First-order reactions are a fundamental concept in chemical kinetics. They describe how reactants transform into products at a rate proportional to their concentration. Understanding these reactions is crucial for predicting chemical behavior and designing efficient processes.

The integrated rate law for first-order reactions allows us to calculate concentrations at any time. We'll explore its derivation, application, and characteristics, including the concept of half-life and how to determine rate constants from experimental data.

First-Order Reactions

Derivation of first-order integrated rate law

• Begins with the differential rate law $rate = -\frac{d[A]}{dt} = k[A]$
  • $[A]$ represents the concentration of reactant A
  • $k$ denotes the rate constant
• Rearranges the differential rate law to separate variables $-\frac{d[A]}{[A]} = kdt$
• Integrates both sides of the equation $\int_{[A]0}^{[A]}-\frac{d[A]}{[A]} = \int{0}^{t}kdt$
  • $[A]_0$ signifies the initial concentration of A
• Solves the integrals to obtain $-ln[A] + ln[A]_0 = kt$
• Rearranges to derive the integrated rate law for a first-order reaction
  • $ln\frac{[A]}{[A]_0} = -kt$
  • $ln[A] = -kt + ln[A]_0$

Application of first-order rate law

• Uses the integrated rate law $ln[A] = -kt + ln[A]_0$ to determine reactant concentration at any time $t$
• Plugs in known values for $k$, $t$, and $[A]_0$
• Solves for $[A]$ by taking the exponential of both sides $[A] = [A]_0e^{-kt}$
• Allows for the calculation of reactant concentration at any point during the reaction (given $k$ and $[A]_0$)

Characteristics in ln(concentration) vs time plots

• Exhibits a linear plot of $ln[A]$ vs time for a first-order reaction
  • The slope of the line equals $-k$
  • The y-intercept corresponds to $ln[A]_0$
• Demonstrates a constant half-life independent of initial concentration
• Enables quick identification of first-order kinetics from experimental data

Rate constant calculation from data

• Collects concentration vs time data for the reaction
• Plots $ln[A]$ vs time to check for linearity (indicating first-order kinetics)
• Determines the rate constant $k$ from the slope of the line
  • $slope = -k$
  • $k = -slope$
• Allows for the calculation of $k$ using experimental data (concentration measurements over time)

Half-life concept for first-order reactions

• Defines half-life ($t_{1/2}$) as the time required for reactant concentration to decrease by half
• Emphasizes the independence of half-life from initial concentration in first-order reactions
• Relates half-life to the rate constant using the equation $t_{1/2} = \frac{ln2}{k} = \frac{0.693}{k}$
• Calculates half-life from the rate constant by plugging $k$ into the equation $t_{1/2} = \frac{ln2}{k}$
• Calculates the rate constant from half-life by rearranging to $k = \frac{ln2}{t_{1/2}}$ and plugging in $t_{1/2}$
• Provides a useful alternative to the integrated rate law for characterizing first-order reactions