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If we take some aqueous solution and dilute it further and further, will the concentration of the solution ever get to zero? I would say no, simply because total dilution implies that all the molecules of the solute have literally disappeared. But, the fact that I am unable to figure out where the molecules have gone doesn't make my argument compelling at all. I am led to believe that there is a far better answer and/or explanation to my question

UPDATE: While it is true that this question closely resembles the linked one, the answer provided in this question is a lot better as it gives a much deeper insight into the dilution process.

Stephen Muga
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    Homeopathy does it all the time :) – Alchimista Aug 07 '20 at 18:45
  • Sorry, Alchimista; that's simply not true. Homeopathy not only recognises but to a large extent depends on the idea that no such thing could ever be possible. – Robbie Goodwin Aug 09 '20 at 00:45
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    @RobbieGoodwin You're taking the the logic of homeopathy at face value and using it to justify homeopathy when you say that, so of course that would be the conclusion. – DKNguyen Aug 09 '20 at 01:30
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    Of course you might find a bottle among others billions of billions containing 1 molecule of the diluted stuff. It does not classify as a solution of that particular stuff as for you can find whatever exist. I did not want to be too serious because the topic doesn't deserve. But there is a nice comment by @ Karl below. Also as a curiosity. Starting from a spoon of NaCl there is not enough water in the known universe to get a C 60 homeopathic solution in one step!!! – Alchimista Aug 09 '20 at 10:39
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    This is one of the questions where "In theory, there is no difference between theory and practice. In practice, there is." – cbeleites unhappy with SX Aug 10 '20 at 07:49
  • @DKNguyen I wasn't taking anything at any value… simply repeating one of the basic tenets of homeopathy. That doesn't, and I didn't mean to, justify anything. When you say "of course that would be the conclusion" what conclusion were you thinking of? – Robbie Goodwin Aug 14 '20 at 19:20

5 Answers5

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It depends how you dilute it. If you take an aqueous solution of A and just add pure water (absolutely 100% water), the concentration of A will never quite be null. In this case however, you will reach a point where the concentration of A is so small that it can be considered null for your applications.

If, however, you dilute the solution, take a sample, then dilute that sample (and so on), you could reach a concentration of exactly 0M. Imagine you have diluted the solution enough so that it contains exactly 1 molecule of A. When you take your sample for the next dilution, if this molecule isn't in the sample, the concentration will be exactly null. If it does happen to be in the sample, it could be left behind when you draw the next sample, or the next, and so on.

In practice though, the water you use for the dilution will likely contain impurities. You will maybe not achieve exactly 0M, but the concentration could be so small that it is undetectable and have no measurable consequence.

Raphaël
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    "...dilute the solution, take a sample, then dilute that sample (and so on), you could reach a concentration of exactly 0M." That's amazing. That's the type of solution I was looking for! – Stephen Muga Aug 07 '20 at 19:00
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    @KaraKirkland It's worth mentioning, explicitly, that the reason this works is because a solution is not a continuous medium—if it were, you could never get to zero. You can only get to zero because molecules are discrete. – theorist Aug 08 '20 at 01:42
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    I get that it wouldn't be in the spirit of the answer, but if you are doing serial dilution with the explicit aim of getting to zero concentration, you could just skip the part where you add any of the original solution to the solvent and you'll get there faster – Paul Aug 08 '20 at 03:05
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    @Paul But water have memory! Ask any homeopath! – d-b Aug 08 '20 at 13:27
  • @Raphaël I suggest it doesn't depend on "how you dilute it"; the difference between your examples is not final concentration but merely time taken; how pure the water solvent is has little to do with how much solute remains…

    "… so small it can be considered null…" is a judgement, not a technicality.

    From a dilute sample diluted further, you could no more reach a concentration of exactly 0M; just more quickly.

    How can you imagine real dilution to 1 molecule? Guess that 12, or 397 or however many dilutions do the trick?

     – Robbie Goodwin Aug 09 '20 at 01:04
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    @RobbieGoodwin if I have a box with 99 red balls in it and one yellow ball, the concentration of yellow balls is 1%. If I add a million red balls, the concentration is about 1 ppm. If I dump half of the contents of that box into another box, I will have one box with about 2 ppm yellow balls and another with exactly 0 ppm yellow balls. The same is true of molecules in solutions, they ARE different processes. – llama Aug 10 '20 at 21:11
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For an analytical chemist, the concept of zero concentration does not exist. The concentration cannot be exactly 0! Only a limit of detection can be developed in terms of statistics. This is why a senior respectable user here has written an entire monograph on this topic. Suppose you have a NaCl solution, and so called "pure water"*, matter how much you dilutions you perform, you cannot say with 100% confidence that there is no single ion sodium left in the solution now.

*Pure water cannot exist in any ordinary laboratory.

Karl
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AChem
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    There's plenty of room at the bottom, as they say, and I suppose this includes room for impurities. The lowest (indirectly) measured concentration I know in a condensed phase solution comes from a dark matter detection study, where the authors propose a 14 yM (that's yoctomolar) concentration of tritium in liquid xenon may have produced enough noise to mask signals from dark matter decay. Somehow it still remains after very stringent purification, and even though tritium itself is a trace radioisotope with modest half-life. – Nicolau Saker Neto Aug 07 '20 at 23:31
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    Very nice example. No reagent is 100% pure. – AChem Aug 07 '20 at 23:32
  • @NicolauSakerNeto unanswered in Physics SE: How will the possibility of tritium contamination decay be removed from XENONnT data? – uhoh Aug 08 '20 at 02:38
  • @uhoh, an unrelated question. Do you anything about echelle grating and cross dispersion. I want to ask a question about it but what about a relevant site? I have a echelle grating (for fun) and I want to see a 2D spectrum in a simple set up which could a simple cardboard box. – AChem Aug 08 '20 at 03:05
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    Searching, there are 10 posts in Astronomy SE and 8 in Physics SE that contain "echelle" and many are interesting. It looks like the ones in Physics SE indicate there is more of a chance of a "DIY-based answer" there but there are some observational astronomers active in Astronomy SE who might really enjoy answering this as long as you mention it might use it to analyze sunlight or something astronomical. – uhoh Aug 08 '20 at 03:16
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    Let us continue this discussion in chat. – uhoh Aug 08 '20 at 03:28
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    You need to go for larger molecules, which are also not ubiquitous in the environment. – Karl Aug 08 '20 at 06:42
  • If you are doing serial dilution, still there is a possibility that at least one single molecule is there. Hence the point that in analytical chemistry, zero concentration does not exist. Nobody can "measure" zero concentration. It has no legal meaning either. – AChem Aug 08 '20 at 06:49
  • Can you provide a reference to back up a general statement such as that made in your first sentence? It sounds like an opinion. I would argue that if you make your samples by diluting with a known buffer not containing the solute (the solute stock solution is the only source of solute), then the buffer would have zero concentration of the solute. Circular, but not controversial. – Buck Thorn Aug 08 '20 at 13:28
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    BTW, it is commonly known as a "blank" measurement. – Buck Thorn Aug 08 '20 at 14:10
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    Buck Thorn, there is a lovely thin book, "Two papers on the limit of detection of a complete analytical procedure" H. Kaiser. – AChem Aug 08 '20 at 14:24
  • @M.Farooq It seems your «Kaiser» (with «ai») is not the same the kayser units prior $\mbox{cm}^{-1}$ / wavenumbers in spectroscopy is named. – Buttonwood Aug 08 '20 at 16:15
  • No two different Kayser and Kaiser. H. Kaiser was at the Institute for Spectrochemistry Dortmund. That Kayser is quite old story! See Zum Problem der Nachweisegrenze in Z. Analyt. Chem. 1965, 209, 1. This book is the translation of his two famous papers. – AChem Aug 08 '20 at 16:18
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    @uhoh, I posted the query on echelle, if you want to check! – AChem Aug 09 '20 at 00:44
  • Thanks Farooq I'll have to take a look. – Buck Thorn Aug 09 '20 at 19:43
  • I'd have formulated a bit differently: it's not that it cannot exist - just that we cannot be sure that (where = in which bottle) we have a zero concentration. Then enter practical considerations like impurities and homogenization. The latter will make it particularly difficult to set a valid upper bound on the concentration one may actually have in one of the "zero conc" bottles. – cbeleites unhappy with SX Aug 10 '20 at 08:00
  • @M.Farooq We can detect the spectral lines from a single isolated atom. Would it not be possible, with a sensitive enough spectrometer, to detect the spectral lines from a single molecule in solution? – probably_someone Aug 10 '20 at 17:20
  • Yes, may be in principle and on a piece of paper, but tell me how would you be "100%" sure that there is a single atom in that solution and how would we practically prepare such a solution? – AChem Aug 10 '20 at 18:04
  • Comments are not intended for extended discussions. There is already a chat room associated with this post: https://chat.stackexchange.com/rooms/111555/discussion-between-uhoh-and-m-farooq Please use this room. – Martin - マーチン Aug 10 '20 at 18:13
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Unequivocally yes! If your solute is something of which you can identify a single molecule in a macroscopic solvent sample (let´s say a fluorescent dye), then you can dilute down to zero, and be sure about it.

(For the nitpickers: This can be done in a finite number of steps, if you split the solution in two after each step, and keep diluting the part which shows a lower concentration. Divide et impera.)

If you´re unable to measure the concentration down to actual zero, then the statement "this has been diluted to zero" is not falsifiable, therefore unscientific (see Karl Popper), and as such not worth pondering. ;)

Karl
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    I'm also a fan of Popper! (and Kuhn, and Lakatos). Anyways: If you knew the initial concentration, and the number of splits, then even if you couldn't measure the final concentration, you could make a scientific statement about its probability of being zero :). – theorist Aug 08 '20 at 01:39
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    I would take a minor issue with your last paragraph: just because you (the hypothetical scientist doing this) are unable to measure the concentration down to zero doesn't mean that the hypothesis that it's zero is unscientific. Maybe someone else can measure it, or people in the future will be able to measure it, or it has some physical consequence that you're not aware of, or so on. – David Z Aug 08 '20 at 02:36
  • @DavidZ Can you explain how any of those would make the statement more "scientific"? – Karl Aug 08 '20 at 10:58
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    @Karl Because if someone in the future, with better technology, could falsify it, then the statement is falsifiable. "There are zero molecules of X in this solution" is a testable prediction when you have a device precise enough to detect whether there is a single molecule of something in a solution. – probably_someone Aug 08 '20 at 18:36
  • @probably_someone Balderdash. A statement that might become correct in five or fifteen years is wrong today, period. – Karl Aug 08 '20 at 20:34
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    @Karl The statement is not "becoming correct", it is becoming measurable. Whether it's correct or not doesn't change. You can make correct statements that aren't measurable at the time (for example, Einstein in 1916: "Gravitational waves exist", or Fermi in 1934: "Neutrinos exist"). The fact that they weren't measurable at the time of proposal doesn't mean they weren't correct. – probably_someone Aug 08 '20 at 21:10
  • @probably_someone Your point being? Esp. in relation to the original topic? – Karl Aug 08 '20 at 21:44
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    @Karl I answered your question: a statement is "scientific" if it is falsifiable. Falsifiable means that someone could conceivably measure it, not that someone can do that right now. And falsifiability doesn't really have anything to do with correctness, so talking about a statement "becoming correct" isn't really relevant here. – probably_someone Aug 08 '20 at 22:17
  • @Karl For example, the falsifiable statement "sterile neutrinos with certain parameters exist" is just like the statement "there are zero molecules of X in this solution". If sterile neutrinos with those parameters exist, there's an identifiable experimental signature, and if they don't exist, we wouldn't see that signature. The fact that our detectors aren't precise enough to measure that signature yet is immaterial. Similarly, there are spectrographic consequences for having a single molecule of X in solution, and an experiment with a precise enough spectrometer would be able to detect that. – probably_someone Aug 08 '20 at 22:24
  • Let us continue this discussion in chat. – Karl Aug 09 '20 at 11:56
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Something like a zero concentration is achieved in the vapor phase in some materials. In this answer the triple-point vapor pressure of gallium is identified as so low that a measurement thereof defaults to zero with a high probability.

Oscar Lanzi
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Hypothetically, I think limitation for zero-dilution would be when you take a sample of diluted sample the possibility of having a solute molecule in there should be zero. To achieve this goal, the concentration of your solution should be less than $10^{-24}$ in power $\left(\frac{1}{6.022 \times 10^{23}}=1.66 \times 10^{-24}\right)$. To do so, if you have $\pu{1 M}$ of solution, you should add another $\pu{1.66 \times 10^{24} L}$ of water to get that dilution. Considering the whole water body of the Earth is about $\pu{1.3859 \times 10^{21} L}$ ($\pu{2.5511 \times 10^6 mile^3}$ according to USGS) you may run out of water before you get right concentration.:-)

That's why Raphaël has said its depends on how you want to do it. So, you may use the procedure called serieal dilution.

Mathew Mahindaratne
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    If you sample substantially less than a litre (say, 1 µL), then the concentration can be proportionately higher and you can still statistically expect to find zero molecules of the solute in some samplings. – Nicolau Saker Neto Aug 07 '20 at 22:29