{\text{A}}$)

Conversion is defined as the ratio of the moles of key reactant (usually the limiting reactant, designated as A) reacted to the moles of reactant fed to the system.

• For a batch reactor (closed system): $$X_{\text{A}} = \frac{N_{\text{A}0} - N_{\text{A}}}{N_{\text{A}0}}$$ where $N_{\text{A}0}$ is the initial number of moles of A, and $N_{\text{A}}$ is the moles of A remaining at time $t$.
• For a flow reactor (open system at steady state): $$X_{\text{A}} = \frac{F_{\text{A}0} - F_{\text{A}}}{F_{\text{A}0}}$$ where $F_{\text{A}0}$ is the molar flow rate of A entering the reactor, and $F_{\text{A}}$ is the molar flow rate of A leaving the reactor.

2. Selectivity ($S$)

In multiple-reaction systems, the reactant may form both desired ($D$) and undesired ($U$) products. Selectivity is a measure of the system's ability to favor the desired reaction over the undesired one.

• Instantaneous Selectivity ($S_{\text{D/U}}$): The ratio of the rate of formation of the desired product to the rate of formation of the undesired product: $$S_{\text{D/U}} = \frac{r_{\text{D}}}{r_{\text{U}}}$$
• Overall Selectivity ($\tilde{S}_{\text{D/U}}$): The ratio of the exit molar flow rates of the desired product to the undesired product (for flow systems): $$\tilde{S}_{\text{D/U}} = \frac{F_{\text{D}}}{F_{\text{U}}}$$

3. Yield ($Y$)

Yield measures the efficiency of converting reactant into the desired product. In the literature, yield can be defined in multiple ways, so pay close attention to NCEES wording:

• Yield based on reactant consumed (Fractional Yield): $$Y_{\text{D}} = \frac{\text{moles of D formed}}{\text{moles of key reactant A consumed}} = \frac{F_{\text{D}} - F_{\text{D}0}}{F_{\text{A}0} - F_{\text{A}}}$$
• Yield based on reactant fed: $$Y_{\text{D, feed}} = \frac{\text{moles of D formed}}{\text{moles of key reactant A fed}} = \frac{F_{\text{D}} - F_{\text{D}0}}{F_{\text{A}0}}$$

Note that the overall yield is equal to the overall selectivity multiplied by the conversion of the key reactant: $Y_{\text{D, feed}} = \tilde{S}_{\text{D/U}} \cdot X_{\text{A}} \cdot \nu$ (where $\nu$ is a stoichiometric factor relating A to D).

Worked Example: Activation Energy Calculation

Problem: A first-order gas-phase reaction has a rate constant of $0.015\text{ s}^{-1}$ at $300\text{ K}$. When the temperature is increased to $350\text{ K}$, the rate constant increases to $0.275\text{ s}^{-1}$. Calculate the activation energy $E_a$ of this reaction in $\text{kJ/mol}$.

Solution:

1. State the given parameters:

   • $T_1 = 300\text{ K}$
   • $k_1 = 0.015\text{ s}^{-1}$
   • $T_2 = 350\text{ K}$
   • $k_2 = 0.275\text{ s}^{-1}$
   • $R = 8.314\text{ J/(mol}\cdot\text{K)}$

2. Set up the integrated two-point Arrhenius equation:

   $$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

3. Substitute the values:

   $$\ln\left(\frac{0.275}{0.015}\right) = \frac{E_a}{8.314} \left(\frac{1}{300} - \frac{1}{350}\right)$$

4. Calculate the left side and the temperature terms:

   $$\ln(18.333) \approx 2.9087$$ $$\left(\frac{1}{300} - \frac{1}{350}\right) = 0.0033333 - 0.0028571 = 0.00047619\text{ K}^{-1}$$

5. Solve for $E_a$:

   $$2.9087 = \frac{E_a}{8.314} \cdot 0.00047619$$ $$E_a = \frac{2.9087 \cdot 8.314}{0.00047619} \approx 50,789\text{ J/mol}$$

6. Convert to $\text{kJ/mol}$:

   $$E_a \approx 50.8\text{ kJ/mol}$$

This step-by-step method ensures accuracy. Pay attention to the temperature unit conversion; if the problem had given the temperatures in Celsius, converting them to Kelvin first is mandatory.