11.3 Half-life and reaction order

Half-life is a key concept in chemical kinetics, measuring how long it takes for a reactant's concentration to halve. It's closely tied to reaction order and rate constants, helping us understand how fast reactions happen and predict their behavior over time.

Knowing how to calculate and interpret half-lives for different reaction orders is crucial. It allows us to determine reaction mechanisms, estimate reaction progress, and solve complex kinetics problems in real-world applications like drug metabolism and environmental processes.

Half-life and Reaction Kinetics

Definition and Relationship to Rate Constant and Reaction Order

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• Half-life (t1/2) represents the time required for the concentration of a reactant to decrease to half of its initial value in a reaction
• For first-order reactions, the half-life is inversely proportional to the rate constant (k) expressed by the equation $t_{1/2} = \frac{ln(2)}{k}$
• For zero-order reactions, the half-life is directly proportional to the initial concentration of the reactant and inversely proportional to the rate constant expressed by the equation $t_{1/2} = \frac{[A]_0}{2k}$
• The relationship between half-life and reaction order varies depending on the specific order of the reaction
  • First-order reactions have a constant half-life independent of the initial concentration of the reactant
  • Zero-order reactions have a half-life that increases as the initial concentration of the reactant increases
  • Second-order reactions have a half-life inversely proportional to the initial concentration of the reactant

Reaction Order and Half-life Relationship

• First-order reactions exhibit a constant half-life regardless of the initial reactant concentration (radioactive decay)
• Zero-order reactions demonstrate a half-life directly proportional to the initial reactant concentration (enzyme-catalyzed reactions with high substrate concentration)
• Second-order reactions have a half-life inversely proportional to the initial reactant concentration (dimerization of cyclopentadiene)
• The relationship between half-life and reaction order helps determine the reaction mechanism and predict the behavior of the reaction over time

Calculating Half-life

First-order Reactions

• For first-order reactions, calculate the half-life using the equation $t_{1/2} = \frac{ln(2)}{k}$, where k is the rate constant
• Example: If the rate constant for a first-order reaction is 0.05 s^-1^, the half-life would be $t_{1/2} = \frac{ln(2)}{0.05 s^{-1}} = 13.9 s$
• The half-life of a first-order reaction is constant and independent of the initial reactant concentration

Zero-order Reactions

• For zero-order reactions, calculate the half-life using the equation $t_{1/2} = \frac{[A]_0}{2k}$, where [A]0 is the initial concentration of the reactant and k is the rate constant
• Example: If the rate constant for a zero-order reaction is 0.02 M s^-1^ and the initial concentration of the reactant is 1.0 M, the half-life would be $t_{1/2} = \frac{1.0 M}{2 \times 0.02 M s^{-1}} = 25 s$
• The half-life of a zero-order reaction increases linearly with increasing initial reactant concentration

Second-order Reactions

• For second-order reactions with a single reactant, calculate the half-life using the equation $t_{1/2} = \frac{1}{k[A]_0}$, where [A]0 is the initial concentration of the reactant and k is the rate constant
• For second-order reactions with two reactants, calculate the half-life using the equation $t_{1/2} = \frac{1}{k[A]_0}$, assuming the initial concentrations of both reactants are equal
• Example: If the rate constant for a second-order reaction is 0.1 M^-1^ s^-1^ and the initial concentration of the reactant is 0.5 M, the half-life would be $t_{1/2} = \frac{1}{0.1 M^{-1} s^{-1} \times 0.5 M} = 20 s$
• The half-life of a second-order reaction is inversely proportional to the initial reactant concentration

Determining Reaction Order

Using Half-life Data

• If the half-life is constant and independent of the initial concentration, the reaction is first-order
• If the half-life increases linearly with increasing initial concentration, the reaction is zero-order
• If the half-life is inversely proportional to the initial concentration, the reaction is second-order
• Analyzing the relationship between half-life and initial concentration helps determine the reaction order

Interpreting Concentration-Time Graphs

• In a concentration-time graph, a first-order reaction will show a straight line when ln[A] is plotted against time
• A zero-order reaction will show a straight line when [A] is plotted against time
• A second-order reaction will show a straight line when 1/[A] is plotted against time
• The linearity of the appropriate plot indicates the reaction order

Concentration Effects on Half-life

First-order Reactions

• For first-order reactions, changing the initial concentration of the reactant does not affect the half-life as the half-life is independent of the initial concentration
• Example: The half-life of a first-order reaction remains constant at 10 minutes, regardless of whether the initial concentration is 1.0 M or 0.5 M
• The rate of a first-order reaction depends only on the concentration of a single reactant

Zero-order Reactions

• For zero-order reactions, increasing the initial concentration of the reactant will increase the half-life as the half-life is directly proportional to the initial concentration
• Doubling the initial concentration of a reactant in a zero-order reaction will double the half-life
• Example: If the half-life of a zero-order reaction is 20 minutes at an initial concentration of 1.0 M, increasing the initial concentration to 2.0 M will result in a half-life of 40 minutes
• The rate of a zero-order reaction is constant and independent of the reactant concentration

Second-order Reactions

• For second-order reactions, increasing the initial concentration of the reactant will decrease the half-life as the half-life is inversely proportional to the initial concentration
• Doubling the initial concentration of a reactant in a second-order reaction will halve the half-life
• Example: If the half-life of a second-order reaction is 30 minutes at an initial concentration of 0.1 M, increasing the initial concentration to 0.2 M will result in a half-life of 15 minutes
• The rate of a second-order reaction depends on the square of the reactant concentration or the product of the concentrations of two reactants

Problem Solving with Half-life

Calculating Half-life

• Use the appropriate equations for half-life based on the reaction order to calculate the half-life when given the rate constant or initial concentration
• Example: Calculate the half-life of a first-order reaction with a rate constant of 0.02 s^-1^
  • Using the equation $t_{1/2} = \frac{ln(2)}{k}$, $t_{1/2} = \frac{ln(2)}{0.02 s^{-1}} = 34.7 s$

Determining Rate Constants

• Determine the rate constant of a reaction by rearranging the half-life equations and substituting the known values of half-life and initial concentration
• Example: Calculate the rate constant of a zero-order reaction with a half-life of 15 minutes and an initial concentration of 0.5 M
  • Rearranging the equation $t_{1/2} = \frac{[A]_0}{2k}$ to solve for k, $k = \frac{[A]_0}{2t_{1/2}} = \frac{0.5 M}{2 \times 15 min} = 0.0167 M min^{-1}$

Time to Reach a Specific Concentration

• Calculate the time required for a reactant to reach a specific concentration using the half-life and the number of half-lives elapsed
• Example: Determine the time required for a reactant in a first-order reaction with a half-life of 20 minutes to reach 25% of its initial concentration
  • The reactant must undergo two half-lives to reach 25% of its initial concentration (100% → 50% → 25%)
  • Time required = Number of half-lives × Half-life = 2 × 20 minutes = 40 minutes

Determining Reaction Order

• Determine the reaction order by analyzing the relationship between the half-life and initial concentration or by interpreting concentration-time graphs
• Example: A reaction has half-lives of 10 minutes, 20 minutes, and 40 minutes at initial concentrations of 1.0 M, 0.5 M, and 0.25 M, respectively. Determine the reaction order.
  • The half-life doubles as the initial concentration is halved, indicating a zero-order reaction

Complex Problem Solving

• Combine the concepts of half-life, rate constants, and reaction order to solve complex problems involving the kinetics of various reaction types
• Example: A second-order reaction has a rate constant of 0.05 M^-1^ s^-1^. If the initial concentration of the reactant is 0.8 M, calculate the concentration of the reactant after 30 seconds.
  • Step 1: Calculate the half-life using the equation $t_{1/2} = \frac{1}{k[A]_0} = \frac{1}{0.05 M^{-1} s^{-1} \times 0.8 M} = 25 s$
  • Step 2: Determine the number of half-lives elapsed in 30 seconds: $\frac{30 s}{25 s} = 1.2$ half-lives
  • Step 3: Calculate the remaining concentration after 1.2 half-lives: $[A] = [A]_0 \times (\frac{1}{2})^{1.2} = 0.8 M \times (\frac{1}{2})^{1.2} = 0.435 M$