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Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to temperature and activation energy: k = Ae−Ea/RT, where A is the frequency factor, Ea is activation energy, R is the gas constant, and T is absolute temperature.

In one line: Arrhenius line slopes downward because higher 1/T means lower temperature and smaller k.

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Understand it

The rate constant usually increases with temperature because a larger fraction of molecules has energy equal to or greater than activation energy. The logarithmic form of the Arrhenius equation gives a straight line for ln k versus 1/T, with slope -Ea/R. A high activation energy means the rate is more sensitive to temperature change.

Example

If a plot of ln k against 1/T has slope -5000 K, then -Ea/R = -5000, so Ea = 5000 × 8.314 = 41570 J mol−1 = 41.57 kJ mol−1.

Rule to remember

Arrhenius equation: k = Ae−Ea/RT.

Log form: ln k = ln A - Ea/(RT).

Two-temperature form: log(k2/k1) = Ea/(2.303R)[(T2 - T1)/(T1T2)].

Use T in K and Ea in J mol−1 with R = 8.314 J mol−1 K−1.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For k1 = 2.0 × 10−3 s−1 at 300 K and k2 = 8.0 × 10−3 s−1 at 320 K, log(k2/k1) = log4 = 0.6021.

  1. 1Ea = [2.303 × 8.314 × 0.6021 × 300 × 320]/20 = 55290 J mol−1 approximately, or 55.3 kJ mol−1.
  2. 2Interpretation: the reaction needs moderate activation energy to speed up over this temperature rise.

Method you can reuse

  1. 1Identify whether the question gives a graph slope or two rate constants.
  2. 2Convert all temperatures to kelvin.
  3. 3Choose ln form for graph and two-point log form for two temperatures.
  4. 4Use R with matching energy units, then convert J mol−1 to kJ mol−1 if required.

Visual tip

An Arrhenius plot of ln k against 1/T is essential. The line has negative slope, and the magnitude of slope gives activation energy.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In a plot of ln k versus 1/T, what does the slope represent?

Think first, then show answer
The slope is -Ea/R, so activation energy Ea = -slope x R.

Common wrong answer

Writing slope = Ea/R instead of -Ea/R, which gives the wrong sign for the Arrhenius plot.

Don't mix this up

Students sometimes use Celsius temperature in Arrhenius calculations. Temperature must be in kelvin.

Exam tip

For two-temperature problems, use the two-point Arrhenius form and check that activation energy is in J mol−1 when R = 8.314 J mol−1 K−1 is used.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions ask for activation energy from slope, comparison of rate constants at two temperatures, explanation of temperature effect, and interpretation of energy distribution or activation barrier diagrams.
  1. 1

    Enough for 1 mark

    The Arrhenius equation relates the rate constant of a reaction to temperature and activation energy: k = Ae−Ea/RT, where A is the frequency factor, Ea is activation energy, R is the gas constant, and T is absolute temperature.

  2. 2

    Add for 2 marks

    Example: If a plot of ln k against 1/T has slope -5000 K, then -Ea/R = -5000, so Ea = 5000 × 8.314 = 41570 J mol−1 = 41.57 kJ mol−1.

  3. 3

    Add for 3 marks

    The rate constant usually increases with temperature because a larger fraction of molecules has energy equal to or greater than activation energy.

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