Arrhenius Equation

Arrhenius equation is the mathematical expression that describes the effects of temperature on the velocity of a chemical reaction. It is an accurate and remarkable formula to show the dependence of rate of reaction on temperature. The higher the temperature, the faster the reaction will be. When temperature is high, chances for molecules to collide is higher and thus increasing the rate of reaction. That means, higher the temperature, higher the kinetic energy which increases the activation energy (energy required to ensure that a reaction happens) of the reaction.

Arrhenius equation is as follows:

k = Aexp (−E/RT). In the equation k= reaction rate constant, A & E= numerical constants of reacting substances , R= thermodynamic gas constant (8.314 x 10^-3kJ mol^-1K^-1), T = absolute temperature. The equation predicts that a small increase in reaction temperature will produce increase in the degree of reaction rate. The equation was formulated by J.J Hood based on studies of variation in reaction rate with different temperature. Later Svante Arrhenius, for whom the equation is named, explained that the relationship is applicable for all kinds of reactions.

When rate constant (Ink) is plotted against inverse of temperature I/T(Kelvin) the slope will be a straight line. The value of the slope is equal to -E_a/R.

Kinetic Theory’s interpretation of Arrhenius equation:

Arrhenius tried to prove that reactants have to acquire a minimum amount of energy called activation energy E_a to transform into products .We can calculate the fraction of molecules that have kinetic energy greater than E_a   from Maxwell Boltzmann Distribution at absolute temperature T and turns out to be proportional to  - E_a/RT.

Arrhenius Plot

The value of Activation energy (E_a) can be calculated. The values of k₁ and k₂ at temperatures T₁ and T₂ can be expressed by the following relation:

In k₁/k₂ = - E_a /R (1/t₁ - 1/T₂)

When the concentrations are constant, rates and rate constants are directly proportional to each other. Then the above equation can be rewritten as:

In (rate₁)/ (rate₂)  = - E_a /R (1/t₁ - 1/T₂)

‘Two point form’ of Arrhenius equation

By substituting two rate constants (k₁  &  k₂) with corresponding temperatures (T₁  &  T₂) in the Arrhenius equation ,activation energy can be found algebraically.

Ink₁ = - E_a/RT₁ + InA   _____   (1)

Ink₂ = - E_a/RT₂ + InA    _____  (2)

Subtract (2) from (1) then:

Ink₁/k₂ = E_a/R (1/T₂ - 1/T₁ )  _____    ( 3)

Rearrange equation (3) to get activation energy.

E_a = RT₁T₂ / (T₁-T₂)    In(k₁/k₂)

Questions:

• Define Arrhenius equation?
• Explain ‘two Point form’ Arrhenius equation?