First Order Reaction Example Problems

Diving Deep into First-Order Reaction Example Problems: A Comprehensive Guide

Understanding first-order reactions is crucial in various fields, from chemical kinetics to pharmaceutical sciences and environmental engineering. This comprehensive guide will delve into the intricacies of first-order reactions, providing a solid foundation through detailed explanations and diverse example problems. We'll explore the underlying principles, solve various problems step-by-step, and address frequently asked questions to solidify your understanding. By the end, you'll be confidently tackling first-order reaction calculations and applying this knowledge to real-world scenarios.

What is a First-Order Reaction?

A first-order reaction is a chemical reaction where the rate of the reaction is directly proportional to the concentration of one reactant. This means that if you double the concentration of that reactant, the rate of the reaction will also double. The rate law for a first-order reaction is expressed as:

Rate = k[A]

Where:

• Rate represents the speed at which the reaction proceeds.
• k is the rate constant, a proportionality constant that depends on temperature and the specific reaction. It's a crucial parameter in understanding reaction speed.
• [A] denotes the concentration of reactant A.

The integrated rate law, which is more useful for calculations involving time and concentration changes, is:

ln([A]t) = -kt + ln([A]0)

Where:

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

Understanding the Rate Constant (k)

The rate constant, k, is a temperature-dependent parameter that provides valuable insights into reaction speed. A larger k value implies a faster reaction, while a smaller k value indicates a slower reaction. The units of k for a first-order reaction are typically reciprocal time (e.g., s⁻¹, min⁻¹, hr⁻¹). Determining k is often the central goal in analyzing first-order reactions.

Example Problem 1: Determining the Rate Constant

Problem: The decomposition of a certain drug follows first-order kinetics. If the initial concentration of the drug is 1.0 M and after 60 minutes, the concentration drops to 0.5 M, calculate the rate constant (k).

Solution:

We'll use the integrated rate law: ln([A]t) = -kt + ln([A]0)

1. Identify known values:

   • [A]0 = 1.0 M
   • [A]t = 0.5 M
   • t = 60 min

2. Substitute the values into the integrated rate law:

   ln(0.5) = -k(60 min) + ln(1.0)

3. Solve for k:

   ln(0.5) - ln(1.0) = -60k -0.693 = -60k k = 0.0115 min⁻¹

Therefore, the rate constant for this first-order reaction is 0.0115 min⁻¹.

Example Problem 2: Determining the Remaining Concentration

Problem: A radioactive isotope decays via first-order kinetics with a rate constant of 0.023 hr⁻¹. If the initial concentration is 2.0 M, what is the concentration remaining after 24 hours?

Solution:

1. Identify known values:

   • k = 0.023 hr⁻¹
   • [A]0 = 2.0 M
   • t = 24 hr

2. Use the integrated rate law: ln([A]t) = -kt + ln([A]0)

3. Substitute the values:

   ln([A]t) = -(0.023 hr⁻¹)(24 hr) + ln(2.0) ln([A]t) = -0.552 + 0.693 ln([A]t) = 0.141

4. Solve for [A]t:

   [A]t = e⁰·¹⁴¹ [A]t ≈ 1.15 M

After 24 hours, approximately 1.15 M of the radioactive isotope remains.

Example Problem 3: Determining the Half-Life

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

t₁/₂ = 0.693/k

Problem: A certain chemical reaction has a rate constant of 0.05 s⁻¹. What is its half-life?

Solution:

Simply substitute the rate constant (k) into the half-life equation:

t₁/₂ = 0.693 / 0.05 s⁻¹ t₁/₂ = 13.86 s

The half-life of this first-order reaction is 13.86 seconds.

Example Problem 4: A More Complex Scenario - Sequential First-Order Reactions

Many real-world reactions involve a sequence of first-order steps. Let's consider a simple case:

Problem: Substance A decomposes into substance B, which further decomposes into substance C. Both steps are first-order. The rate constant for A → B is k₁ = 0.1 min⁻¹, and for B → C is k₂ = 0.05 min⁻¹. If we start with 1 M of A, what are the concentrations of A, B, and C after 10 minutes? This requires solving a system of differential equations, which is beyond the scope of simple substitution into the integrated rate law. However, numerical methods or specialized software can be employed for accurate solutions.

Explaining the Scientific Basis

The first-order rate law arises from the statistical probability of reactant molecules colliding with sufficient energy to overcome the activation energy barrier. In a first-order reaction, the probability of a successful collision depends only on the concentration of the single reactant involved. This leads to the direct proportionality between the rate and the concentration. The integrated rate law is derived from calculus, integrating the rate law with respect to time. The half-life equation is a direct consequence of the integrated rate law when [A]t is set to half of [A]0.

Frequently Asked Questions (FAQ)

• Q: How can I determine if a reaction is first-order experimentally?

  • A: Plot ln([A]t) versus time. If the result is a straight line, the reaction is first-order, and the slope of the line is -k.

• Q: Are all reactions first-order?

  • A: No, reactions can be zero-order, first-order, second-order, or even higher orders, depending on the reaction mechanism and rate-limiting step.

• Q: What happens if the temperature changes?

  • A: The rate constant, k, is temperature-dependent. The Arrhenius equation describes this relationship: k = A * exp(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. Increasing temperature generally increases k and speeds up the reaction.

Conclusion

Understanding first-order reactions is fundamental in chemistry and many related disciplines. By grasping the rate law, the integrated rate law, and the concept of the half-life, you can effectively analyze and predict the behavior of various first-order processes. The example problems provided illustrate the practical application of these principles in diverse scenarios. Remember that while simple substitution works for many cases, more complex situations might necessitate numerical methods or specialized software for accurate calculations. Continued practice and exploration of different problem types will solidify your understanding and build confidence in tackling even the most challenging first-order reaction problems.