Arrhenius Behavior
(1) Reaction profile for a reaction that proceeds via a transition state
(2) Reaction profile for a reaction that proceeds via an intermediate: Each step has its own activation energy (E_{a}(1) and E_{a}(2), respectively.
Not all molecules at the same temperature have the same energy, instead there is a distribution of molecules with certain energies. A plot of the relative number of molecules with a given energy versus kinetic energy is called the Maxwell distribution.
Maxwell distribution - Fraction of molecules with certain energy versus kinetic energy:
In order to undergo reaction, molecules must possess an energy that is equal or higher that the activation energy E_{a}. At low temperature, only a few molecules have sufficient energy - the reaction will proceed, but at a slow rate. At higher temperate, more molecules are able to surpass the energy barrier: the reaction proceeds at a faster rate. The kinetic energy of a molecule is directly proportional to the temperature.
R is the ideal gas constant, T is temperature in Kelvin, m is the mass of the molecule and v is the velocity. Thus, the higher the temperature, the greater the speed of the molecules, and the higher their kinetic energy.
The rate equation for a chemical reaction states that the rate is proportional to the rate constant k and the concentration of the reactant(s):
Looking at the rate equation, we see that neither the reactant concentration [A] or [B] nor the order of the reaction (x or y) changes with increasing temperature. So, we need to explore the rate constant itself.
Svante Arrhenius found that the fraction of molecules whose energy equals or exceeds the activation energy is proportional to e^{-Ea/RT}. Therefore, the rate constant k must be proportional to the same factor:
The proportionality factor "A" is the so-called collision factor and describes the frequency of collisions with correct orientation. The exponential term e^{-Ea/RT} describes the fraction of molecules with minimum energy for the reaction. R is the ideal gas constant with a value of 8.314 J/K mol.
Arrhenius equation can be used to
The Arrhenius equation becomes more useful if we take the natural logarithm
and rearrange the equation to resemble a straight-line equation:
Rearranged Arrhenius equation:
Straight line equation:
If we plot the natural logarithm of k (ln k) versus the inverse of the
temperature (1/T) a straight line with a negative slope will result. The
slope of the line equals E_{a}/R.
The activation energy of a chemical reaction can also be determined algebraically using Arrhenius equation. In that case, the rate constant for a reaction needs to be determined at two different temperatures. For each rate constant we can write the Arrhenius equation:
Taking the difference leads to
Arrangement of the equation yields
which allows us to calculate the activation energy for a chemical reaction.
Example:
For the reaction
2 HI(g) è H_{2}(g) + I_{2}(g)
the rate constants have been determined as:
k_{1 }= 2.15x10^{-8} L/mol s at 650 K
k_{2} = 2.39x10^{-7} L/mol s at 700 K
Therefore:
from which the and the activation energy is calculated:
E_{a} = 182 kJ/mol
Exercise: Use Arrhenius euqation to determine the activation energy for the two reactions below. Part A uses the graphical method, Part B uses the algebraic method.
EFFECT OF TEMPERATURE ON THE RATE CONSTANT
(Arrhenius Equation)
PART A:
In the hydrolysis of an ester, when reactant
concentrations are held constant and temperature increases, the rate of
the reaction and the rate constant increase.
EXPT | [ESTER] | [H2O] | T (K) | Rate (mol/L s) | k
(L/mol s) |
1/T | Ln k |
1 | 0.1 | 0.2 | 288 | 1.04 x 10^{-3} | 0.0521 | ||
2 | 0.1 | 0.2 | 298 | 2.02 x 10^{-3} | 0.101 | ||
3 | 0.1 | 0.2 | 308 | 3.68 x 10^{-3} | 0.184 | ||
4 | 0.1 | 0.2 | 318 | 6.64 x 10^{-3} | 0.332 |
where k = rate constant, e = the base of the
natural logarithms, T is the absolute temperature, and R is the universal
gas constant (R= 8.314 J/K mol). Constant A is related to the orientation
of the colliding molecules, and Ea is the activation energy of the reaction.
The activation energy is the minimum energy that the molecules must have
to react.
PART B:
Understanding the characteristics of high temperature
formation and breakdown of the nitrogen oxides is essential for controlling
the pollutants generated by car engines. The second order breakdown of
nitric oxide to its elements has rate constants of 0.0796 L /mol s at 737^{o}C
and 0.0815 L/mol s at 947^{o}C . What is the activation energy
of this reaction?
SUMMARY: