Chemical Kinetics - Half-life of first order reactions

• Definition of half-life
• First order reactions
• Rate law for first order reactions
• Integrated rate equation for first order reactions
• Deriving the half-life equation for first order reactions

Definition of half-life

• Half-life (t1/2) is the time it takes for the concentration of a reactant to decrease to half of its initial value
• It is a characteristic property of first order reactions

First order reactions

• First order reactions occur when the rate of reaction is directly proportional to the concentration of a single reactant
• The rate equation for a first order reaction is given by: Rate = k[A]

Rate law for first order reactions

• The rate law for first order reactions is given by: Rate = k[A]
• [A] represents the concentration of the reactant A at any given time
• k is the rate constant, which is specific to the reaction and temperature

Integrated rate equation for first order reactions

• The integrated rate equation for a first order reaction is: ln([A]t/[A]0) = -kt
• [A]t is the concentration of reactant A at time t
• [A]0 is the initial concentration of reactant A

Deriving the half-life equation for first order reactions

• At half-life (t1/2), [A]t is equal to [A]0/2
• By substituting these values into the integrated rate equation and solving for t1/2, we get:
  • ln(2) = -kt1/2
  • t1/2 = ln(2)/k
• The half-life of a first order reaction depends only on the rate constant and is independent of the initial concentration of the reactant.

11. Factors affecting the rate of reactions

• Concentration: An increase in the concentration of reactants generally leads to an increase in the rate of reaction
• Temperature: Higher temperatures usually result in faster reaction rates due to increased molecular collisions
• Catalysts: Catalysts can increase the rate of reaction by providing an alternative pathway with lower activation energy
• Surface area: Reactions involving solids tend to proceed faster when the surface area of the solid is increased
• Nature of reactants: The chemical nature and structure of the reactants can affect the reaction rate

12. Collision theory

• The collision theory states that for a reaction to occur, particles must collide with sufficient energy and the correct orientation
• Activation energy: The minimum amount of energy required for a successful collision to occur
• Effective collisions: Collisions that result in a reaction due to sufficient energy and proper orientation
• Frequency factor: A measure of how often the reactant molecules collide with each other

13. Arrhenius equation

• The Arrhenius equation relates the rate constant (k) of a reaction to the temperature (T) and activation energy (Ea)
• The equation is given by: k = Ae^(-Ea/RT)
• A is the pre-exponential factor, R is the ideal gas constant, and T is the absolute temperature

14. Activation energy and reaction rates

• Increasing the temperature increases the average kinetic energy of particles, leading to a greater number of particles possessing the minimum required energy for a reaction
• This results in an increased reaction rate
• Activation energy determines the rate at which the reaction occurs once the particles possess the necessary energy for a reaction to occur

15. Effect of concentration on reaction rate

• For reactions involving multiple reactants, the rate is often determined by the slowest step, known as the rate-determining step
• Increasing the concentration of one reactant will increase the rate of reaction if it is present in the rate-determining step
• The rate equation reflects the effect of concentration on the rate of reaction

16. Rate constant and rate equation

• The rate constant (k) is the proportionality constant that relates the concentration of reactants to the rate of reaction
• The rate equation is an expression that relates the rate of reaction to the concentrations of the reactants
• The rate equation is determined experimentally

17. Determining the order of a reaction

• The order of a reaction is determined by the sum of the exponents in the rate equation
• The order can be zero, first, second, or even fractional
• The order can only be determined experimentally, not based on the stoichiometry of the balanced equation

18. Half-life in radioactive decay

• Radioactive decay is a first order reaction
• The half-life of a radioactive substance is the time it takes for half of the nuclei to decay
• The half-life can be determined using the equation t1/2 = 0.693/k, where k is the rate constant for the radioactive decay

19. Half-life equation for second order reactions

• Second order reactions have a rate equation of the form: Rate = k[A]^2 or Rate = k[A][B]
• The integrated rate equation for a second order reaction is given by: 1/[A]t - 1/[A]0 = kt
• The half-life equation for a second order reaction is: t1/2 = 1/(k[A]0)

20. Summary

• Half-life is a useful concept in understanding the kinetics of reactions
• It is a characteristic property of first order reactions, allowing us to determine the time it takes for the concentration to decrease to half
• The half-life equation for first order reactions is t1/2 = ln(2)/k, where k is the rate constant
• Factors such as concentration, temperature, catalysts, surface area, and nature of reactants can affect reaction rates
• The Arrhenius equation relates the rate constant to temperature and activation energy

Slide 21

• Reaction mechanism
  • Step-by-step sequence of elementary reactions that comprise an overall reaction
  • Reaction intermediates: Transient species formed and consumed during the reaction
• Rate-determining step
  • Slowest step in the reaction mechanism
  • Determines the overall rate of the reaction

Slide 22

• Activation energy and reaction rate
  • The energy barrier that must be overcome for a reaction to occur
  • Higher activation energy leads to slower reaction rates
• Effect of catalysts on activation energy
  • Catalysts provide an alternative pathway with lower activation energy
  • Decrease in activation energy increases the reaction rate

Slide 23

• Order of reaction from initial rates
  • Using the initial rate method, the order of a reaction can be determined
  • Varying the initial concentrations of reactants and measuring the corresponding reaction rates
  • Deriving the rate equation and determining the order of reaction

Slide 24

• Zero order reactions
  • Rate is independent of the concentration of reactants
  • Rate equation: Rate = k
  • Integrated rate equation: [A]t = [A]0 - kt
  • Examples of zero order reactions: decomposition of ozone, photolysis of nitrogen pentoxide

Slide 25

• Second order reactions
  • Rate is proportional to the square of the concentration of a single reactant or the product of the concentrations of two reactants
  • Rate equation: Rate = k[A]^2 or Rate = k[A][B]
  • Integrated rate equation: 1/[A]t - 1/[A]0 = kt or 1/[A]t - 1/[A]0 = kt / [B]0
  • Example of a second order reaction: formation of nitrogen dioxide from dinitrogen tetroxide

Slide 26

• Pseudo-first order reactions
  • A reaction that appears to be first order because one reactant is present in large excess
  • The concentration of the excess reactant remains essentially constant
  • Pseudo-first order rate equation and integrated rate equation

Slide 27

• Determining the rate constant
  • Experimental methods for determining the rate constant
  • Initial rate method
  • Method of initial rates with varying initial concentrations
  • Half-life method for first order reactions

Slide 28

• Factors affecting reaction rate
  • Temperature: Higher temperature increases the rate of reaction due to increased molecular collisions
  • Concentration: Increased concentration of reactants leads to a higher reaction rate
  • Catalysts: Catalysts increase the reaction rate by providing an alternative pathway with lower activation energy

Slide 29

• Collision frequency and collision theory
  • Collision frequency: The frequency of effective collisions between reacting particles
  • Collision theory: Particles must collide with sufficient energy and correct orientation for a reaction to occur
  • Activation energy, effective collisions, and reaction rate

Slide 30

• Summary
  • Half-life is a characteristic property of first-order reactions
  • The half-life equation for first-order reactions is t1/2 = ln(2)/k
  • Reaction mechanisms involve stepwise elementary reactions
  • Catalysts decrease activation energy and increase reaction rate
  • Determining the order of a reaction through initial rates or integrated rate equations
  • Factors such as temperature, concentration, and catalysts affect reaction rates
  • Collision theory explains the factors necessary for a reaction to occur