Integrated Rate Laws Summary

Integrated Rate Laws

Overview

• Integrated rate laws relate reactant concentration to time.

• They can be used to determine reaction orders and calculate reactant concentrations at any time.

• Focus on zero, first, and second-order reactions.

Half-Life

• The half-life ($t_{1/2}$) is the time it takes for the reactant concentration to fall to half its initial value.

• Half-life depends on the reaction order.

Zero Order Reactions

• Rate law: Rate = $k$

• Integrated rate law: $[A] = -kt + [A]_0$

• Graph of $[A]$ vs time is linear with slope $-k$

• Half-life: $t<em>{1/2} = [A]</em>0 / 2k$ (directly proportional to $[A]_0$).

First Order Reactions

• Rate law: Rate = $k[A]$

• Integrated rate law: $ln[A] = -kt + ln[A]_0$

• Graph of $ln[A]$ vs time is linear with slope $-k$

• Half-life: $t<em>{1/2} = 0.693 / k$ (independent of $[A]</em>0$).

Second Order Reactions

• Rate law: Rate = $k[A]^2$

• Integrated rate law: $1/[A] = kt + 1/[A]_0$

• Graph of $1/[A]$ vs time is linear with slope $+k$

• Half-life: $t<em>{1/2} = 1 / (k[A]</em>0)$(inversely proportional to $[A]_0$).

Graphical Determination of Reaction Order

• Plotting $[A]$, $ln[A]$, and $1/[A]$ versus time can determine reaction order.

  • Linear $[A]$ vs $t$: zero order; Rate = $k$

  • Linear $ln[A]$ vs $t$: 1st order; Rate = $k[A]$

  • Linear $1/[A]$ vs $t$: 2nd order; Rate = $k[A]^2$

Summary Table