Integrated Rate Equation For First Order Reaction

The integrated rate equation for a first-order reaction is a cornerstone of chemical kinetics, providing a mathematical relationship that describes how the concentration of a reactant changes over time. It allows us to predict the amount of reactant remaining after a certain period, determine the rate constant of the reaction, and understand the factors that influence the speed of chemical transformations.

Understanding Reaction Orders and Rate Laws

Before diving into the integrated rate equation for first-order reactions, it's crucial to understand the concept of reaction order and its relationship to the rate law.

• Reaction Order: The reaction order defines how the rate of a reaction is affected by the concentration of the reactants. It is experimentally determined and cannot be predicted solely from the balanced chemical equation. The most common reaction orders are zero-order, first-order, and second-order.

• Rate Law: The rate law expresses the relationship between the rate of a reaction and the concentrations of the reactants. For a general reaction:

  aA + bB -> cC + dD

  The rate law takes the form:

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

  Where:

  • Rate is the reaction rate, usually expressed in units of M/s (molarity per second).
  • 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, respectively.
  • m and n are the reaction orders with respect to reactants A and B, respectively. The overall reaction order is the sum of m and n.

Defining First-Order Reactions

A first-order reaction is a chemical reaction where the rate of the reaction is directly proportional to the concentration of only one reactant. Mathematically, this means:

Rate = k[A]

Where:

• Rate is the reaction rate.
• k is the rate constant.
• [A] is the concentration of reactant A.

The defining characteristic of a first-order reaction is that doubling the concentration of reactant A will double the rate of the reaction. The units of the rate constant, k, for a first-order reaction are s^-1 (per second).

Examples of First-Order Reactions

First-order reactions are prevalent in various chemical processes, including:

• Radioactive Decay: The decay of radioactive isotopes follows first-order kinetics. The rate of decay is proportional to the amount of the radioactive substance present. For example, the decay of uranium-238 to thorium-234.
• Unimolecular Decomposition: Some molecules decompose into smaller fragments in a unimolecular process, where the rate depends only on the concentration of the decomposing molecule. An example is the decomposition of dinitrogen pentoxide (N2O5) into nitrogen dioxide (NO2) and oxygen (O2).
• Isomerization Reactions: The conversion of one isomer of a molecule to another isomer can sometimes follow first-order kinetics.
• Enzyme-Catalyzed Reactions (under certain conditions): When the substrate concentration is much lower than the Michaelis constant (Km), enzyme-catalyzed reactions can approximate first-order kinetics.

Derivation of the Integrated Rate Equation for a First-Order Reaction

The integrated rate equation for a first-order reaction is derived from the differential rate law using calculus. Let's consider a simple first-order reaction:

A -> Products

The differential rate law is:

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

Where:

• -d[A]/dt represents the rate of disappearance of reactant A with respect to time. The negative sign indicates that the concentration of A is decreasing.

To derive the integrated rate equation, we need to separate the variables and integrate both sides of the equation:

1. Separate Variables:

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

2. Integrate both sides:

   ∫(d[A]/[A]) = ∫(-k dt)

   The integration is performed from an initial time t = 0 to a time t, and from an initial concentration [A]₀ to a concentration [A] at time t.

   ln[A] |from [A]₀ to [A] = -kt |from 0 to t

3. Evaluate the integrals:

   ln[A] - ln[A]₀ = -kt

4. Rearrange the equation:

   ln([A]/[A]₀) = -kt

This is one form of the integrated rate equation for a first-order reaction. It can also be expressed in exponential form:

[A] = [A]₀ * e^(-kt)

Where:

• [A] 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.
• t is the time elapsed.
• e is the base of the natural logarithm (approximately 2.71828).

Forms of the Integrated Rate Equation

The integrated rate equation for a first-order reaction can be expressed in several equivalent forms:

1. Natural Log Form:

   ln([A]/[A]₀) = -kt

   This form is useful for determining the rate constant, k, from experimental data by plotting ln([A]/[A]₀) versus time. The slope of the resulting straight line will be equal to -k.

2. Exponential Form:

   [A] = [A]₀ * e^(-kt)

   This form is useful for calculating the concentration of reactant A at any given time, t, if the initial concentration, [A]₀, and the rate constant, k, are known.

3. Logarithmic Form (Base 10):

   log([A]/[A]₀) = -kt / 2.303

   This form uses the base-10 logarithm instead of the natural logarithm. It is less commonly used than the natural log form.

Half-Life of a First-Order Reaction

The half-life (t₁/₂) of a reaction is the time it takes for the concentration of a reactant to decrease to one-half of its initial value. For a first-order reaction, the half-life is constant and independent of the initial concentration.

To derive the half-life equation for a first-order reaction, we set [A] = [A]₀/2 and t = t₁/₂ in the integrated rate equation:

ln(([A]₀/2) / [A]₀) = -k t₁/₂

ln(1/2) = -k t₁/₂

ln(0.5) = -k t₁/₂

-0.693 = -k t₁/₂

Therefore, the half-life of a first-order reaction is:

t₁/₂ = 0.693 / k

This equation highlights a crucial characteristic of first-order reactions: the half-life depends only on the rate constant, k, and is independent of the initial concentration of the reactant. This makes first-order reactions particularly useful in fields like nuclear chemistry, where the rate of radioactive decay (a first-order process) is used for dating materials.

Determining the Rate Constant (k)

The rate constant, k, is a crucial parameter that reflects the intrinsic speed of a reaction. For a first-order reaction, the rate constant can be determined using several methods:

1. Using the Integrated Rate Equation and Experimental Data:

   • Measure the concentration of the reactant [A] at different times, t.
   • Plot ln([A]/[A]₀) versus time.
   • The slope of the resulting straight line will be equal to -k. Therefore, k = -slope.

2. Using the Half-Life:

   • Determine the half-life (t₁/₂) experimentally. This can be done by measuring the time it takes for the concentration of the reactant to decrease to half of its initial value.
   • Calculate the rate constant using the formula: k = 0.693 / t₁/₂

3. Using Initial Rates (for more complex reactions that approximate first-order behavior):

   • Measure the initial rate of the reaction at different initial concentrations of the reactant.
   • If the reaction is truly first-order, a plot of the initial rate versus the initial concentration will be a straight line passing through the origin. The slope of the line will be equal to the rate constant, k. This method is less precise for true first-order reactions, as it's usually applied in situations where other factors might influence the rate, and first-order kinetics is an approximation.

Applications of the Integrated Rate Equation

The integrated rate equation for first-order reactions has numerous applications in various fields:

• Chemical Kinetics: Predicting the concentration of reactants and products at any given time. Determining the rate constant of a reaction. Understanding the factors that influence the rate of a reaction.
• Radioactive Dating: Determining the age of ancient artifacts and geological formations by measuring the amount of radioactive isotopes remaining. Carbon-14 dating, for example, relies on the first-order decay of carbon-14.
• Pharmacokinetics: Studying the absorption, distribution, metabolism, and excretion of drugs in the body. The elimination of many drugs from the body follows first-order kinetics.
• Environmental Science: Modeling the degradation of pollutants in the environment. The breakdown of some pollutants can be approximated as a first-order process.
• Enzyme Kinetics: Analyzing enzyme-catalyzed reactions, particularly at low substrate concentrations where the reaction approaches first-order kinetics.

Limitations of the Integrated Rate Equation

While the integrated rate equation for first-order reactions is a powerful tool, it's important to be aware of its limitations:

• Only Applies to First-Order Reactions: The equation is specifically derived for reactions that follow first-order kinetics. It cannot be directly applied to reactions of other orders.
• Assumes Constant Temperature: The rate constant, k, is temperature-dependent. The integrated rate equation assumes that the temperature remains constant throughout the reaction. If the temperature changes significantly, the rate constant will also change, and the equation will no longer be accurate.
• Ideal Conditions: The derivation of the integrated rate equation assumes ideal conditions, such as a homogeneous reaction mixture and no side reactions. In reality, these conditions may not always be met, which can lead to deviations from the predicted behavior.
• Reversible Reactions: The simple integrated rate equation does not account for reversible reactions, where the products can react to reform the reactants. For reversible reactions, a more complex rate equation is required.
• Complex Mechanisms: Many reactions involve multiple steps, and the overall reaction order may not be easily determined. Even if a reaction appears to be first-order under certain conditions, the underlying mechanism might be more complex.

Examples and Practice Problems

To solidify your understanding of the integrated rate equation for first-order reactions, let's work through a few examples:

Example 1:

The decomposition of dinitrogen pentoxide (N₂O₅) in the gas phase follows first-order kinetics:

N₂O₅(g) -> 2NO₂(g) + 1/2 O₂(g)

The rate constant for this reaction at 338 K is 4.82 x 10⁻³ s⁻¹. If the initial concentration of N₂O₅ is 0.0250 M, what will be the concentration of N₂O₅ after 10 minutes?

Solution:

1. Convert time to seconds:

   t = 10 minutes * 60 seconds/minute = 600 seconds

2. Use the integrated rate equation:

   [N₂O₅] = [N₂O₅]₀ * e^(-kt)

   [N₂O₅] = (0.0250 M) * e^(-(4.82 x 10⁻³ s⁻¹) * (600 s))

   [N₂O₅] = (0.0250 M) * e^(-2.892)

   [N₂O₅] = (0.0250 M) * 0.0554

   [N₂O₅] = 0.00139 M

Therefore, the concentration of N₂O₅ after 10 minutes will be 0.00139 M.

Example 2:

A radioactive isotope has a half-life of 15 hours. How long will it take for 80% of the isotope to decay?

Solution:

1. Calculate the rate constant:

   k = 0.693 / t₁/₂

   k = 0.693 / 15 hours

   k = 0.0462 hours⁻¹

2. Determine the fraction of isotope remaining:

   If 80% has decayed, then 20% remains. [A]/[A]₀ = 0.20

3. Use the integrated rate equation:

   ln([A]/[A]₀) = -kt

   ln(0.20) = -(0.0462 hours⁻¹) * t

   -1.609 = -(0.0462 hours⁻¹) * t

   t = -1.609 / -0.0462 hours⁻¹

   t = 34.8 hours

Therefore, it will take 34.8 hours for 80% of the isotope to decay.

Common Mistakes to Avoid

When working with the integrated rate equation for first-order reactions, be sure to avoid these common mistakes:

• Using the wrong rate equation: Make sure you are using the correct integrated rate equation for the specific reaction order. Using the first-order equation for a second-order reaction will lead to incorrect results.
• Incorrect units: Pay close attention to the units of the rate constant and time. Ensure that they are consistent. For example, if the rate constant is in s⁻¹, the time must be in seconds.
• Forgetting the negative sign: The integrated rate equation often involves a negative sign, particularly in the natural log form. Forgetting this sign will lead to incorrect calculations.
• Assuming first-order kinetics without verification: Do not assume that a reaction is first-order without experimental evidence. Determine the reaction order experimentally before applying the integrated rate equation.
• Ignoring temperature effects: Remember that the rate constant is temperature-dependent. If the temperature changes significantly during the reaction, the integrated rate equation may not be accurate.

Conclusion

The integrated rate equation for first-order reactions is a powerful tool for understanding and predicting the behavior of chemical reactions. By understanding its derivation, applications, and limitations, you can confidently apply it to solve a wide range of problems in chemistry, and related fields. Mastering this equation is essential for anyone studying chemical kinetics, reaction mechanisms, and the factors that influence the speed of chemical transformations. Remember to practice with examples, pay attention to units, and always verify that the reaction truly follows first-order kinetics before applying the equation.