"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 ΔHhyd
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:
M + 1/2 X2
(aq) + X-
ΔH = ΔHf
(aq)) + ΔHf
(aq)) - ΔHf
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:
(g) + X-
(aq) + X-
ΔH = U(MX) + ΔHhyd
(g)) + ΔHhyd
(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 ΔHhyd
may make ΔHsol
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+
), 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:
= A + B/(r+
) - C/r+
where A, B, C and D are constants.
(Can you sketch the form of this curve if you vary r-
while keeping r+
If we differentiate by r-
, we find that this curve has a maximum at r-
/(sqrt(B/D)-1). This suggests that a good choice for the x axis would be the radius ratio 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.)