Integrated Rate Law For Second Order Reaction

The study of chemical kinetics unveils the rates at which reactions occur and the factors influencing these rates. Among the various types of reactions, second-order reactions hold a significant place due to their prevalence in numerous chemical processes. Understanding the integrated rate law for second-order reactions is crucial for predicting reactant concentrations over time and determining the reaction rate constant.

Introduction to Second-Order Reactions

Second-order reactions are chemical reactions where the rate of the reaction is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. This differentiates them from first-order reactions, where the rate is proportional to the concentration of a single reactant, and zero-order reactions, where the rate is independent of reactant concentrations.

Mathematically, a second-order reaction can be represented as:

A + A → Products (Rate = k[A]²)

or

A + B → Products (Rate = k[A][B])

where:

• k is the rate constant
• [A] and [B] are the concentrations of reactants A and B, respectively

The integrated rate law, derived from the differential rate law, provides a more practical way to determine the concentration of reactants at a given time.

Derivation of the Integrated Rate Law for Second-Order Reactions

Let's consider the simpler case of a second-order reaction:

2A → Products

The differential rate law for this reaction is:

Rate = -d[A]/dt = k[A]²

To derive the integrated rate law, we need to separate variables and integrate:

-d[A]/[A]² = k dt

Integrating both sides:

∫ -d[A]/[A]² = ∫ k dt

The integral of -1/[A]² is 1/[A], so:

1/[A] = kt + C

where C is the integration constant.

To find the value of C, we use the initial conditions. At time t = 0, the concentration of A is [A]₀ (initial concentration):

1/[A]₀ = k(0) + C

Therefore, C = 1/[A]₀

Substituting C back into the equation, we get the integrated rate law:

1/[A] = kt + 1/[A]₀

This equation can be rearranged to solve for [A]:

[A] = [A]₀ / (1 + kt[A]₀)

Integrated Rate Law for A + B → Products (When [A]₀ ≠ [B]₀)

When the initial concentrations of A and B are not equal ([A]₀ ≠ [B]₀), the derivation becomes slightly more complex. The rate law is:

Rate = -d[A]/dt = -d[B]/dt = k[A][B]

Let x be the concentration of A that has reacted at time t. Then:

[A] = [A]₀ - x

[B] = [B]₀ - x

Substituting these into the rate law:

dx/dt = k([A]₀ - x)([B]₀ - x)

Separating variables:

dx / (([A]₀ - x)([B]₀ - x)) = k dt

To integrate the left side, we use partial fraction decomposition:

1 / (([A]₀ - x)([B]₀ - x)) = (1/([B]₀ - [A]₀)) * (1/([A]₀ - x) - 1/([B]₀ - x))

Now, the integral becomes:

∫ (1/([B]₀ - [A]₀)) * (1/([A]₀ - x) - 1/([B]₀ - x)) dx = ∫ k dt

(1/([B]₀ - [A]₀)) * (∫ 1/([A]₀ - x) dx - ∫ 1/([B]₀ - x) dx) = kt + C

Integrating each term:

(1/([B]₀ - [A]₀)) * (-ln|A₀ - x| + ln|B₀ - x|) = kt + C

(1/([B]₀ - [A]₀)) * ln|(B₀ - x)/(A₀ - x)| = kt + C

At t = 0, x = 0:

(1/([B]₀ - [A]₀)) * ln|B₀/A₀| = C

Substituting C back:

(1/([B]₀ - [A]₀)) * ln|(B₀ - x)/(A₀ - x)| = kt + (1/([B]₀ - [A]₀)) * ln|B₀/A₀|

Rearranging to get the integrated rate law:

ln|(B₀ - x)/(A₀ - x)| = ln|B₀/A₀| + k([B]₀ - [A]₀)t

This can be further simplified to:

ln([B]/[A]) = ln([B]₀/[A]₀) + k([B]₀ - [A]₀)t

Half-Life of a Second-Order Reaction

The half-life (t₁/₂) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For a second-order reaction, the half-life can be derived from the integrated rate law.

Using the integrated rate law for 2A → Products:

1/[A] = kt + 1/[A]₀

At t = t₁/₂, [A] = [A]₀/2

1/([A]₀/2) = kt₁/₂ + 1/[A]₀

2/[A]₀ = kt₁/₂ + 1/[A]₀

kt₁/₂ = 2/[A]₀ - 1/[A]₀

kt₁/₂ = 1/[A]₀

t₁/₂ = 1/(k[A]₀)

Unlike first-order reactions, the half-life of a second-order reaction is dependent on the initial concentration of the reactant. As the initial concentration increases, the half-life decreases, and vice versa. This is a key characteristic that distinguishes second-order reactions from other reaction orders.

Characteristics of Second-Order Reactions

Several characteristics distinguish second-order reactions from reactions of other orders:

1. Rate Law: The rate of the reaction is proportional to the square of one reactant's concentration or the product of two reactants' concentrations.
2. Integrated Rate Law: The integrated rate law is different from that of first-order and zero-order reactions, allowing for the determination of reactant concentrations at specific times.
3. Half-Life: The half-life of a second-order reaction is inversely proportional to the initial concentration of the reactant. This is a crucial factor in determining the reaction's behavior over time.
4. Units of Rate Constant: The rate constant k for a second-order reaction has units of M⁻¹s⁻¹ (or L mol⁻¹ s⁻¹), which differs from the units for first-order (s⁻¹) and zero-order (M s⁻¹) reactions.
5. Graphical Representation: Plotting 1/[A] versus time yields a straight line for a second-order reaction, with the slope equal to the rate constant k.

Factors Affecting Reaction Rates

Several factors can affect the rates of second-order reactions, including:

• Temperature: Increasing the temperature generally increases the rate of the reaction. According to the Arrhenius equation, the rate constant k is exponentially dependent on temperature: k = A * exp(-Ea/RT) where:

  • A is the pre-exponential factor
  • Ea is the activation energy
  • R is the gas constant
  • T is the temperature in Kelvin

• Concentration: As indicated by the rate law, increasing the concentration of the reactants increases the reaction rate. This is because a higher concentration leads to more frequent collisions between reactant molecules.

• Catalysts: Catalysts can increase the rate of a reaction by providing an alternative reaction pathway with a lower activation energy. Catalysts do not change the equilibrium constant but only affect the rate at which equilibrium is reached.

• Surface Area: For heterogeneous reactions (reactions occurring at the interface between two phases), increasing the surface area of a solid reactant can increase the reaction rate. This is because a larger surface area provides more sites for the reaction to occur.

• Pressure: For gas-phase reactions, increasing the pressure can increase the concentration of the reactants, leading to an increase in the reaction rate.

Examples of Second-Order Reactions

Second-order reactions are common in various chemical and industrial processes. Here are a few examples:

1. Diels-Alder Reaction: The Diels-Alder reaction is a cycloaddition reaction between a conjugated diene and a dienophile to form a cyclohexene derivative. This reaction is second order overall, first order with respect to each reactant.

2. Saponification: Saponification is the alkaline hydrolysis of esters (such as triglycerides) to form soap and glycerol. The reaction between an ester and a hydroxide ion (OH⁻) is typically second order.

3. NO₂ Decomposition: The decomposition of nitrogen dioxide (NO₂) into nitrogen monoxide (NO) and oxygen (O₂) is a second-order reaction.

   2NO₂ (g) → 2NO (g) + O₂ (g)

4. Reaction of Ozone with Oxygen Atoms: The reaction between ozone (O₃) and oxygen atoms (O) in the stratosphere is a second-order reaction that helps to regulate the ozone layer.

   O₃ + O → 2O₂

5. The reaction of ethyl acetate with sodium hydroxide: This is a classic saponification reaction.

   CH₃COOC₂H₅ + NaOH → CH₃COONa + C₂H₅OH

Applications of Integrated Rate Law

The integrated rate law for second-order reactions has numerous applications in chemical kinetics and related fields:

• Determining Reaction Order: By analyzing experimental data and comparing it with the integrated rate laws of different reaction orders, one can determine the order of a reaction.

• Calculating Rate Constants: The integrated rate law can be used to calculate the rate constant k of a reaction by measuring reactant concentrations at different times.

• Predicting Reactant Concentrations: Given the rate constant and initial concentrations, the integrated rate law can predict the concentration of reactants at any given time.

• Designing Chemical Reactors: In chemical engineering, the integrated rate law is used to design and optimize chemical reactors by predicting reaction rates and product yields under various conditions.

• Studying Reaction Mechanisms: Analyzing the kinetics of a reaction can provide valuable insights into the reaction mechanism, including the identification of rate-determining steps and intermediates.

Limitations and Considerations

While the integrated rate law for second-order reactions is a powerful tool, it is essential to be aware of its limitations and considerations:

• Ideal Conditions: The integrated rate law is derived under ideal conditions, assuming that the reaction is elementary and that the rate law is followed throughout the reaction. In reality, many reactions are complex and involve multiple steps, which can deviate from the simple second-order kinetics.

• Reversible Reactions: The integrated rate law discussed here applies to irreversible reactions. For reversible reactions, the rate law becomes more complex and must take into account the reverse reaction.

• Temperature Dependence: The rate constant k is temperature-dependent, so the integrated rate law is only valid at a constant temperature. If the temperature changes during the reaction, the rate constant must be adjusted accordingly.

• Experimental Errors: Experimental errors in measuring reactant concentrations and time can affect the accuracy of the integrated rate law. It is essential to minimize these errors through careful experimental design and data analysis.

Examples of Problem Solving Using the Integrated Rate Law

Example 1:

The decomposition of acetaldehyde (CH₃CHO) is a second-order reaction:

CH₃CHO → CH₄ + CO

The rate constant k is 0.105 M⁻¹s⁻¹ at 500°C. If the initial concentration of acetaldehyde is 0.200 M, what is the concentration after 100 seconds?

Solution:

Using the integrated rate law:

1/[A] = kt + 1/[A]₀

1/[A] = (0.105 M⁻¹s⁻¹)(100 s) + 1/(0.200 M)

1/[A] = 10.5 M⁻¹ + 5 M⁻¹

1/[A] = 15.5 M⁻¹

[A] = 1/15.5 M

[A] ≈ 0.0645 M

Therefore, the concentration of acetaldehyde after 100 seconds is approximately 0.0645 M.

Example 2:

For the reaction 2A → Products, the initial concentration of A is 0.500 M, and the half-life is 50 seconds. Calculate the rate constant k.

Solution:

Using the half-life equation for a second-order reaction:

t₁/₂ = 1/(k[A]₀)

50 s = 1/(k * 0.500 M)

k = 1/(50 s * 0.500 M)

k = 1/25 M s

k = 0.04 M⁻¹s⁻¹

Therefore, the rate constant k for the reaction is 0.04 M⁻¹s⁻¹.

Conclusion

The integrated rate law for second-order reactions is a crucial tool for understanding and predicting the behavior of chemical reactions. By grasping the underlying principles and derivations, chemists and engineers can analyze experimental data, calculate rate constants, predict reactant concentrations, and design chemical reactors effectively. While it is essential to acknowledge the limitations and considerations associated with the integrated rate law, its applications in various fields underscore its significance in chemical kinetics. From industrial processes to environmental studies, the understanding of second-order reactions plays a vital role in advancing scientific knowledge and improving technological applications.