Writing The Rate Law Implied By A Simple Mechanism

Unraveling the secrets of chemical reactions often feels like piecing together a complex puzzle. One crucial piece of this puzzle is understanding the rate law, which mathematically describes how the speed of a reaction depends on the concentration of reactants. While determining rate laws experimentally is a common approach, a deeper understanding can be gained by examining the reaction mechanism. This article will explore how to write the rate law implied by a simple mechanism, providing a comprehensive guide for chemists and students alike.

Introduction to Reaction Mechanisms and Rate Laws

Chemical reactions rarely occur in a single step. Instead, they usually proceed through a series of elementary steps, collectively known as the reaction mechanism. Each elementary step represents a single molecular event, such as the collision of two molecules or the decomposition of a molecule.

A rate law expresses the relationship between the rate of a reaction and the concentrations of the reactants. For a simple, one-step reaction:

aA + bB -> cC + dD

The rate law is often written as:

Rate = k[A]^m[B]^n

Where:

• k is the rate constant, a proportionality constant that reflects the intrinsic speed of the reaction.
• [A] and [B] are the concentrations of reactants A and B.
• m and n are the reaction orders with respect to reactants A and B, respectively. These exponents must be determined experimentally and are not necessarily equal to the stoichiometric coefficients a and b.

However, when dealing with multi-step reaction mechanisms, determining the overall rate law requires a more nuanced approach. This is because the overall rate is often dictated by the slowest step in the mechanism, known as the rate-determining step.

Identifying Elementary Steps and Intermediates

The first step in deriving a rate law from a mechanism is to identify the elementary steps and any intermediates involved.

Elementary Steps: These are the individual steps that make up the reaction mechanism. Each elementary step is a single molecular event. The molecularity of an elementary step refers to the number of molecules that participate in that step. Common types of elementary steps include:

• Unimolecular: A single molecule undergoes a change, such as decomposition or isomerization. (A -> products)
• Bimolecular: Two molecules collide and react. (A + B -> products or 2A -> products)
• Termolecular: Three molecules collide simultaneously and react. (Rare due to the low probability of a three-body collision) (A + B + C -> products)

Intermediates: These are species that are formed in one elementary step and consumed in a subsequent step. Intermediates do not appear in the overall balanced equation for the reaction. They are transient species that exist only during the reaction process.

Example:

Consider the following hypothetical reaction mechanism:

Step 1: A + B -> I (slow)

Step 2: I + C -> D (fast)

In this mechanism:

• Step 1 and Step 2 are the elementary steps.
• I is an intermediate because it is formed in Step 1 and consumed in Step 2.
• If A + B + C -> D is the overall reaction, this shows that I does not appear in the overall equation.

Determining the Rate Law from the Rate-Determining Step

The rate-determining step (RDS), also known as the rate-limiting step, is the slowest step in the reaction mechanism. Because it's the slowest, it acts as a bottleneck and determines the overall rate of the reaction.

The rate law for the overall reaction is determined solely by the rate-determining step. To write the rate law:

1. Identify the Rate-Determining Step: This is usually given in the problem or can be deduced from experimental data.
2. Write the Rate Law for the RDS: The rate law for an elementary step is directly determined by its stoichiometry. For example, if the rate-determining step is A + B -> products, then the rate law for that step is Rate = k[A][B].
3. Check for Intermediates in the Rate Law: If the rate law contains any intermediates, they must be eliminated by using the equilibrium expressions from the fast, pre-equilibrium steps.

Example (Continued):

In the mechanism above, Step 1 (A + B -> I) is given as the slow, rate-determining step. Therefore, the rate law for the overall reaction is:

Rate = k[A][B]

Notice that this rate law only involves the reactants A and B and does not include the intermediate I or the reactant C, which appears only in the fast step.

Dealing with Intermediates in the Rate Law: The Pre-Equilibrium Approach

Often, the rate-determining step involves an intermediate. Because intermediates are not reactants or products in the overall reaction, their concentrations cannot be easily measured. Therefore, the rate law must be expressed in terms of the reactants only. This is where the pre-equilibrium approach comes into play.

The pre-equilibrium approach is used when a fast, reversible step precedes the rate-determining step. This fast step is assumed to reach equilibrium quickly, allowing us to relate the concentration of the intermediate to the concentrations of the reactants using the equilibrium constant (K).

Steps for Using the Pre-Equilibrium Approach:

1. Identify the Fast, Reversible Step: This step precedes the rate-determining step and is assumed to be at equilibrium.

2. Write the Equilibrium Constant Expression (K): For the reversible step aA + bB <=> cC + dD, the equilibrium constant is:

   K = [C]^c[D]^d / [A]^a[B]^b

3. Solve for the Concentration of the Intermediate: Rearrange the equilibrium expression to solve for the concentration of the intermediate that appears in the rate-determining step.

4. Substitute into the Rate Law: Substitute the expression for the intermediate's concentration into the rate law obtained from the rate-determining step. This will eliminate the intermediate from the rate law, expressing it solely in terms of reactants.

Example:

Consider the following mechanism:

Step 1: 2A <=> I (fast, equilibrium)

Step 2: I + B -> C (slow)

In this mechanism:

• I is an intermediate.
• Step 2 is the rate-determining step.

The rate law based on the slow step is:

Rate = k[I][B]

However, this rate law contains the intermediate I. To eliminate it, we use the pre-equilibrium approach:

1. Fast, Reversible Step: 2A <=> I

2. Equilibrium Constant Expression: K = [I] / [A]^2

3. Solve for [I]: [I] = K[A]^2

4. Substitute into the Rate Law: Rate = k(K[A]^2)[B] = kK[A]^2[B]

   Since k and K are both constants, we can combine them into a single rate constant k':

   Rate = k'[A]^2[B]

This is the final rate law, expressed only in terms of the reactants A and B.

Practice Problems and Examples

To solidify understanding, let's work through some practice problems:

Problem 1:

Given the following mechanism, determine the rate law:

Step 1: NO + O2 <=> NO3 (fast, equilibrium)

Step 2: NO3 + NO -> NO2 + NO2 (slow)

Solution:

1. Rate-Determining Step: Step 2: NO3 + NO -> NO2 + NO2
2. Rate Law from RDS: Rate = k[NO3][NO]
3. Identify Intermediate: NO3
4. Fast, Reversible Step: Step 1: NO + O2 <=> NO3
5. Equilibrium Constant Expression: K = [NO3] / ([NO][O2])
6. Solve for [NO3]: [NO3] = K[NO][O2]
7. Substitute into Rate Law: Rate = k(K[NO][O2])[NO] = kK[NO]^2[O2]
8. Simplify: Rate = k'[NO]^2[O2]

Problem 2:

Determine the rate law for the following mechanism:

Step 1: A + B <=> C (fast, equilibrium)

Step 2: C + D -> E (slow)

Solution:

1. Rate-Determining Step: Step 2: C + D -> E
2. Rate Law from RDS: Rate = k[C][D]
3. Identify Intermediate: C
4. Fast, Reversible Step: Step 1: A + B <=> C
5. Equilibrium Constant Expression: K = [C] / ([A][B])
6. Solve for [C]: [C] = K[A][B]
7. Substitute into Rate Law: Rate = k(K[A][B])[D] = kK[A][B][D]
8. Simplify: Rate = k'[A][B][D]

Problem 3:

Consider the following mechanism for the reaction between hydrogen and iodine to form hydrogen iodide:

Step 1: I2 <=> 2I (fast, equilibrium) Step 2: H2 + 2I -> 2HI (slow)

Determine the rate law.

Solution:

1. Rate-Determining Step: Step 2: H2 + 2I -> 2HI
2. Rate Law from RDS: Rate = k[H2][I]^2
3. Identify Intermediate: I
4. Fast, Reversible Step: Step 1: I2 <=> 2I
5. Equilibrium Constant Expression: K = [I]^2 / [I2]
6. Solve for [I]^2: [I]^2 = K[I2]
7. Substitute into Rate Law: Rate = k = kK[H2][I2]
8. Simplify: Rate = k'[H2][I2]

These examples illustrate the general procedure for deriving rate laws from reaction mechanisms using the pre-equilibrium approach.

The Steady-State Approximation

Another method for dealing with intermediates in rate laws is the steady-state approximation. This approximation is used when the pre-equilibrium approach is not applicable, often because there is no fast, reversible step preceding the rate-determining step.

The steady-state approximation assumes that the concentration of the intermediate remains relatively constant during the reaction. This means that the rate of formation of the intermediate is approximately equal to the rate of its consumption.

Steps for Using the Steady-State Approximation:

1. Identify the Intermediate: Determine the species that are formed and consumed during the reaction but do not appear in the overall balanced equation.
2. Write Expressions for the Rate of Formation and Rate of Consumption of the Intermediate: This involves examining all the elementary steps in which the intermediate participates.
3. Set the Rate of Formation Equal to the Rate of Consumption: This is the core of the steady-state approximation.
4. Solve for the Concentration of the Intermediate: This will give you an expression for the intermediate's concentration in terms of the reactants and products.
5. Substitute into the Rate Law: Substitute the expression for the intermediate's concentration into the rate law obtained from the rate-determining step. This will eliminate the intermediate from the rate law, expressing it solely in terms of reactants and products.

Example:

Consider the following mechanism:

Step 1: A -> B + C (slow)

Step 2: B + D -> E (fast)

Step 3: E -> A + F (fast)

Let's assume the rate law we want is for the formation of F.

1. Identify Intermediate: B, and E are intermediates
2. Write Expressions for the Rate of Formation and Rate of Consumption of the Intermediate:

• For B:
  • Rate of formation: k1[A]
  • Rate of consumption: k2[B][D]
• For E:
  • Rate of formation: k2[B][D]
  • Rate of consumption: k3[E]

3. Set the Rate of Formation Equal to the Rate of Consumption:

• k1[A] = k2[B][D]
• k2[B][D] = k3[E]

4. Solve for the Concentration of the Intermediates:

• [B] = (k1[A]) / (k2[D])
• [E] = (k2[B][D]) / k3 = (k2 * (k1[A] / k2[D]) * [D]) / k3 = (k1[A]) / k3

5. Determine the Rate Law for the Formation of F: The rate of formation of F is given by step 3:

Rate = k3[E]

6. Substitute the expression for [E] into the Rate Law:

Rate = k3 * (k1[A] / k3) = k1[A]

The rate law for the formation of F is Rate = k1[A]. This result indicates that the rate of formation of F depends only on the concentration of A and the rate constant of the first step.

Important Considerations When Using the Steady-State Approximation:

• The steady-state approximation is most accurate when the intermediate is highly reactive and its concentration is low.
• The approximation may not be valid if the rate of formation and consumption of the intermediate are not approximately equal.

Limitations and Complexities

While deriving rate laws from mechanisms is a powerful tool, it's important to acknowledge its limitations:

• Mechanisms are Hypotheses: A proposed mechanism is just a hypothesis that needs to be supported by experimental evidence. It is not a definitive description of the reaction.
• Complex Mechanisms: Some reactions have very complex mechanisms with many elementary steps and intermediates, making it difficult to derive a simple rate law.
• Approximations: The pre-equilibrium and steady-state approximations are based on assumptions that may not always be valid, leading to inaccuracies in the derived rate law.
• Experimental Verification: The derived rate law must always be verified by experimental data. If the derived rate law does not agree with the experimental rate law, the proposed mechanism may be incorrect.

Conclusion

Determining the rate law implied by a simple mechanism is a fundamental skill in chemical kinetics. By understanding elementary steps, identifying intermediates, and applying the rate-determining step concept along with the pre-equilibrium or steady-state approximation, one can unravel the relationship between reaction mechanisms and reaction rates. Remember that while these methods provide valuable insights, experimental verification is crucial to validate the proposed mechanism and the derived rate law. Mastering these concepts allows for a deeper understanding of chemical reactions and their underlying dynamics.