Integrated rate laws are crucial tools in chemical kinetics, helping us understand how reactant concentrations change over time. They provide mathematical equations for different reaction orders, allowing us to calculate concentrations, determine reaction rates, and predict half-lives.

By applying these laws, we can analyze experimental data to determine reaction orders and rate constants. This knowledge is essential for understanding reaction mechanisms and predicting the behavior of chemical systems in various applications.

Integrated Rate Laws

Deriving Integrated Rate Laws for Different Reaction Orders

• Derive integrated rate law for zero-order reactions
  • Equation: $[A] = -kt + [A]₀$
    • $[A]$ concentration of reactant A at time t
    • $k$ rate constant
    • $[A]₀$ initial concentration of A
• Derive integrated rate law for first-order reactions
  • Equation: $ln[A] = -kt + ln[A]₀$
    • $[A]$ concentration of reactant A at time t
    • $k$ rate constant
    • $[A]₀$ initial concentration of A
• Derive integrated rate law for second-order reactions
  • Equation: $1/[A] = kt + 1/[A]₀$
    • $[A]$ concentration of reactant A at time t
    • $k$ rate constant
    • $[A]₀$ initial concentration of A

Applying Integrated Rate Laws to Calculate Concentrations

• Calculate concentration of reactant at given time using integrated rate law for specific reaction order
  • Requires knowledge of initial concentration and rate constant
• Zero-order reaction: $[A] = -kt + [A]₀$
  • Example: If $k = 0.5 M/min$ and $[A]₀ = 2 M$, calculate $[A]$ at $t = 3 min$
• First-order reaction: $[A] = [A]₀e^(-kt)$
  • Example: If $k = 0.2 min^(-1)$ and $[A]₀ = 1.5 M$, calculate $[A]$ at $t = 5 min$
• Second-order reaction: $[A] = 1/(kt + 1/[A]₀)$
  • Example: If $k = 0.1 M^(-1)min^(-1)$ and $[A]₀ = 4 M$, calculate $[A]$ at $t = 2 min$

Determining Reaction Order

Analyzing Concentration-Time Data

• Determine reaction order by analyzing relationship between concentration and time
• Plot concentration vs. time for zero-order, ln(concentration) vs. time for first-order, or 1/concentration vs. time for second-order
  • Straight line indicates reaction follows specific order
• Slope of straight line determines rate constant (k)
• Y-intercept corresponds to initial concentration ($[A]₀$) for zero-order, $ln([A]₀)$ for first-order, or $1/[A]₀$ for second-order

Determining Rate Constant from Appropriate Plot

• Use slope of straight line obtained from appropriate plot to determine rate constant (k)
  • Zero-order: $k = -slope$
  • First-order: $k = -slope$
  • Second-order: $k = slope$
• Example: Plot concentration vs. time for a reaction and obtain a straight line with slope $-0.03 M/min$. Reaction is zero-order with $k = 0.03 M/min$
• Example: Plot ln(concentration) vs. time for a reaction and obtain a straight line with slope $-0.15 min^(-1)$. Reaction is first-order with $k = 0.15 min^(-1)$

Half-Life and Integrated Rate Laws

Relationship between Half-Life and Integrated Rate Laws

• Half-life ($t_{1/2}$) time required for concentration of reactant to decrease to half of initial value
• Zero-order reaction: $t_{1/2} = [A]₀/(2k)$
  • $[A]₀$ initial concentration
  • $k$ rate constant
  • Half-life depends on initial concentration
• First-order reaction: $t_{1/2} = ln(2)/k$
  • $k$ rate constant
  • Half-life independent of initial concentration
• Second-order reaction: $t_{1/2} = 1/(k[A]₀)$
  • $[A]₀$ initial concentration
  • $k$ rate constant
  • Half-life depends on initial concentration

Calculating Half-Life for Different Reaction Orders

• Use half-life equations for zero-order, first-order, and second-order reactions to solve problems
  • Example: For a zero-order reaction with $k = 0.02 M/min$ and $[A]₀ = 1.6 M$, calculate $t_{1/2}$
  • Example: For a first-order reaction with $k = 0.1 min^(-1)$, calculate $t_{1/2}$
  • Example: For a second-order reaction with $k = 0.5 M^(-1)min^(-1)$ and $[A]₀ = 0.8 M$, calculate $t_{1/2}$

Solving Integrated Rate Law Problems

Applying Integrated Rate Laws

• Use appropriate integrated rate law based on given information (initial concentration, rate constant, time) to:
  • Calculate concentration of reactant at specific time
  • Calculate time required to reach certain concentration
• Example: For a first-order reaction with $k = 0.2 min^(-1)$ and $[A]₀ = 2 M$, calculate $[A]$ at $t = 10 min$
• Example: For a second-order reaction with $k = 0.05 M^(-1)min^(-1)$ and $[A]₀ = 1.2 M$, calculate time required for $[A]$ to reach $0.3 M$

Using Half-Life to Solve Problems

• Use half-life equations for zero-order, first-order, and second-order reactions to solve problems related to:
  • Time required for reactant to reach half of initial concentration
  • Determining rate constant from given half-life
• Example: For a zero-order reaction with $[A]₀ = 2.4 M$ and $t_{1/2} = 30 min$, calculate rate constant $k$
• Example: For a first-order reaction with $k = 0.08 min^(-1)$, calculate time required for reactant to reach one-fourth of its initial concentration

Analyzing Concentration-Time Data to Determine Reaction Order and Rate Constant

• Plot appropriate graph (concentration vs. time, ln(concentration) vs. time, or 1/concentration vs. time) to determine reaction order
• Determine rate constant from slope and y-intercept of resulting straight line
• Example: Given concentration-time data, plot ln(concentration) vs. time and obtain a straight line with slope $-0.12 min^(-1)$ and y-intercept $-0.22$. Determine reaction order and calculate rate constant and initial concentration