# How to interpret the Maxwell-Boltzmann distribution to find the activation energy?

I understand activation energy to be the relative difference in energy between that of the activated complex/transition state (at some temperature) and the average energy of the reactants like shown here:

If this is correct, I have questions about graph like this one: (Source: webchem.net (not available anymore, Internet Archive), retrieved via the Internet Archive)

1. Is the location marked activation energy supposed to be the energy level of the transition state provided the energy axis is absolute?

2. Can the Maxwell-Boltzmann distribution be done for heterogeneous systems (i.e., two chemical reactants)?

3. Unrelated to the graph, why does negative activation energy result in a decrease in reaction rate (per Arrhenius equation)?

1. The marked location corresponds to a level of kinetic energy in the reactants sufficient to result in a successful collision (energy wise, it says nothing about orientation). The energy required for a successful collision is the gap in potential energy between the reactants and transition state.

2. A heterogeneous system is a system where the reactants are in different phases. A common example is a gaseous reactant that collides with a solid catalyst. In that case the Maxwell Boltzmann distribution can be applied to the gaseous reactant.

3. Can you clarify this? What reactions have negative activation energies? The reactions you'll commonly encounter have positive activation energies.

• For Q1: is that basically the same as the energy level of the transition state? For Q2: what about for combustion reactions (two gases), in this case is it possible to represent the Maxwell Boltzmann Distribution for both gases combined? Is that what the graph I linked is doing? For Q3: I don't have a specific reaction in mind and I was reading this page: en.wikipedia.org/wiki/…. Is it theoretically possible to heat any substance hot enough such that activation energy is negative? Oct 8, 2014 at 3:29
• Q3: The original wiki is referring to empirical activation energy, i.e. a value that you would got if you calculate it from temperature dependent kinetic data. For that kind of temperature dependence you need formally a negative energy value if you use this simple model. Therefore, it is more like of an artifact.
– Greg
Oct 9, 2014 at 8:06
• @brinnb For Q2, if a heterogeneous Maxwell Boltzmann Distribution is possible (say reactants A and B), is it correct to interpret it as the sum of the energies of A and B molecules which collide and the distribution is the representation of the sum? Oct 11, 2014 at 1:43

In question 3, it is never possible in a unimolecular or bimolecular reaction to have an negative activation energy. No matter how you draw two potential energies as intersecting parabolas, with the transition state at the intersection, there is always a positive activation energy.

In reactions with a complex mechanism, one reaction following on from another, then the measured activation energy for however many reactions are present can be negative.

The Boltzmann distribution applies to all reactants and products, but on average we may take the difference of average energy of reactants and the average at the transition state to be the activation energy. As the Boltzmann distribution applies this is the reason that reactions with large activation energies (>> kBT) are slow; only very occasionally do the reactants transiently acquire enough energy from the solution, or from a collision if in the gas phase, to react. (There are other effect that may need to be considered in particular cases, such as translational diffusion limiting reaction or reactants having such complex structures that only certain orientations permit reaction.)