How many atoms/molecules are in a mole? There are precisely 6.022 × 10^23 units (Avogadro’s number). While many are familiar with this number, the true enormity of its size is hard to fully grasp.

**In this article, I explore the concept of Avogadro’s limit.****Through this, I convey the immense multiplicity of molecules making up the world around us.**__Avogadro’s limit__

Avogadro’s limit refers to the concentration below which a solution becomes so dilute it is unlikely that any of the original molecules are present. More precisely, we can define the limit as the concentration at which there is only one molecule per litre of solution. Since there are 6.022 × 10^23 atoms in a mole, this gives a limit of 1.66 × 10^-24 mol/L.

__How to dilute a solution to Avogadro’s limit__

The true size of Avogadro’s constant is clarified by how challenging it is to dilute a solution to Avogadro’s limit. I illustrate this by exploring here how a solution of table salt (sodium chloride) at Avogdro’s limit could be prepared, starting with a single grain of salt.

First, we must estimate the mass of a grain of salt. An average grain of regular table salt has dimensions of roughly 0.3 mm × 0.3 mm × 0.3 mm [1]. This corresponds to a volume of 2.7 × 10^-5 mL. The density of NaCl is 2.16 g/mL and so the grain contains roughly 5.83 x 10^-5 g. The molecular weight of NaCl is 58.44 g/mol, giving 9.98 × 10^-7 moles (

First, we must estimate the mass of a grain of salt. An average grain of regular table salt has dimensions of roughly 0.3 mm × 0.3 mm × 0.3 mm [1]. This corresponds to a volume of 2.7 × 10^-5 mL. The density of NaCl is 2.16 g/mL and so the grain contains roughly 5.83 x 10^-5 g. The molecular weight of NaCl is 58.44 g/mol, giving 9.98 × 10^-7 moles (

*n*). Now we calculate what volume (*V*) we’d have to dissolve the salt grain in to reach Avogadro’s limit concentration (*C*).To put this in prospective, this is roughly the volume of the Black Sea (5.47 × 10^17 L) [2]. Imagine dissolving a single grain of salt into such a volume! At this concentration, 1 L of the resulting solution typically would contain a single molecule of NaCl. Obviously, this would not be a very practical way to prepare such a solution. (Of course, attaining water truly free of NaCl would itself be unachievable. This concept works only as a thought experiment).

__Reaching Avogadro’s limit by serial dilution__

A much more efficient way to achieve the required, remarkable dilution is to do a series of repeated dilutions. Though more steps are required, much less water is needed. Each dilution decreases the initial concentration (

*Ci*) to a final concentration (*Cf*) in proportion to the dilution factor (*D*).For instance, suppose we dissolve the single grain of table salt into 1 L of water. This gives a molarity of roughly 10^-6 mol/L. We proceed to dilute 500 mL of this solution into a fresh 500 mL of water. How many times would we have to repeat this process to reach Avogadro’s limit? Each dilution achieves a 2-fold dilution factor.

We would have to repeat this serial dilution 59 times to reach Avogadro’s limit! This is a lot, though it is certainly easier than using a Black Sea’s volume of water. The serial dilution can be made more efficient by using a higher dilution factor:

Suppose we instead dilute our initial 1 L salt grain solution by adding 100 mL of it into a fresh 900 mL of water. Each dilution now achieves a 10-fold dilution factor.

Suppose we instead dilute our initial 1 L salt grain solution by adding 100 mL of it into a fresh 900 mL of water. Each dilution now achieves a 10-fold dilution factor.

We’d now only have to repeat this dilution 18 times to go below Avogadro’s limit. I hope that imagining the physical preparation of these solutions will give my readers pause at the size of Avogadro’s number.

__Avogadro’s limit in homeopathy__

It is interesting to consider Avogadro’s limit in the context of homeopathy. This pseudoscientific approach to alternative medicine is based on the bizarre premise that dilution

The principles of homeopathy hopefully surprise and confuse you, as it goes against our most basic understanding of physics, chemistry and biology. All scientifically understood drugs exert their effects through physical or chemical interactions with the biological system. Surely a drug cannot interact with/modulate a biological system if none is administered. The most commonly proposed mechanism for homeopathy is the concept of “water memory,” that water molecules are permanently imprinted with the memory of compounds they’ve previously interacted with. Proponents even claim that this information is transmitted to new water molecules upon dilution. Of course, there are serious problems with these ideas. The structural features of water have been heavily studied and are well understood. Water molecules dynamically associate in a hydrogen-bonding network. Presumably, it is through this that the water could “remember.” Unfortunately for homeopathy though, it seems that water is rather forgetful. Experimentally, this network’s has been shown to be remarkably short-lived with all memory gone after only 50 femtoseconds [4].

The concept of water memory does not even require a scientific rebuttal to expose its absurdity, however. If water is imprinted with all chemicals it has ever meet, then just about all water on Earth would already have memory of near every conceivable medicine or toxin. Furthermore, recall the central premise that potency increases with dilution. It follows that the addition and subsequent dilution of a specific medicine would be ineffective at developing that particular activity over all others already present.

*enhances*the potency of medicine. Usually homeopathic preparations are diluted well below Avogadro’s Limit, commonly to around 10^-60 mol/L (37 orders of magnitude below the limit!). Such a preparation would certainly contain none of the original substance. Not surprisingly, a wealth of scientific evidence suggests that homeopathic remedies’ benefits are merely due to the placebo effect [3].

The principles of homeopathy hopefully surprise and confuse you, as it goes against our most basic understanding of physics, chemistry and biology. All scientifically understood drugs exert their effects through physical or chemical interactions with the biological system. Surely a drug cannot interact with/modulate a biological system if none is administered. The most commonly proposed mechanism for homeopathy is the concept of “water memory,” that water molecules are permanently imprinted with the memory of compounds they’ve previously interacted with. Proponents even claim that this information is transmitted to new water molecules upon dilution. Of course, there are serious problems with these ideas. The structural features of water have been heavily studied and are well understood. Water molecules dynamically associate in a hydrogen-bonding network. Presumably, it is through this that the water could “remember.” Unfortunately for homeopathy though, it seems that water is rather forgetful. Experimentally, this network’s has been shown to be remarkably short-lived with all memory gone after only 50 femtoseconds [4].

The concept of water memory does not even require a scientific rebuttal to expose its absurdity, however. If water is imprinted with all chemicals it has ever meet, then just about all water on Earth would already have memory of near every conceivable medicine or toxin. Furthermore, recall the central premise that potency increases with dilution. It follows that the addition and subsequent dilution of a specific medicine would be ineffective at developing that particular activity over all others already present.

__The probability of finding a molecule below Avogadro’s limit__

When doing calculations of moles or concentration, we normally needn’t consider probability at all. This is broadly true under regular concentration scales as long as the solution is well-mixed. The abundance of atoms allows the law of large numbers to hold. There are so many molecules that the probabilistic aspects of where any particular one is located averages out, macroscopically the solution is perfectly homogenous. For instance, a small portion taken of a 1 mol/L solution would also be at 1 mol/L.

An interesting aspect of Avogadro’s limit is that here the law of large numbers no longer applies. Calculations of moles or concentrations near or below Avogadro’s limit must consider probability. Under these conditions, finding a molecule of interest in a given volume of a solution is probabilistic. Not only this, there is even uncertainty in the concentration of a prepared solution, as long as it is made by serial dilution. I am unaware of any attempts to account for the probabilistic nature of this situation, and so I conclude this article by developing a numerical solution below.

Broadly, the probability of an event is given by the ratio of how many ways there are for the desired outcome to occur over the total number of possible outcomes. We can calculate the probability of the stated event by developing expressions for these two quantities. In doing so, it is helpful to divide the total solution into imaginary

We can evaluate the total number of possible outcomes from our random sample by noticing that this is equivalent to a classic combinatorics problem: sorting

An interesting aspect of Avogadro’s limit is that here the law of large numbers no longer applies. Calculations of moles or concentrations near or below Avogadro’s limit must consider probability. Under these conditions, finding a molecule of interest in a given volume of a solution is probabilistic. Not only this, there is even uncertainty in the concentration of a prepared solution, as long as it is made by serial dilution. I am unaware of any attempts to account for the probabilistic nature of this situation, and so I conclude this article by developing a numerical solution below.

*Suppose we have a solution of a molecule of interest at concentration C with volume VT. We withdraw a sample of volume VS. What is the probability of the sample containing nS molecules?*Broadly, the probability of an event is given by the ratio of how many ways there are for the desired outcome to occur over the total number of possible outcomes. We can calculate the probability of the stated event by developing expressions for these two quantities. In doing so, it is helpful to divide the total solution into imaginary

*Vs*-sized portions, with the molecules randomly distributed among them.We can evaluate the total number of possible outcomes from our random sample by noticing that this is equivalent to a classic combinatorics problem: sorting

*x*identical objects into*r*distinct groups. The number of possible outcomes is given by the following combination [5]:First, the number of objects is given by the total number of molecules (

*nT*) in the solution:The number of distinct groups is the number of imaginary

*Vs*-sized portions we divided the solution into:The total number of outcomes is given by:

Next, how many ways are there for our sample volume to specifically contain

*nS*molecules? This is actually an equivalent combinatorics problem to the above. However, now we must begin with our selected sample volume containing precisely*nS*molecules. This leaves*nS*fewer objects to arrange and one less group to distribute them among. We now have:Therefore, the total number of desired outcomes is given by:

Now we calculate the probability of our desired outcome by taking the ratio of the desired outcomes to all possible outcomes:

This equation robustly allows calculation of the probability of withdrawing

*nS*molecules from a solution of known concentration. To gain further insight on Avogadro’s limit, we can use it to evaluate the probability of our sample containing at least one molecule of interest. The probability equation simplifies greatly in the case of no molecules (*nS*= 0):It is now easy to calculate the probability of there being at least one molecule of interest in the sample through the complement:

To clarify the effect of concentration on the probability of a molecule of interest being present, the equation is plotted below for

*VT*=10 L and*VS*=1 L.At concentrations well above Avogadro’s limit (1.66 x 10^-24 mol/L) the probability of obtaining at least one molecule tends towards unity, but falls off quickly below the limit. The exact curve shape depends on both total and sample volumes, which together affect how much uncertainty there is in the outcome.