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I just read:

In $\ce{NaCl}$, atoms don't aggregate so as to form discrete molecules but they are held together in a network structure. The ratio of the $\ce{Na}$ to $\ce{Cl}$ atoms in a sodium chloride crystal is $1:1$.

Here I have a $\ce{NaCl}$ crystal. Assume that I can break this crystal in all possible ways. So that I have its two pieces every time with a different mass ratio. Would these pieces always contain even number of atoms of $\ce{Na}$ and $\ce{Cl}$. Or, is this ratio going to be same for all conditions?

Suppose I have a $\ce{NaCl}$ crystal of mass $a$. I break it into two pieces of masses $x$ and $y$ such that $x,y<a$ and $x+y=a$. $a$ is a constant by its definition whereas $x,y$ are variables. Then, is the following true?

$$R(x)=R(y)=1:1$$

Where $R(x)$ and $R(y)$ denotes the ratio of $\ce{Na}$ to $\ce{Cl}$ atoms for all values of $x$ and $y$ respectively.

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    $\begingroup$ So that's what happens when mathematician ask about chemistry ;) 1:1 means that there's no significant inherent tendency for non-stoichiometry. Macroscopic crystals are never ideal. $\endgroup$ – Mithoron Dec 12 '15 at 18:19
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    $\begingroup$ :) I don't know what you mean by ideality of crystal. $\endgroup$ – Sufyan Naeem Dec 12 '15 at 18:49
  • $\begingroup$ They are never pure and undefected $\endgroup$ – Mithoron Dec 12 '15 at 19:10
  • $\begingroup$ What if I assume it ideal? Then, there must be an answer to my question. $\endgroup$ – Sufyan Naeem Dec 12 '15 at 19:15
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    $\begingroup$ I removed the answer for now since I need to elaborate on some stuff and address your issues comprehensively, hopefully. $\endgroup$ – M.A.R. Dec 12 '15 at 19:41
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Tl;dr: Theoretically yes, practically no.


Needless to say this is a very mathematical question that almost calls for a philosophical answer. The key point to consider is usually the crystal’s surface. Here, there are most defects and the least ideal structure since there is at least one row of atoms which are lacking partners on one side. But I’ll keep that out of my reasoning for a second.

Consider an ideal crystal and a Subtle Knife. You can use the knife’s almost infinitely thin blade to cut exactly between two sheets of atoms. The initial crystal had a $1:1$ distribution, every single sheet has a $1:1$ distribution and thus so will your two bits.

Then consider cutting one of the fragments again, but three atom rows away from where you cut it before. If our ideal crystal is infinitely large, we still get a $1:1$ distribution. Now continue to consider cutting the fragments into smaller fragments until you arrive at a theoretical body of $3 \cdot 3$ size. This cube has to violate the $1:1$ condition. It will be charged depending on which atom type is present in the corners. (The same is true for any other cube of an odd side length when measured in atoms.) However, we have a charged particle (consisting of nine atoms) now. It will attrackt particles of the other charge and pull them towards it. In the long run, one of the following may happen:

  • An ion of the opposite charge is drawn towards the crystal and attaches to it or inserts into it.

  • An ion of the extraneous charge is removed.

Both will make us regain our $1:1$ crystal — but it is no longer ideal. However, see below for the evolution of a more realistic view.


Now let’s consider a slightly more realistic case. We no longer have an infinitely large ideal crystal but one with a finite size. Let’s have a cube of even atom length just for simplicity ($1:1$ ratio). And because we want to see our cube, let’s assume a macroscopic size, e.g. a cubic centimetre. Now cut it with your Subtle Knife so that the even side length is split into to cuboids with one odd length. One must have a $+1$ charge, the other must have $-1$. Thus, neither have $1:1$.

One could now say that maybe a chloride ion ‘jumps’ to the other side or something, again creating a non-ideal crystal. Or maybe moisture from the air supplies one side with a proton and the other with a hydroxide. But we could also say that we have a charge of $+1$ and one of $-1$ each delocalised over some $10^{20}$ atoms — no big deal if you think of it.


And let’s go a step further and consider a real crystal. It’s going to have defects, impurities, non-aligned domains and a surface. If it ever had a mathematically exact $1:1$ ratio, then that was purely by chance. We can still delocalise the effective charge over billions of billions of atoms. We would need an extensive charge in a very small area to cause a problem at all.

Even worse if our crystal is put into a saturated $\ce{NaCl}$ solution. (Saturated, because I don’t want it to dissolve.) All the time ions will be entering and leaving solvation in a non-coordinated manner. Both random in time and random in location. The numeric ratio of atoms will change all the time. If we consider a very sub-saturated solution, of course there will be more ions dissolving into solution than recrystallising. But also, those that end up in solution also dissociate from each other. Thus, there is nothing actually keeping an atom of a charge of one (or two, or three, see other metals) close to its counter-ion — charge dissociation is not as bad as it seems on the paper.

So real world crystals will only have the ideal stoichiometry in a short-lived transition state between being too positive and being too negative — and being philosophical I could say that that almost be like a person’s mood.

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