# How does the relative valency of the impurity and host metals affect solubility in a substitutional solution?

I have a book that claims:

A metal will have more of a tendency to dissolve another metal of higher valency than one of a lower valency.

I interpret this as saying that a metal with higher valency will dissolve more easily into a host metal than it would if it had a lower valency. For example, $\textrm{Fe}^{3+}$ as impurity atoms and $\textrm{Na}^{+}$ as host will work better than vice versa.

But then in the summary of the chapter it says:

A valence that is the same as or less than the host material will have more appreciable solubility.

Which of the two statements is correct?

• Which book is "my book?" It can be helpful to know the source. – starrise Dec 12 '15 at 18:01

## Summary

Your book is correct. The same valence (+3/+3 solvent/solute) will be most likely to have large solubility, while higher valences (+3/+4) will be more likely to have high solubility than lower valences (+3/+2).

Putting them in order as an example:

Solvent / Solute

+3  /  +3       Highest Likelihood of Solubility
+3  /  +4
+3  /  +2       Lowest Likelihood of Solubility


## Explanation

The Hume-Rothery rules for solubility are necessary, but not sufficient, conditions for complete solid-solution solubility. That means if any of the rules are violated, complete solubility does not occur. However, even if all of the rules are met, complete solubility is not guaranteed. Mixing always occurs due to entropy, though possibly in infinitesimally small ratios.

For posterity, the rules are as follows:

1. Difference of less than 15% in the ratio of the radii of the solute and the solvent, so that

$$\left( \frac{\left|r_{solute} - r_{solvent}\right|}{r_{solvent}} \right)\leq 0.15$$

1. There must be the same or similar crystal structures between the solute and solvent.

2. Complete solubility occurs when the solvent and solute have the same valency. A metal will be more likely to dissolve a solute of higher valency than a solute of lower valency.

3. The solvent and solute should have similar electronegativity. Intermetallics tend to form when the difference is large.

The specific part you are interested in is in bold.

These rules, more or less stated the same way, can be found from several sources, including The Wiki and The University of Cambridge Department of Material Science and Metallurgy. Note that the Cambridge link states the opposite of the bolded portion of rule 3. The Cambridge link is possibly incorrect, but it is difficult to tell based on the literature whether or not rule 3 is even useful.

A thorough review of the literature surrounding the rules can be found at this link. The paper is unfortunately downloadable in Word Document form, but seems to be clean. A brief summary of the relevant part is that the relative valence factor rule (rule 3), specifically the bolded part, seems to only be valid when monovalent Cu, Ag, and Au are alloyed with B-subgroup elements of higher valence (using the Old IUPAC nomenclature, so the right-hand-block of the periodic table). The explanation involves Fermi surface and Brillouin zone interactions in the B-subgroup elements. What I take away from the review is that this particular part of rule 3 is not terribly useful in practice.

• But then 'appreciable solubility is possible only... (other 3 rules).. and when impurity atoms have a valence that is the same as or less than the host material' is wrong? You say both are correct, but then you state that if the solute has a lower valence than the solvent it dissolves less than if it would have had a higher valence than the solvent. Your example does not stroke with the link you gave of cambridge university? Or what am I missing? So what I mean is, what you have in bold agrees with your example but the summary in my book and the piece in the link does not? – strateeg32 Dec 12 '15 at 17:47
• After researching this further, there seems to be a lot of confusion about what the rules actually state. The original research was published in 1935 and in a German publication in German. It might be helpful if we were able to see and understand that work. Unfortunately I don't understand German, though I may be able to get a copy of the publication. – starrise Dec 12 '15 at 17:55
• ''A metal will be more likely to dissolve a solute of higher valency than a solute of lower valency. '' This bold part in your answer is the complete opposite of ''A metal with a lower valency is more likely to dissolve in one which has a higher valency, than vice versa'', which is stated in the link to cambridge source. This same contradiction is also in my book. – strateeg32 Dec 12 '15 at 17:56
• Ah okey, so it isn't just me? – strateeg32 Dec 12 '15 at 17:57
• Oh I'm sorry I misread part of your question. Let me correct my answer. Also no, it isn't just you. Now I wish I had access to the original publication. Things tend to get lost and changed when repeated without question over the course of 80 years. – starrise Dec 12 '15 at 18:00