# Activation Parameters

#### Determination of Activation Parameters

It seems intuitive that a reaction goes faster as the temperature is raised, as more reactant molecules have the energy needed to overcome the activation barrier to the reaction. The Arrhenius equation relates the reaction rate constant (k) and temperature. One of the forms of the Arrhenius equation is given below:

\[ \ln k = -{E_a \over RT} + \ln A \]

where E_{a} is the activation energy for the reaction, T is the absolute temperature (in Kelvin) at which a corresponding k is determined, R is the gas constant, and A is a pre-exponential factor. The activation energy may then be extracted from a plot of ln k vs. 1/T, which should be linear. This plot is called an "Arrhenius plot."

Example ARK1.1.

Recall that y = mx + b.

- In an “Arrhenius plot” plot, what is the slope?
- What is the intercept?

Example ARK1.2.

Using the following data, construct an Arrhenius plot and determine the activation energy (in both kcal/mol and kJ/mol) and the pre-exponential factor.

| ln k (unitless) |

0.00152 | 3.7 |

0.00157 | 3.2 |

0.00160 | 2.9 |

0.00165 | 2.2 |

0.00170 | 1.6 |

Example ARK1.3.

Using the following data, construct an Arrhenius plot and determine the activation energy (in both kcal/mol and kJ/mol) and the pre-exponential factor.

T (°C) | k (mol L^{-1} s^{-1}) |

40 | 1.3 x 10^{-4} |

50 | 2.2 x 10^{-4} |

60 | 4.0 x 10^{-4} |

70 | 7.5 x 10^{-4} |

80 | 1.4 x 10^{-3} |

In practice, activation energies are not often cited in the current literature. Instead, a similar but more useful equation called the Eyring equation is used. The Eyring equation is:

\[ \ln \left ( {k \over T} \right ) = - {\Delta H^{\ddagger} \over RT} + \ln \left ( {k_B \over h}\right ) + {\Delta S^{\ddagger} \over R}\]

where k, T and R are the same as in the Arrhenius equation, k_{B} is Boltzmann’s constant, h is Planck’s constant and ΔH^{‡} and ΔS^{‡} are the enthalpy and entropy of activation, respectively.

Example ARK1.4.

- What information is required to make an Eyring plot?
- What is the slope?
- What is the intercept?

Note that the activation parameters (ΔH^{‡} and ΔS^{‡} ) are not the same as the entropy and enthalpy of the reaction, which can usually be calculated from tables of values. Since they depend on how the reaction proceeds, not just the initial and final states of the reaction, they must be determined experimentally. Once that has been done, interpretation of the numerical values provides insight into the mechanism of the reaction.