Rate Laws And Integrated Rate Laws

Let's delve into the heart of chemical kinetics, exploring how reaction rates are expressed and used to predict reactant concentrations over time: rate laws and integrated rate laws.

Understanding Rate Laws

A rate law is an equation that connects the reaction rate with the concentrations or partial pressures of the reactants and certain catalysts. It's an experimental determination, meaning it can't be derived simply by looking at the balanced chemical equation. The rate law must be determined through experimentation.

General Form of a Rate Law

For a general reaction:

aA + bB → cC + dD

The rate law typically takes the form:

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

Where:

• Rate is the speed at which reactants are consumed or products are formed (usually in units of M/s).
• k is the rate constant, a proportionality constant that reflects the intrinsic speed of the reaction at a given temperature. The units of k depend on the overall order of the reaction.
• [A] and [B] represent the molar concentrations of reactants A and B, respectively.
• m and n are the reaction orders with respect to reactants A and B, respectively. These exponents indicate how the concentration of each reactant affects the rate of the reaction. Crucially, m and n are not necessarily equal to the stoichiometric coefficients a and b in the balanced chemical equation.

Reaction Orders

The reaction order with respect to a specific reactant tells us how the rate changes as the concentration of that reactant changes.

• Zero Order (m or n = 0): The rate is independent of the concentration of the reactant. Changing the concentration of the reactant has no effect on the reaction rate.
• First Order (m or n = 1): The rate is directly proportional to the concentration of the reactant. Doubling the concentration doubles the rate.
• Second Order (m or n = 2): The rate is proportional to the square of the concentration of the reactant. Doubling the concentration quadruples the rate.
• Higher Orders (m or n > 2): While less common, reactions can have orders higher than 2.

The overall order of the reaction is the sum of the individual orders with respect to each reactant (m + n in the example above). This overall order helps determine the units of the rate constant, k.

Determining Rate Laws Experimentally

Since rate laws must be determined experimentally, several methods can be used:

1. Method of Initial Rates: This is a common technique. Several experiments are run where the initial concentrations of reactants are varied, and the initial rate of the reaction is measured for each set of concentrations. By comparing how the initial rate changes with changes in initial concentrations, the reaction orders can be determined.

   • For example, if doubling the concentration of A doubles the initial rate while keeping [B] constant, the reaction is first order with respect to A.
   • If doubling the concentration of B quadruples the initial rate while keeping [A] constant, the reaction is second order with respect to B.

2. Graphical Methods: By plotting concentration vs. time data in different ways, we can often determine the order of a reaction and the rate constant. This method ties in closely with integrated rate laws (explained later).

3. Isolation Method: If a reaction involves multiple reactants, the isolation method simplifies the process. All reactants except one are present in large excess. This means their concentrations remain essentially constant during the reaction. This allows us to isolate the effect of the single non-constant reactant on the rate.

Example of Determining a Rate Law

Consider the reaction:

2NO(g) + O2(g) → 2NO2(g)

We perform three experiments and obtain the following initial rate data:

Let the rate law be: Rate = k[NO]^m[O2]^n

• Comparing Experiments 1 and 2: [NO] doubles, [O2] is constant, and the rate quadruples (increases by a factor of 4). This indicates that the reaction is second order with respect to NO (m = 2).
• Comparing Experiments 1 and 3: [O2] doubles, [NO] is constant, and the rate doubles. This indicates that the reaction is first order with respect to O2 (n = 1).

Therefore, the rate law is: Rate = k[NO]^2[O2]

To find the rate constant k, we can use the data from any of the experiments. Let's use Experiment 1:

0. 020 M/s = k(0.10 M)^2(0.10 M) k = 0.020 M/s / (0.01 M^2 * 0.10 M) k = 20 M^-2 s^-1

The complete rate law is: Rate = (20 M^-2 s^-1)[NO]^2[O2]

Integrated Rate Laws: Concentration Changes Over Time

While rate laws tell us how the rate depends on concentration, integrated rate laws tell us how the concentration of reactants changes over time. They are derived from the differential rate laws using calculus. Integrated rate laws allow us to:

• Predict the concentration of reactants or products at any given time during the reaction.
• Determine the time required for a certain amount of reactant to be consumed.
• Calculate the half-life of a reaction.

Zero-Order Integrated Rate Law

• Differential Rate Law: Rate = k
• Integrated Rate Law: [A]t = -kt + [A]0

Where:

• [A]t is the concentration of reactant A at time t.
• [A]0 is the initial concentration of reactant A at time t = 0.
• k is the rate constant.

Graphical Representation: A plot of [A]t vs. time gives a straight line with a slope of -k and a y-intercept of [A]0.

Half-Life (t1/2): The half-life is the time it takes for the concentration of a reactant to decrease to half its initial value. For a zero-order reaction:

t1/2 = [A]0 / 2k

First-Order Integrated Rate Law

• Differential Rate Law: Rate = k[A]
• Integrated Rate Law: ln[A]t = -kt + ln[A]0 or [A]t = [A]0 * e^(-kt)

Where:

• ln is the natural logarithm.

Graphical Representation: A plot of ln[A]t vs. time gives a straight line with a slope of -k and a y-intercept of ln[A]0.

Half-Life (t1/2): For a first-order reaction, the half-life is constant and independent of the initial concentration:

t1/2 = 0.693 / k (where 0.693 ≈ ln(2))

This is a key characteristic of first-order reactions. The half-life remains the same no matter how much reactant you start with.

Second-Order Integrated Rate Law (with respect to one reactant)

• Differential Rate Law: Rate = k[A]^2
• Integrated Rate Law: 1/[A]t = kt + 1/[A]0

Graphical Representation: A plot of 1/[A]t vs. time gives a straight line with a slope of k and a y-intercept of 1/[A]0.

Half-Life (t1/2): The half-life for a second-order reaction depends on the initial concentration:

t1/2 = 1 / (k[A]0)

As the initial concentration decreases, the half-life increases.

Summary Table of Rate Laws

Using Integrated Rate Laws: Examples

1. First-Order Decomposition: The decomposition of N2O5(g) into NO2(g) and O2(g) is a first-order reaction. At 45 °C, the rate constant k = 6.2 x 10^-4 s^-1. If the initial concentration of N2O5 is 0.40 M, what will the concentration be after 10.0 minutes?

   • First, convert time to seconds: 10.0 min * 60 s/min = 600 s
   • Use the integrated rate law: ln[N2O5]t = -kt + ln[N2O5]0
   • ln[N2O5]t = -(6.2 x 10^-4 s^-1)(600 s) + ln(0.40)
   • ln[N2O5]t = -0.372 - 0.916 = -1.288
   • [N2O5]t = e^(-1.288) = 0.276 M

2. Second-Order Reaction: The dimerization of butadiene (C4H6) to form C8H12 is a second-order reaction. The rate constant k = 4.0 x 10^-2 M^-1s^-1 at a certain temperature. If the initial concentration of butadiene is 0.200 M, how long will it take for the concentration to decrease to 0.040 M?

   • Use the integrated rate law: 1/[C4H6]t = kt + 1/[C4H6]0
   • 1/0.040 M = (4.0 x 10^-2 M^-1s^-1)t + 1/0.200 M
   • 25 M^-1 = (4.0 x 10^-2 M^-1s^-1)t + 5 M^-1
   • 20 M^-1 = (4.0 x 10^-2 M^-1s^-1)t
   • t = 20 M^-1 / (4.0 x 10^-2 M^-1s^-1) = 500 s

Complex Reactions

The rate laws and integrated rate laws discussed above apply primarily to elementary reactions – reactions that occur in a single step. Many reactions, however, are complex and involve multiple steps. These complex reactions have more complicated rate laws.

• Rate-Determining Step: In a multi-step reaction, the slowest step is called the rate-determining step. The rate of the overall reaction is limited by the rate of this slowest step. The rate law for the overall reaction is often (but not always) determined by the rate law of the rate-determining step.

• Reaction Mechanisms: A reaction mechanism is a step-by-step sequence of elementary reactions that describes the overall reaction. Determining the reaction mechanism can be challenging but provides a detailed understanding of how the reaction occurs at a molecular level. Experimental data, including rate laws, are crucial for proposing and validating reaction mechanisms.

Factors Affecting Reaction Rates

Several factors influence reaction rates besides concentration:

• Temperature: Generally, increasing the temperature increases the reaction rate. This is because higher temperatures provide more molecules with the activation energy needed to overcome the energy barrier for the reaction. The Arrhenius equation quantifies the relationship between temperature and the rate constant.
• Catalysts: Catalysts speed up reactions without being consumed in the process. They provide an alternative reaction pathway with a lower activation energy.
• Surface Area: For reactions involving solids, increasing the surface area of the solid reactant generally increases the reaction rate. This is because more of the solid reactant is exposed to the other reactants.
• Pressure (for gases): Increasing the pressure of gaseous reactants generally increases the reaction rate, as it effectively increases the concentration of the reactants.

FAQ: Rate Laws and Integrated Rate Laws

• Q: Can I determine the rate law from the balanced chemical equation?

  • A: No! The rate law must be determined experimentally. The stoichiometry of the balanced equation does not necessarily reflect the reaction orders.

• Q: What are the units of the rate constant k?

  • A: The units of k depend on the overall order of the reaction. Here are some examples:
    • Zero order: M/s
    • First order: s^-1
    • Second order: M^-1s^-1

• Q: How do I know which integrated rate law to use?

  • A: You can determine the order of the reaction experimentally (e.g., using the method of initial rates) or by graphing concentration data. The integrated rate law that gives a straight line when the appropriate plot is made (as shown in the summary table above) corresponds to the correct order.

• Q: What is the difference between a rate law and a rate constant?

  • A: The rate law is an equation that relates the rate of a reaction to the concentrations of reactants. The rate constant k is a proportionality constant within the rate law that reflects the intrinsic speed of the reaction at a given temperature.

• Q: Why is the rate-determining step important?

  • A: The rate-determining step is the slowest step in a multi-step reaction. It limits the overall rate of the reaction, and its rate law often determines the rate law for the overall reaction.

Conclusion

Rate laws and integrated rate laws are fundamental tools in chemical kinetics. Rate laws describe the relationship between reaction rates and reactant concentrations, while integrated rate laws describe how reactant concentrations change over time. By understanding these concepts, we can predict reaction rates, determine reaction orders, calculate half-lives, and gain valuable insights into reaction mechanisms. These principles are essential for chemists, engineers, and anyone working with chemical reactions. The experimental determination of rate laws and the application of integrated rate laws are crucial for understanding and controlling chemical processes in various fields, from industrial chemistry to environmental science.