Chemistry / Chem 1660 · Procedure · 60–90 seconds
Solving the First-Order Integrated Rate Law for Time or Concentration
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Solving the first-order integrated rate law for an unknown time or concentration means substituting the known rate constant and the two related concentration values into ln[A]t = ln[A]0 − kt, then rearranging to isolate whichever quantity is unknown, taking care that both concentrations are expressed in the same consistent unit before the natural logarithm is taken.
Cyclobutane decomposes at 500 °C with k = 9.2 × 10⁻³ s⁻¹. Treating the initial concentration as x, 80.0% decomposition leaves 0.200x remaining; substituting both expressions into the integrated law, taking the logarithm of the ratio, and solving for t gives about 175 seconds for that much decomposition to occur.
Substituting the percent decomposed, 0.800, for [A]t instead of the fraction remaining, 0.200, silently inverts the ratio inside the logarithm and gives a wrong time.
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