# Freezing point depression of salt solutions

I have recently come across this particular equation which is used to estimate the freezing point depression of a salt solution:

$$\theta_\mathrm{B} = \frac{\Delta H^\mathrm{fus}_{0,T_\mathrm{F}} - 2RT_\mathrm{F}\ln (a_\mathrm{liq}) - \sqrt{ 2\Delta C^\mathrm{fus}_{P}T^2_\mathrm{F}R\ln (a_\mathrm{liq})+(\Delta H^\mathrm{fus}_{0,T_\mathrm{F}})^2 }}{2\left(\Delta H^\mathrm{fus}_{0,T_\mathrm{F}}/T_\mathrm{F} + 0.5\Delta C_P^\mathrm{fus} - R\ln (a_\mathrm{liq})\right)}$$

I have read the research paper of the authors that have published this equation, and apparently they have been able to obtain results with it. It should be noted that the solvent I am using is water and the solute is $$\ce{NaCl}$$, and the only value in the equation that I don't already have a value for is $$\ln (a_\mathrm{liq})$$.

Regardless of my difficulty in finding a value for $$\ln (a_\mathrm{liq})$$, when I plot a graph of this equation I observed that the domain (range of value $$a$$) is only within $$0.958 - 1$$, freezing point depression equals $$0$$ when $$a = 1$$ suggests that that is when the solution is pure water; the range of the function is approximately $$0 - 60$$.

Given that $$a$$ is supposedly the "activity of solvent in the solution" as quoted from the research paper, I find it extremely doubtful that precise values of $$a$$ can be calculated for various salts and that it would have a close estimation of experimental freezing point depression values.

I know of a way to calculate the activity of the salt in the solution, but not of the water. Can I get some advice on whether this equation is applicable?

• The wikipedia page you quote says "The solvent activity can be calculated from the Pitzer model or modified TCPC model, which typically requires 3 adjustable parameters." You can also measure the osmotic pressure of the solution, or the water vapor pressure, but you said you wanted to calculate it. For binary water NaCl mixtures, tabulated activities or activity coefficients can be readily found through a quick internet search. – Karsten Theis Feb 6 at 12:44
• @TryHard I have edited the equation, thanks for spotting that. – Goldsphere Feb 8 at 15:56
• @KarstenTheis I am aware that the solvent activity can be calculated from the osmotic coefficient, and the osmotic coefficient can be calculated from the Pitzer equations for various salts. However, as mentioned in the question, the range of the value (a) is extremely small (0.958-1) in order for the equation to achieve results. Also note that the solvent activity is not the same as the activity of the solution, I have tried using the activity coefficients from this website: kayelaby.npl.co.uk/chemistry/3_9/3_9_6.html, where the activity of 1molal NaCl is 0.657 (not in domain) – Goldsphere Feb 8 at 16:13
• I think $a_{liq}$ is the activity of water. So for example, for a 7% NaCl solution in water, the water activity is 0.96, i.e. within the domain. Source – Karsten Theis Feb 8 at 18:07
• @KarstenTheis I don't think that is the case, when I enter the corresponding values for 1M NaCl, I get a freezing point depression of 26.8 degrees celcius, which is unrealistic. I will try to get my hands on the research paper concerning pitzer coefficients for various salts and try them out to see if I get results. Thank you for your help so far and I will edit this post if I am able to achieve results. – Goldsphere Feb 9 at 8:15