Predict melting point of water CaCl2 solution

Question

I'm doing a science project for my school. I experimentally measured the melting point of water-salt solutions, and I want to compare my results to theoretical melting points, but I encountered a problem.

I used this equation to calculate the melting point depression: $\Delta T = b \cdot i \cdot K_\mathrm{f}$. I also compared my data to a phase diagram. I expected my measured melting point would be on the liquid line. But these two methods give me a different prediction. A phase diagram tells me that $\ce{CaCl2}$ would be much more effective at lowering the freezing point of water while the above equation tells me otherwise.

Here is a comparison: Purple indicates the melting point calculated with the equation. The black line is the liquid line on the phase diagram.

Which method is correct and why is there such a big difference?

(I didn't encounter this problem with $\ce{NaCl}$ solutions, the two methods gave more or less the same predictions)

EDIT: my math was wrong on the graph i posted, here is the corrected graph:

Answer 1

I think there are three issues here. The first is that your purple line of calculating the freezing point doesn't appear to be correct. Let's arbitrarily select $b=1 ~ \mathrm{mol/kg}$ to compare. $i=3$ and $K_\mathrm{f}=1.86 ~ \mathrm{^{o}C/mol}$. The molecular weight of $\ce{CaCl2}$ is $111 ~ \mathrm{g/mol}$. Then, our mass concentration is $b·\mathrm{MW}/(b·\mathrm{MW}+1 ~ \mathrm{kg})=0.1 ~ \mathrm{kg}_\mathrm{salt}/\mathrm{kg}_\mathrm{total}$. The freezing point is $b·i·K_\mathrm{f}=1·3·1.86=5.58 ~ \mathrm{^{o}C}$. This is much closer to the red/black line than the purple one.

The second problem is that that relation is only valid at low concentrations. As the concentration of salt goes up, it has an increasing effect on the activity of the water. As we can see from the phase diagram, once you get to a 6:1 mol ratio of $\ce{Ca^{2+}}$ to $\ce{H2O}$, the interaction is extremely strong. I'm just speculating here, but I would guess six water molecules are forming a weak octahedral coordination complex with the $\ce{Ca^{2+}}$, with the electronegative oxygen ends oriented toward the ion. This effect becomes so strong, that the hydrate becomes the relevant element for consideration above that concentration.

Third, you're ignoring the existence of hydrates. If you find the melting points and the cryoscopic constants of the hydrates, you can actually calculate the right side of the plot the same way.