I get negative value under the root because ln(a

_{liq}) is negative when calculated with the following formula:

ln(a

_{liq})=-(v*m)*φ/55.51

The link of this formula:

https://en.wikipedia.org/wiki/Activity_coefficient#Theoretical_calculation_of_activity_coefficientsWhere v is the number of ions produced from dissociation of one molecule of the dissolved salt, m is the molality of the salt dissolved in water, φ is the osmotic coefficient of water, and the constant 55.51 represents the molality of water.

since none of the constants (v, m or φ) will never be negative, the value of ln(a

_{liq}) will always be negative by definition.

This is a calculation for one of my results:

There is a concentration of 1.084mol dm

^{-3} of BaCl

_{2} in water.

Finding ln(a

_{liq}):

v=3

m=1.084

φ=1 (i don't know the value of the osmotic constant so i assume it is 1)

With these values:

ln(a

_{liq})=-(3*1.084*1)/55.51=-0.0586

Knowing ln(a

_{liq}) we can apply the freezing point depression formula by Ge and Wang (this formula can be found here:

https://en.wikipedia.org/wiki/Freezing-point_depression#Calculation)

The rest of the constants are:

ΔH

^{fus}_{Tf}=333.6 J/g

R=8.314 J/(K*mol)

T

_{f}=273.15

ΔC

^{fus}_{p}=2.11 J/(g*K)

Plugging these values in the root we get:

[2*ΔC

^{fus}_{p}*T

_{f}^{2}*R*ln(a

_{liq})+(ΔH

^{fus}_{Tf})

^{2}]

^{(1/2)}=[2*2.11*273.15

^{2}*8.314*(-0.0586)+333.6

^{2}]

^{(1/2)}=[-42110.03]

^{(1/2)}This is a negative value and therefore the formula can't work for these values.

The only possible solution I think is that ΔC

^{fus}_{p} is a negative value. Therefore i retried the calculation plugging the values in the full formula, with ΔC

^{fus}_{p}=-2.11 J/(g*K) and it works. The value i get is -65.54K (this is how much the freezing point is depressed in that solution).

This value however is very far from the experimental value which is about -6K freezing point depression.

From here I think the only thing that could change the theoretical value is the osmotic coefficient (φ) which i do not know the values for.

However possibly i am doing something else wrong when calculating, or does anybody maybe know other possible reasons why the theoretical value is so different from the experimental value?

Hope this makes it more clear

Alessandro