# Equilibrium for very small amounts of reactants

Is there any difference in the concept of equilibrium when it comes down to dealing with extremely small amounts of reactants?

Say we have $$\ce{A + B <=> C + D}$$ and $K_c$ is $100,000$. If we only have $10$ atoms of each $\ce{A}$ and $\ce{B}$ in a beaker, we would expect the container to be totally full of $\ce{C}$ and $\ce{D}$. But $K_c$ is not infinite, so there must be some unreacted $\ce{A}$ and $\ce{B}$. However, if there was even one atom of reactants, the reaction would quickly shift back to satisfy the equilibrium.

Does the concept of equilibrium only apply to macro-sized reactions? Is there an extendable concept for the micro scale?

• Equilibrium constants are more related to thermodynamic functions then numbers of molecules. – Mithoron Aug 26 '18 at 23:21
• chemistry.stackexchange.com/a/50496/9961 – Mithoron Aug 27 '18 at 0:33
• Can you estimate the minimum number of molecules of A and B you should have initially so that, with an equilibrium constant of 100,000, the statistics at equilibrium would be valid? – Chet Miller Aug 27 '18 at 12:25

Thermodynamics describe an average state for a large number of items. However, for a time, that state will vary from that average. So if you take a snapshot of the system at an instant, it would likely not be exactly as equilibrium would predict, yet over time that would be the average state.

"The larger your sample size, the smaller the standard deviation" from the expected value. However, by increasing the number of samples over time, even a small number of molecules would behave as expected.

However, the question is not unreasonable, in that it describes the problem of reconciling macroscopic behavior with quantum effects. Boltzmann explored this issue, and Planck introduced quanta to harmonize statistical thermodynamics with observed black-body radiation.

Thermodynamics is only applicable to the systems of size in between astronomic and microscopic scales. Systems with a small number of particles can spontaneously abandon equilibrium state due to fluctuations. Statistical physics tells us [1] that the relative magnitude of the fluctuations of thermodynamic quantities $$x$$ (e.g. pressure, temperature, volume, internal energy, entropy etc.) is inversely proportional to the root of the number of particles in the system:

$$\frac{ \sqrt{\left\langle\left(\Delta x \right)^2\right\rangle} }{ \left\langle x \right\rangle } \sim \frac{1}{\sqrt{N}}$$

Lets arbitrarily set the detection limit for experimental determination of the relative change of $$x$$ to optimistic value of $$10^{-8}$$. In this case we would need at least $$N = (10^{-8})^{-2} = 10^{16}$$ particles belonging to the thermodynamic system before we can speak of equilibrium at all.

### References

1. Landau, L. D.; Lifshitz, E. M. Statistical Physics, 3rd Edition, Part 1: Vol. 5; Butterworth-Heinemann: Amsterdam u.a, 1980.

If you observe a small number of reactants and for a short time then their behaviour will not conform to that of many millions as there will be some variation simply due to the small number of reactants and the order in which they react with one another etc. From statistics we know that the fewer the number of measurements the larger is the standard deviation of that measurement.

However, if you repeat the experiment on your small number of molecules many thousands of times then the average behaviour is recovered. This means that the time average is the same as the ensemble average and is expressed in the Ergodic Hypothesis. This is sometimes expressed as 'an ensemble of particles (atoms ,molecules) will pass through all the states available to it given sufficient time'.