What does the rate of a reaction depend on?

Just because a reaction is feasible, it doesn’t mean that it will happen at a rate by which we are able to observe reactants changing into products.

The rate of a reaction depends on a number of factors:

1. collisions between the reactant molecules
2. collisions between the reactant molecules with the correct orientation
3. collisions between the reactant molecules with sufficient energy to overcome the activation energy barrier

Let’s explore each of them in a little more detail …

1. The number of collisions in a certain timeframe (rate) depends on the concentration of the molecules in question, the speed at which they are moving and their size (the larger the molecule, the greater the chance of a collision). In the reaction:

CH₃CH₂Br + OH^– ⇾ CH₃CH₂OH + Br^–

the rate of collisions is proportional to the concentration of the reactants

rate ∝ [CH₃CH₂Br] [OH^–]

2. In many reactions (particularly organic ones) the reactants must collide with a certain geometric orientation in order for the reaction to be successful and products form.

3. There is an energy barrier, E_a, to overcome for most reactions which is the minimum amount of energy that two colliding molecules must possess in order to react. We can illustrate this on a Maxwell-Boltzmann distribution curve:

The area under the curve is proportional to the total number of particles involved and the blue shaded area is proportional to the number of particles with energy greater than E_a (activation energy).  Hence the fraction of particles with energy greater than E_a is given by the ratio:

In the mathematical expression shown above that Maxwell and Boltzmann derived for this fraction, R is the gas constant (8.31 JK^-1mol^-1) and T is the temperature in K. The implication of the ‘RT’ term is that the higher the temperature, the greater the probability that a given molecule possesses a particular energy.

We can use the expression to calculate the number of molecules in 1.0 mol of gas at 25°C (298K) with energy greater than 55.0 kJ mol^-1:

• firstly, calculate the fraction of molecules with energy greater than 55.0 kJ mol^-1

  Fraction of molecules with energy > E_a  =  e ^{-Ea / RT}    

where R = 8.31 J K^-1 mol, T = 298K and  E = 55000 J mol^-1

=  e ^{– (55000 / 8.31 x 298)}  =  e ^-22.2  =  2.28 x 10^-10

• secondly, calculate the total number of molecules with energy greater than 55.0 kJ mol^-1. The total number of molecules present in 1.0 mol of gas is given by Avogadro’s number, 6.02 x 10²³ mol^-1, therefore the number of molecules in 1.0 mol of gas with E_a > 55.0 kJ mol-1 =  

(6.02 x 10²³)  x  2.28 x 10^-10  =   1.37 x 10¹⁴

If we repeat this exercise for a slightly higher temperature, say 35°C (308K), the number of molecules in 1.0 mol of gas with E_a > 55.0 kJ mol-1 = 2.77 x 10¹⁴.

A 10°C rise in temperature leads to a doubling of the number of molecules that exceed E_a!

But why do reactions have an activation enthalpy?

Let’s look at the reaction profile for the reaction between CH₃CH₂Br and OH^–. In order for the reaction to proceed, the bond between C and Br must lengthen and break as a new bond between C and O forms, and the methyl group and hydrogen atoms around the carbon also must move to a less than ideal geometry to accommodate this.

It’s important to realise that the transition state is not a real molecule! The reaction merely passes through this arrangement of atoms on its way from reactants to products, in 10^-12 seconds or less.

In the next post, I’ll be building on these ideas to introduce rate laws and orders of reaction …