# Calculating the density of a magnesium sulphate solution

I am trying to calculate the density of a magnesium sulphate solution and came across this paper which determines an equation from experiment to do that. However, I am having trouble understanding the paper, as parameters A, B, C and D are mentioned but I don't understand how to calculate these from Table 2 in the paper. Can someone please explain how this works?

EDIT:

I am adding the data from the above paper:

The paper mentions the equation

to calculate the relative density where the parameters A, B, C and D are said to come from the table. m stands for the molality (I assume mol/kg) t is temperature (°C) and d is in g/cm^3. As you can see there is no mention of A, B, C or D in the table, leading to the confusion.

• Unfortunately behind the paywall... – Poutnik Nov 15 '20 at 9:24
• Sorry I didn't notice but I still seem to have some kind of access from my university. But there are ways around the paywall – hanai Nov 15 '20 at 9:31
• I did not get a clear answer on the SE policies when googling but I'd also prefer not to post potentially illegal links or copyrighted material. – hanai Nov 15 '20 at 9:39
• Note that it seems there is an error in the exponent in the formula ( 2/3 versus 3/2 ). Rather check it within the paper context and/or the calculated values. – Poutnik Nov 15 '20 at 12:33
• Mistake from my side. corrected – hanai Nov 15 '20 at 16:05

Each of parameters A-D is a polynomial function of t:

$$A = \text{"m"} + \text{"mt"}.t + \text{"mt2"}.t^2 + \text{"mt3"}.t^3 + \text{" mt4"}.t^4$$

$$B = \text{"m3/2“} + \text{"m3/2t"}.t + \text{"m3/2t2"}.t^2$$

$$C = \text{"m2“} + \text{"m2t"}.t$$

$$D= \text{"m5/2"}$$

The terms in quote means the equally denoted numerical values from the provided table.

The complete final function $$1000 \cdot \Delta \rho=f(m,t)$$ has for each term $$a_{b,c} \cdot m^a \cdot t^b$$ the value for $$a_{b,c}$$ equal to the particular table value named "$$m^bt^c$$".

So one can the complicated function use in 2 ways:

1. The full blown function $$1000 \cdot \Delta \rho=f(m,t) = \sum {\left( a_{b,c} \cdot m^a \cdot t^b \right)}$$ without the need of coefficients A-D.
2. Temperature specific $$1000 \cdot \Delta \rho=A \cdot m + B \cdot m^{3/2} + C \cdot m^2 + D \cdot m^{5/2}$$ with precalculated A-D values for given temperature.

BTW, it looks like a very interesting numerical approximation. Maybe even better could be rational functions like Padé approximant, that have smaller deviations and better extrapolation behaviour, compared to polynomials.