3.3 Graphical methods for rate law analysis

Graphical methods help us figure out how fast chemical reactions happen. We use plots and equations to determine reaction rates and orders. These techniques make complex kinetics easier to understand and analyze.

By looking at how concentrations change over time, we can tell if a reaction is first-order, second-order, or zero-order. This info helps predict how long reactions take and how they behave under different conditions.

Graphical Methods for Rate Law Analysis

Graphical methods for kinetic data

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• Integrated rate law method involves integrating the differential rate law equation to produce a linear equation relating concentration and time, with different reaction orders yielding different linear equations (first-order, second-order, zero-order)
• Half-life method utilizes the time required for the reactant concentration to decrease by half, with the relationship between half-life and rate constant depending on the reaction order (first-order: constant half-life, second-order: half-life inversely proportional to initial concentration, zero-order: half-life directly proportional to initial concentration)

Linearization of rate law equations

• First-order reactions:
  • Integrated rate law: $ln[A]_t = -kt + ln[A]_0$
  • Plot $ln[A]$ vs. $t$ to obtain a straight line with slope $-k$ and y-intercept $ln[A]_0$
• Second-order reactions:
  • Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • Plot $\frac{1}{[A]}$ vs. $t$ to obtain a straight line with slope $k$ and y-intercept $\frac{1}{[A]_0}$
• Zero-order reactions:
  • Integrated rate law: $[A]_t = -kt + [A]_0$
  • Plot $[A]$ vs. $t$ to obtain a straight line with slope $-k$ and y-intercept $[A]_0$

Interpretation of rate law plots

• First-order reactions: slope of $ln[A]$ vs. $t$ plot gives $-k$, y-intercept provides $ln[A]_0$
• Second-order reactions: slope of $\frac{1}{[A]}$ vs. $t$ plot gives $k$, y-intercept provides $\frac{1}{[A]_0}$
• Zero-order reactions: slope of $[A]$ vs. $t$ plot gives $-k$, y-intercept provides $[A]_0$
• Reaction order determined by the linearity of the corresponding plot (first-order: linear $ln[A]$ vs. $t$, second-order: linear $\frac{1}{[A]}$ vs. $t$, zero-order: linear $[A]$ vs. $t$)

Half-life in reaction analysis

• First-order reactions:
  • Half-life is constant and independent of initial concentration
  • $t_{1/2} = \frac{ln2}{k}$ allows calculation of rate constant from half-life
• Second-order reactions:
  • Half-life is inversely proportional to initial concentration
  • $t_{1/2} = \frac{1}{k[A]_0}$ allows calculation of rate constant from half-life and initial concentration
• Zero-order reactions:
  • Half-life is directly proportional to initial concentration
  • $t_{1/2} = \frac{[A]_0}{2k}$ allows calculation of rate constant from half-life and initial concentration
• Determine reaction order by observing the relationship between half-life and initial concentration (first-order: constant half-life, second-order: half-life decreases with increasing initial concentration, zero-order: half-life increases with increasing initial concentration)