Book Cover Chemistry 8th Edition / Chang
Student Study Guide

Chapter 13: Chemical Kinetics


Index | 13.1 | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 |

ACTIVATION ENERGY AND TEMPERATURE DEPENDENCE OF REACTION RATES (13.4)

STUDY OBJECTIVES
  1. Describe three fundamental tenents to the collision theory.
  2. Calculate the activation energy for a reaction when given rate constants at several different temperatures.
  3. Describe the concept of activation energy and show how it explains the variation of reaction rate with temperature.
  4. Describe a reaction energy profile, including the activation energy and energy change of reaction.

Collision Theory. The collision theory of chemical reactions provides a general explanation of how reaction rates are affected by reactant concentrations, and temperature. The basic ideas of the theory are:

  1. In order for atoms, molecules, or ions to react they must first collide with each other. The rate of reaction is proportional to the rate of collisions, called the collision frequency. The more concentrated the reactants, the greater the collision frequency, and the reaction rate.
  2. For molecules to react they must come together in the proper orientation. Molecules can be complex and often it is just one atom in a molecule that reacts upon collision with another molecule.
  3. When reactant molecules collide, they must possess a minimum amount of kinetic energy in order for an effective collision —a reaction— to occur. Without this necessary energy two molecules will just bump each other and bounce back without reacting. The minimum amount of energy required to initiate a chemical reaction is called the activation energy.

Effect of Temperature. The temperature of a reaction system is an important variable because of its strong effect on reaction rates. As a rough rule, reaction rates approximately double with a 10°C rise in temperature. In general, the rate equation is rate = k[A]x[B]y. Since the concentrations [A] and [B] are unaffected by temperature, it is the rate constant that changes with temperature. In 1889, S. Arrhenius found that a plot of the natural logarithm of the rate constant (ln k) versus the reciprocal of the absolute temperature 1/T gave a straight line. See visual aid Figure 13.16 (text). Arrhenius identified the slope of the line as being related to an energy term

where R is the ideal gas constant (in units of joules) and Ea is the activation energy. The logarithmic form of the Arrhenius equation is

where ln A is the intercept. Both the Arrhenius A and Ea are constants for a particular reaction. Taking the antilog of both sides gives the Arrhenius equation which relates the rate constant to the temperature.

k = A e–Ea/RT

From the Arrhenius equation we can point out that reactions for which Ea is large will be much slower than those for which Ea is small. As Ea increases, the negative exponent increases, and so k decreases.

A convenient equation that can be used to calculate the activation energy is

where k1 is the rate constant at temperature T1, and k2 is the rate constant at temperature T2. Use of this equation requires that rate constants k1 and k2 be measured at two temperatures T1 and T2, respectively.

EXAMPLE Calculating Activation Energy

For the reaction

NO + O3 NO2 + O2

the following rate constants have been obtained:
Temperature, °C k(M–1 s–1)
10.0° 9.30 x 106
30.0° 1.25 x 107

Calculate the activation energy for this reaction.
Ea = kJ/mol

The Meaning of A and Ea. The parameter A is called the frequency factor. It is related to the frequency of molecular collisions, and the fraction of collisions that have the correct orientation. As discussed above, the collision frequency is important because molecules must collide in order to react.

The activation energy Ea is related to the formation of the activated complex. The activated complex is the high-energy intermediate species that dissociates into the products. Figure 13.3 shows that the activation energy is the difference in energy between the reactants and the activated complex. The activation energy is provided by the kinetic energy of rapidly moving molecules during collisions. Reactants must "get over the barrier" before they become products. The factor e–Ea/RT that appears in the Arrhenius equation is the fraction of molecules with energies equal to or greater than the activation energy. This factor changes significantly with temperature. As temperature increases a greater fraction of molecules has an energy equal to, or greater than, the activation energy.

Figure 13.3 Molecules of A and B with average energies must first acquire an energy Ea before they can react.

EXAMPLE Reaction Energy Profile

Draw a reaction energy profile for the following endothermic reaction:

2HI(g) H2(g) + I2(g)      H°rxn = 12.5 kJ

Given the activation energy Ea = 185 kJ/mol. Label the activation energy and the activated complex. What is the activation energy for the reverse reaction?
Ea = kJ/mol

OBJECTIVE CHECK

Complete the following questions to check your understanding of the material. Select the check button to see if you answered correctly.

  1. Calculate the activation energy for a reaction given that the rate constant is 4.60 x 10–4 s–1 at 350°C and 1.87 x 10–4 s–1 at 320°C.
  2. The rate constant for a first-order reaction is 4.60 x 10–4 s–1 at 250°C. If the activation energy is 100.0 kJ/mol, calculate the rate constant at 300°C.
  3. A substance decomposes according to first-order kinetics, the rate constants at various temperatures being as follows:
    4. Temp (°C) k (s–1)
    15.0 4.41 x 10–6
    25.0 1.80 x 10–5
    30.0 2.44 x 10–5
    1. If one were to make an Arrhenius plot what would its slope be equal to according to the Arrhenius theory?
    2. Estimate the numerical value of the slope by using the first and third points.
    3. Calculate the activation energy from your value for the slope.
  4. The reaction 2NOCl 2NO + Cl2 has an Ea of 102 kJ/mol and a Hrxn of 75.5 kJ. Sketch the reaction energy diagram and determine the activation energy for the reverse reaction.


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