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A topic from the subject of Chemical Kinetics in Chemistry.

  • 1 year ago
  • 6 min read
Half-Life of a Reaction
Introduction

The half-life of a reaction is the time it takes for the concentration of a reactant to decrease to half its initial value. This concept is crucial in understanding reaction kinetics and predicting the rate at which a reaction proceeds. Different reaction orders exhibit different relationships between half-life and initial concentration.

First-Order Reactions

For first-order reactions, the half-life (t1/2) is independent of the initial concentration [A]0 and is given by:

t1/2 = 0.693 / k

where k is the rate constant.

Second-Order Reactions

For second-order reactions, the half-life is dependent on the initial concentration and is given by:

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

where k is the rate constant.

Zero-Order Reactions

For zero-order reactions, the half-life is also dependent on the initial concentration:

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

where k is the rate constant.

Determining Reaction Order and Half-Life

The reaction order and rate constant can be determined experimentally by measuring the concentration of the reactant at different times. Plotting the appropriate data (e.g., ln[A] vs. time for first-order, 1/[A] vs. time for second-order) will yield a straight line, allowing for the determination of k. From k, the half-life can then be calculated.

Applications of Half-Life

The concept of half-life has numerous applications, including:

  • Pharmacokinetics: Determining the elimination rate of drugs from the body.
  • Nuclear Chemistry: Predicting the decay rate of radioactive isotopes.
  • Chemical Engineering: Designing and optimizing chemical reactors.
  • Environmental Science: Modeling the degradation of pollutants.
Conclusion

Understanding the half-life of a reaction is fundamental to understanding reaction kinetics and has wide-ranging applications in various scientific and engineering disciplines. The relationship between half-life, reaction order, and rate constant provides valuable tools for predicting reaction behavior and designing chemical processes.

Half-Life of a Reaction

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half its initial value. It's a measure of the reaction rate and is often used to compare the rates of different reactions.

Key Points
  • For first-order reactions, the half-life is independent of the initial concentration of the reactants.
  • The half-life of a reaction is constant for a given reaction at a given temperature.
  • The half-life of a reaction can be used to calculate the rate constant for the reaction, and vice versa.
  • The half-life varies with reaction order. For example, it is concentration-dependent for second-order reactions.
Main Concepts

The half-life of a reaction is based on the concept of exponential decay. This means the concentration of a reactant decreases exponentially with time. The following equations describe this:

First-Order Reactions:

The integrated rate law for a first-order reaction is:

$$[A] = [A]_0 e^{-kt}$$

where:

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

The half-life for a first-order reaction is:

$$t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$$

Second-Order Reactions:

For a second-order reaction with the rate law rate = k[A]2, the integrated rate law is:

$$ \frac{1}{[A]} = kt + \frac{1}{[A]_0} $$

The half-life for a second-order reaction is:

$$t_{1/2} = \frac{1}{k[A]_0}$$

Note that the half-life of a second-order reaction depends on the initial concentration [A]0.

These equations show the relationship between half-life and the rate constant. A faster reaction (larger k) will have a shorter half-life.

Understanding half-life is crucial in various applications, including radioactive decay (nuclear chemistry) and pharmaceutical kinetics (drug metabolism).

Half-Life of a Reaction Experiment

Objective: To determine the half-life of a first-order reaction by monitoring the disappearance of a colored reactant.

Materials:

  • 2 Beakers (100mL or larger)
  • Analytical Balance
  • Stopwatch
  • Spectrophotometer (or colorimeter) with cuvettes
  • Phenolphthalein solution (prepare a dilute solution)
  • Sodium hydroxide solution (e.g., 0.1 M NaOH)
  • Dilute hydrochloric acid solution (e.g., 0.1 M HCl)
  • Graduated cylinders (or pipettes) for accurate volume measurements

Procedure:

  1. Prepare a known concentration of phenolphthalein solution. Record the concentration and the method of preparation.
  2. Using a graduated cylinder, measure and add a precise volume (e.g., 25 mL) of the sodium hydroxide solution to a beaker. Record this volume.
  3. Using a graduated cylinder, add a precise volume (e.g., 25 mL) of the prepared phenolphthalein solution to the beaker containing the sodium hydroxide. Mix gently to ensure homogeneity. This is your initial solution.
  4. Zero the spectrophotometer (or colorimeter) using a blank cuvette filled with distilled water.
  5. Immediately measure the absorbance of the solution at a suitable wavelength (e.g., 550 nm). Record this absorbance as the initial absorbance (A₀).
  6. Using a graduated cylinder, add a precise volume (e.g., 5 mL) of dilute hydrochloric acid solution. Mix gently.
  7. Record the time. Immediately measure and record the absorbance of the solution. Repeat steps 6 and 7, adding further increments of acid (e.g., another 5 mL) at regular intervals (e.g., every 30 seconds). Continue until the absorbance becomes very low.
  8. Repeat steps 2-7 using different initial volumes of either phenolphthalein or sodium hydroxide. Remember to keep the total volume consistent across runs.

Data Analysis:

1. Plot the absorbance (A) against time (t). The reaction is first-order if a plot of ln(A) vs. time is linear.

2. Determine the rate constant (k) from the slope of the ln(A) vs. time plot (slope = -k). The half-life (t1/2) of a first-order reaction is given by: t1/2 = 0.693/k

3. Alternatively, you can find the half-life graphically by determining the time it takes for the absorbance to decrease to half its initial value (A₀/2).

Report the calculated half-life with appropriate units (e.g., seconds).

Significance:

The half-life of a reaction is crucial for understanding reaction kinetics. It helps determine the reaction rate constant, reaction order, and provides insight into the reaction mechanism. In many real-world applications, including drug metabolism and radioactive decay, the half-life is a critical parameter.

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