[10042]

Hi, I've been researching Alkali Halides and their heat of solution when they are dissolved in water. I have just recently noticed that the trend is not what i expected. As can be seen in the attached image, the Enthalpy of solution of sodium halides decrease as you go down the periodic table, whilst, the enthalpy goes up for potassium and rubidium halides. I've been looking into things like kosmotropes and chaotropes but I'm having a hard time comprehending. Any help would be appreciated! Thanks!

[Enthalpy]

The figure suggests that it's a matter of compared ionic radii.

[Corribus]

Could you describe better what the x-axis is? It's the difference between the radius of the cation and the radius of the anion?

[10042]

Thanks for the reply. Well according to the book that I took the image from, it says that the x axis refers to the difference between Pauling radii of the cation and anion. I have also attached the Pauling radii that they gave.

[mjc123]

I have just recently noticed that the trend is not what i expected.

What trend did you expect, and why?

I've been looking into things like kosmotropes and chaotropes but I'm having a hard time comprehending

I wouldn't bother with stuff like that. It can be understood in much simpler terms.
What factors contribute to the heat of solution? (Draw a Hess's Law cycle.) How do these vary with ionic radius?

[10042]

I have just recently noticed that the trend is not what i expected.

What trend did you expect, and why?

I expected that as the anions went from Chloride to Iodide, the heat of solution would get more exothermic because the energy of formation is becoming less each time. In saying that though, the hydration enthalpy would also become less with anions like iodide. I just feel as though I am either overthinking it extremely or there is something more deep, because potassium and sodium have opposite trends.

I've been looking into things like kosmotropes and chaotropes but I'm having a hard time comprehending
I wouldn't bother with stuff like that. It can be understood in much simpler terms.
What factors contribute to the heat of solution? (Draw a Hess's Law cycle.) How do these vary with ionic radius?

So the factors affecting heat of solution are the formation energy and hydration energy. Since the formation energy is a positive value, as the ionic radii increases, the number will be smaller. As for hydration, the value is negative, and it will become more positive when the radii is increased. I think I am correct on this, am I not?

[mjc123]

"Formation energy" of what? The salt? The ions? Why do you think is it positive?
The enthalpy of hydration is actually positive for the halide ions except fluoride. (I assume this is because the ion disrupts the existing hydrogen bonded structure of water.) You are right that it becomes more positive with increasing radius. (Note: I have just found another set of ΔH_hyd values where the cation and anion values are much more similar, and all negative. But the slopes - see 1st and 2nd graphs below - are virtually the same, which is what matters. Since we can only ever measure a cation and anion together, to get individual values we need to define some arbitrary zero, and evidently different people have done it differently.)
This is quite vague. Did you do a Hess's law cycle?
There are various ways we can do this, e.g by heats of formation:
MX(s)  M + 1/2 X₂  M⁺(aq) + X^-(aq)
ΔH = ΔH_f(M⁺(aq)) + ΔH_f(X^-(aq)) - ΔH_f(MX(s))
But "formation" is not a simple process that can be easily compared, e.g. you will get the effect of the different states of the elemental halogens. It is more useful to compare simple processes involving only the ions:
MX(s)  M⁺(g) + X^-(g)  M⁺(aq) + X^-(aq)
ΔH = U(MX) + ΔH_hyd(M⁺(g)) + ΔH_hyd(X^-(g)) where U is the lattice energy of crystalline MX.
It is relatively easy to compare trends in lattice and hydration energies, but you should bear in mind
(i) The heat of solution is generally a small difference between large quantities; a relatively small uncertainty in a lattice or hydration energy value can mean a relatively large uncertainty in the heat of solution. Likewise a relatively small departure from a linear trend in U or ΔH_hyd may make ΔH_sol look quite non-linear.
(ii) Ionic radii are not well defined - all we actually have is interatomic distances, and people differ as to how to divide these between the ions. See for example the "crystal radii" and "effective radii" (equivalent to Pauling's) in https://en.wikipedia.org/wiki/Ionic_radius. In any case, the hard-sphere ionic model is only an approximation. (In what follows I have used "crystal radii", but the principle is the same whichever set you use, only the numbers are different.)

First, qualitatively, the heats of hydration of cations and anions become more positive (or less negative) with increasing ionic radius. The lattice energies also become less positive with increasing radius, but (comparing a series of halides of the same metal) the decrease with anion radius is greater the smaller the cation; in fact for Li it is greater than the increase in anion hydration energy, but for K and Ru it is less, while it is about the same for Na. Hence you get the observed qualitative trends - for Li, solution energy decreases with increasing anion radius, while for Ru it increases, and for Na the values are all fairly similar. But can we be a bit more quantitative about this, and in particular explain the non-linear trends?

Hydration energy, as we have said, increases with ionic radius. In fact we find that for both anions and cations we get good straight line plots if we plot hydration energy versus the reciprocal of the ionic radius (see first graph below). The lattice energy depends on both the cation and anion radii; in fact if you remember the Kapustinskii equation it goes as 1/(r₊+r_-), and if we plot this we find that all the values fall on the same straight line. So if we add these equations we get an estimate for the heat of solution of the form:
ΔH_sol = A + B/(r₊+r_-) - C/r₊ - D/r_- where A, B, C and D are constants.
(Can you sketch the form of this curve if you vary r_- while keeping r₊ constant?)

If we differentiate by r_-, we find that this curve has a maximum at r_- = r₊/(sqrt(B/D)-1). This suggests that a good choice for the x axis would be the radius ratio r_-/r₊, as we should see the maxima coming at the same x value for all the metals, and this graph is shown below. We see there is a pretty good, though not perfect, fit. It's not so good for Cs, but maybe that's because the Cs salts (apart from CsF) have a different crystal structure. Still, this simple model gives us fairly good estimates of the actual values, accounts for the non-linear trends by having a term that varies inversely with anion radius and another that varies inversely with the sum of the radii, and shows that the apparently different qualitative trends (e.g. of the Li halides and the Ru halides) are part of the same overall picture - it just depends where you are relative to the curve maximum. (This is why the plot versus radius ratio is more useful than your plot versus radius difference.)

[Enthalpy]

Thanks mjc123, nicely done!

Meditating about the heat of hydration of M+ and X-, whose sum is measurable but not the individual ones...

Could we at least take a hydrogen electrode as a reference that is already known for other purposes, and use it for this purpose as well? Because we can measure energies:
- From metallic M to gaseous M atoms
- From gaseous M atoms to M+ ions in vacuum
- And from solid M to hydrated M+, and least if referring to the hydrogen electrode
By closing this loop of transformations, we would have a value from metallic M to hydrated M+. It's still conventional, but at least the convention would be consistent with an existing one.

And what would an endothermic hydration mean for X-?

That the X- ions spontaneously jump out of the water at supersonic speed until the solution is so positively charged that it attracts them back? It doubt that, since we would have a measurable (indirectly, through an electron beam for instance) potential between the solution and an electrode above it, wouldn't we?

I just intuite - not formally, not safely - that the hydration is exothermic both for X- and M+, and that measures other than thermochemical can in principle observe it, but that 400kJ/mol make the ions over the solution too scarce for an easy measure.

Maybe by thermoionic emission. If X- or M+ is the charged species that leaves most easily the solution towards the atmosphere, and the gas doesn't meddle in, then the voltage-to-current and temperature-to-current relations would give an extraction energy. It's just that at +100°C the current is tiny.

[mjc123]

I got my original hydration energies by comparing the heats of formation of the gaseous and aqueous ions given in Johnson, Thermodynamic Aspects. The baseline is ΔH_f ≡ 0 for H⁺(aq). ΔH_f for H⁺(g) is 1536 kJ/mol, which is clearly half the bond enthalpy of H₂ plus the ionisation energy of H. This would imply ΔH_hyd = -1536 kJ/mol. Atkins, however, says "there is some agreement [doesn't say how it's measured or calculated] that its value is about -1090 kJ/mol" and gets my second set of hydration energies which, I agree with you, look more plausible. I think the problem for single ions is that (e.g.) 1536 kJ/mol is the heat of formation of H⁺ plus an electron, and what happens to the electron on hydration? Does it go into solution - then what is its hydration energy? Does it not - then do you get a positively charged solution and a negatively charged gas? Of course for a real compound, cation and anion, it cancels out.