The Effect of Temperature on the Rate of a Reaction:

Arrhenius Behavior

Collision Theory | Activation Energy | Maxwell Distribution | Arrhenius Equation | Graphical method | Algebraic method |

A general observation is that most reactions proceed faster as the temperature is raised. An increase of 10oC typically doubles the rate of a reaction. Kinetic molecular theory (KMT) describes molecules much like defective billiard balls: if the balls collide at a low speed, then they simply bounce off each other. If they collide at very high speeds, they may break apart. A similar behavior is found in molecules: at low temperatures the molecules collide with each other, but bounce apart. If, however, molecules collide at high temperatures, bonds may be broken and new are molecules formed, i.e. a chemical reaction has taken place. Collision theory states that three conditions must be met for a reaction to occur:
• Molecules must collide with one another
• Molecules must have the right orientation
• Molecules must have sufficient energy
Every reaction has an energy barrier. The fact that a reaction increases with increasing temperature suggests that only molecules with sufficient energy are able to react. The energy barrier or minimum energy a molecule must possess to overcome this barrier is called activation energy (Ea). Principally, there are two types of reaction profiles: (1) the reaction proceeds via a transition state, and (2) the reaction proceeds via a very short-lived intermediate:

(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 (Ea(1) and Ea(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 Ea. 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.

The Arrhenius Equation:

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

• Calculate Ea from the temperature dependency of k
• Calculate k for a given T if Ea is known.
The Arrhenius equation is not very useful in its exponential form, since the collision factor "A" is neither known, nor can it be controlled. There are two ways to analyze Arrhenius equation:
• Graphical method
• Algebraic method
Graphical method:

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 Ea/R.

Algebraic method:

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 H2(g) + I2(g)

the rate constants have been determined as:

k1 = 2.15x10-8 L/mol s at 650 K

k2 = 2.39x10-7 L/mol s at 700 K

Therefore:

from which the and the activation energy is calculated:

Ea = 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

• Use a piece of graphing paper or your pocket calculator to plot of the rate constant k versus temperature T (it should show a smoothly increasing curve).
• Using graphing paper or your pocket calculator, plot ln k versus 1/T, which should result in a straight line. Determine the intercept and slope of that line. How does the slope and intercept of the straight line relate to Arrhenius equation?
Arrhenius equation is

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 737oC and 0.0815 L/mol s at 947oC . What is the activation energy of this reaction?

SUMMARY:

• Particles must collide in order to react
• Increasing temperature à increasing collision frequency
• Energy threshold = Actication energy (Ea)
• Temperature rise increases fraction of molecules to overcome activation energy threshold
• Reaction nearly doubles with 10oC temperature increase
• The larger Ea, the smaller the rate constant k, the slower the reaction.