The Temperature Dependence of Reaction Rates
Consider the reaction
Consider the reaction
Only collisions with enough energy react to form products. The energy of the system changes as the reactants approach each other. The critical amount of energy to make the reaction proceed is called the Activation Energy.
The Reaction Coordinate is the 'distance' along the path of the reaction, and is plotted along the horizontal axis. The energy of interaction of the reactive system is plotted vertically, and is called the Chemical potential, or just potential energy. You fight gravitational potential energy when you try to roll a boulder over a mountain.
A chemical potential of interaction usually looks like something like the graph above, which is similar to the 'pushing a boulder over a hill' graph above. The Graph above is drawn for the isomerization of an isonitrile that we discussed before. The barrier to the isomerization keeps the unstable CH3NC from reacting away quickly at low temperature, even though energy is released upon the net reaction.
A typical organic reaction mechanism is the SN2 reaction, as seen in the replacement of the Br- leaving group with the OH- nucleophile. Here is a movie of an SN2 reaction.
The reaction potential depends greatly on the reaction mechanism. Changing the mechanism of a reaction changes the reaction coordinate, which can affect both the height of the barrier and its 'shape'. Kinetic analysis can tell us the nature of this potential surface and thus give clues as to the reaction pathway. But there is something that we can say about the reaction potential curve that is always true: The 'barrier' in the forward direction is different from the barrier in the reverse direction by the difference in energy (enthalpy) of the reactants and products. No matter what the shape of the barrier is, or no matter what we do to change the reaction path, the forward and reverse barrier to reaction are related by the properties of the reactants and products alone.
We need a way to measure the activation energy of a reaction. The activation energy is different for the forward and reverse reactions, and the difference is the reaction exoergicity, as we said before. But to measure the height of the barrier, we need to run the reaction with different amounts of energy, so we can see how much energy it takes to make the reaction go.
Where does the chemical activation energy come from anyway? It comes from the thermal excitation of the molecules. Remember that at a given temperature, the molecules in any sample have a distribution of energies, with a well defined average, but with a probability that any given molecule has a given energy. This energy distribution is the same distribution as the kinetic energy distribution in a gas, which we know looks like this:
Reactions in a gas get energy from the only source available,
the energy of the collisions in the gas. If we draw a line at the
energy required to activate the reaction, the rate of the reaction
should depend on the fraction of collisions that have energy in excess of this
critical value. From the for of the energy distribution, the
fraction of highly excited molecules (molecules with energy above a critical value)
is exponentially dependent on the temperature. So then, the reaction rate should
be exponentially dependent on temperature.
The following shows a typical temperature dependence of the rate constant for a reaction as the temperature is varied. (What is the order of the reaction in this example?)
The formula that describes the temperature dependence of the rate constant is attributed to Arrhenius, but is a consequence of our arguments above:
The relative rate constants for the same reaction at two different temperatures may be calculated taking into account the exponential temperature dependence of the rate constant. This expression looks very much like the equation we used to determine the vapor pressure as a function of temperature (The Clausius - Clapeyron equation).
Interpreting the Arrhenius Factors: A and Ea
The temperature dependence of the absolute rate constant appears to depend on two quantities (A and Ea), whereas the relative rate constants at two temperatures depend only on one quantity (Ea). What do these qunatities represent in the reaction mechanism?
Ea, the Activation Energy
The quantity Ea is simply the energy of activation for the reaction, and solely determines the increase in rate constant with an increase in temperature. The temperature dependence of the rate of a reaction is used to determine Ea in the laboratory!
The Arrhenius A factor
Not all molecules that that have energy greater than or equal to Ea will lead to a reaction, because they must have a collision. The reaction rate must include the collision frequency somewhere, even for reactions in solution. The factor A includes the collision frequency, which must be multiplied by the fraction of collisions with sufficient energy to get an estimate of the reaction rate. For this reason, the A factor is sometimes called the Approach Factor.
Molecular Steric Effects
Not all collisions that have sufficient energy to react will react, beacuse the geometry of the molecules may not be 'favorable' for reaction. The figure below shows the importance of molecular orientation in an effective collision. Only one of the five orientations shown for the collision between NO and NO3 leads to product. (Nitrogen is Blue, Oxygen is Red) In the effective orientation, those atoms collide that will become bonded in the product.
Molecular steric effects (like only certain collisions leading to reaction products) AND the Frequency of Approach (collision frequency) are both included in the value of the the Arrhenius A (pre-exponential) factor. This makes the A factor much more difficult to interpret than the Energy of Activation.
The A factor is equal to the high temperature limiting rate constant for the reaction. This can be seen by inserting a very large (infinite) temperature into the Arrhenius equation above. This is true because at very high temperature, ALL the molecules have enough energy to react, but they must have a collision and the proper geometry even so.
Catalysts can greatly influence the energy of activation. They change the nature of the activated complex, usually by influencing the electronic structure of the reactants and lowering their bond dissociation energy. The following shows how a metal can influence the rate of hydrogentation of an alkene:
Catalysis of a reaction influences the shape of the potential energy curve connecting the reactants and products. Thus the Energy of activation can be lowered and the rate of a reaction increased at a given temperature. Sometimes they even stabilize the activated complex so much that it becomes a real reactive intermediate.
Enzymes are natures catalysts for the reactions that make life work. Enzymes are great catalysts because they can increase the A factor and decrease the energy of activation at the same time, both of which increase the reaction rate without increasing the temperature. They can affect A by grabbing the reactants out of solution by binding to the active site of the enzyme, thus increasing the probability the reactants will approach each other. They are also good at orienting the reactants just right for the reaction, improving the steric factor considerably.