When are transition state's energy barrier "reasonable" at a certain temperature?

Question

Every computational chemist who does a little bit of mechanistic studies has asked himself this question. When you obtain your first transition state and look at the activation barrier needed to access it, when would you tell that this required energy is reasonable and can be reached at a certain temperature ?

I have heard (never found a proof yet) that a biological process can be performed under standard conditions ($\pu{37 ^\circ C}$) with energy barrier as high as $\pu{15 kcal/mol}$.

Does anyone has a guideline for this?

Answer 1

Your question is really about what you consider to be a reasonable rate constant and so is somewhat subjective. The fastest a bimolecular reaction can be in solution is given by how fast the reactants can collide and then react on first collision. i.e reaction is diffusion controlled. In this case as reaction is at first meeting the activation energy must be small, $\le \pu{5 kJ mol-1}$. In water the diffusion controlled rate constant is in the region of a few times $\pu{10^10 dm3 mol-1 s-1}$. Most reactions are far slower than this.

The simplest, but very approximate, approach for other reactions is to use the Arrhenius equation to estimate the rate constant. The equation is $$ k = A \exp(-E_a/RT),$$ where $E_a$ can be taken as your activation energy and you have then to estimate $A$. This can be done for simple reactions using statistical mechanics and calculating the partition functions, but for anything complex you are left only with guessing values based on experimental data from similar reactions. However, because of the exponential dependence of the rate constant on activation energy even a small change in this can have a dramatic influence on the rate constant, far bigger than any similar change in $A$.

If you do have a potential energy surface then you can calculate trajectories along this surface by solving the (classical) equations of motion, repeatedly using energy sampled from a Boltzmann energy distribution, and so work out a theoretical rate constant.

Answer 2

If you don't have any idea what $k$ value would be appropriate, arguably the best option is to identify a catalyst that is experimentally verified to do the reaction of interest and is assumed to involve the same mechanistic step you computed. Identify the lowest temperature the experiments are successfully run at. Determine the rate constant for the elementary step with known catalyst, either from previously published work or by running the calculation yourself (preferably at the same level of theory as your predicted catalyst). With $k$ known for the experimentally tested catalyst, you can back out the required $T$ for your catalyst by plugging in your catalyst's $E_{\text{a}}$ and $A$ into the Arrhenius equation (or you can just assume the prefactors are identical since this is all an estimate anyway).

This process is a very rough approximation though for many reasons. It assumes the mechanism is identical to the reference material, it assumes the DFT-computed energetics are fairly accurate (even small inaccuracies can greatly change the required $T$), and it neglects the effect of concentrations on the rate, among other things. Depending on your needs, it may be okay for a ballpark estimate though. For instance, if you are studying a biological system and get that $T$ must be greater than 400 °C, you probably need to look for a new catalyst.

I also feel obligated to point out the following two limitations to this kind of estimate. The first is that it tells you little about the thermodynamic feasibility of the reaction. It would be worth making sure that the reaction is not predicted to be highly endothermic (although a low barrier necessary implies this won't be the case). In addition, if you are studying a mechanism where the reactant for the elementary step involves a bound adsorbate, it's always possible the energetically favorable binding mode is different between the reference catalyst and your own, which would further complicate matters.

Answer 3

Using this website, input the activation energy and temperature and it gives half life. A reasonable half-life is up to the user, but more than 1 day is going to be super slow. Usually assume the transmission coefficient is less than 1 (e.g. 0.5).

For example, an upper bound at room temperature is usually less than 25 kcal/mol $\approx$ 105 kJ/mol. With a 0.5 transmission coefficient the half-life is 4.2 days.

Answer 4

The other answers have given you approaches to estimating the overall rate constant as a function of temperature, which in turn requires you get an estimate of the prefactor, $A$. Knowing rate as a function of temperature is of course useful.

However (and I write my answer not just for you, but for others who might be searching for this topic), your question wasn't about estimating the overall rate; rather it was about estimating the temperature at which the activation barrier could be 'reasonably' overcome by thermal energy: "When are transition state's energy barrier “reasonable” at a certain temperature?"

In response, let me offer a very simple general guideline commonly used by chemists and physicists. The activation barrier is comparable in height to the amount of available thermal energy when:

$$RT \approx E_a$$

I.e., when:

$$T\approx E_a/R$$

At this temperature, the exponential term in the Arrhenius rate expression is simply $e^{(-E_a/RT)}=e^{-1} \approx 0.37 $.

A good number to memorize is that, at room temperature (298 K), $RT\approx 2.5\,\, \text{kJ/mol.}$ This tells you that the average thermal energy at room temperature is within the range needed to break water-water or water-protein hydrogen bonds. It also says you're in the regime where changes in temperature have a significant effect on the stabilty of these bonds. [The stability of stronger (e.g., covalent) bonds isn't significantly affected by temperature changes around room temp, and weaker associations (those due to van der Walls forces) are thoroughly disrupted at much lower temperatures.]

[This isn't to say VDW forces are unimportant at room temp — they are! Rather, it's saying these bonds are in a continuous state of dissociation and re-association at room temp.]