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Arrhenius: Activation Energy from k(T)

First-order decay of an unknown reaction. Measure [A] at several temperatures, recover k = −ln([A]/[A]₀)/t, then fit ln k versus 1/T to get E_a.

University / research· 26 min·Related simulator: ChemistryReaction Rate

Goal

Determine E_a from the Arrhenius law k = A e^(−E_a/RT). The slope of ln k versus 1/T is −E_a/R.

Equipment

  • Thermostatted cell
  • Spectrophotometer ([A])
  • Stopwatch (t)
  • Known [A]₀

Experiment

Theory

For a first-order reaction [A] = [A]₀ e^(−kt), so k = −ln([A]/[A]₀)/t. Arrhenius then says ln k = ln A − (E_a/R)(1/T). The bench hides E_a, the prefactor, live k and t₁/₂; you only read a spectrophotometer.

Procedure

  1. [A]₀ is known. The reaction is first-order; E_a is hidden. You set the bath temperature T and the sampling time t.
  2. Record the spectrophotometer [A]. Small readout noise is added. There is no live k or half-life.
  3. The notebook computes k = −ln([A]/[A]₀)/t, ln k and 1/T. Pick t so [A] is not tiny and not almost [A]₀.
  4. Repeat for at least 6 temperatures from about 280 K to 325 K.
  5. Fit ln k versus 1/T; E_a (kJ/mol) = −slope · R / 1000. Compare with the reference.

Conclusion

The fitted activation energy agrees with the hidden E_a. Main uncertainties: spectrophotometer noise, using a single-time first-order k, and the Arrhenius assumption over a modest T range.