Lesson 4b: the ZdanovskiiStokesRobinson relationship 
1^{st}, 2^{nd} and 3^{rd} Calculations 

Note: the above should be entered on the
"simple" calculations page of Model III

No.  moles NH4^{+}  moles Na^{+}  moles NO3^{−}  moles H2O  mNO3^{−} 
1  1.0  0  1.0  5.52  10.05 
2  0  1.0  1.0  6.966  7.969 
3  1.0  1.0  2.0  12.47  8.905 
We saw in Lesson 4a that the water uptake of the mixture (12.467 moles), which contained 2 moles of salt, differed significantly from that of the same amount of NH4NO3 (2 × 5.5234 = 11.0468 moles). However, we can see from the table above that the amount of water in the mixture is very close to the sum of the water contents of the two single salt solutions at the same relative humidity. Zdanovskii published this relationship in 1948 for salt solutions, and Stokes and Robinson in 1966 for nonsalt solutions, so this is called the ZdanovskiiStokesRobinson or "ZSR" relation. For an arbitrary number of solutes it is written as follows:
(m_{1} / m_{0,1}) + (m_{2} / m_{0,2}) + (m_{3} / m_{0,3}) + ... = 1.0
where each m_{i} is the molality of solute "i" in the mixture, and m_{0,i} the molality of solute "i" in a solution containing only "i" at the same water activity (equilibrium relative humidity) as the mixture. The value of m_{i} is equal to the number of moles of solute "i" (n_{i}) in the mixture, divided by the number of kg of water present. Thus, replacing all m_{i} by n_{i} in the above equation the right hand side becomes equal to the number of kg of water in the mixed solution.
The relationship above is similar to Amagat's law for gas mixtures. ZSR is used in the EAIM models to calculate a first guess for the water content of mixed solutions, but the final answer is calculated more accurately by other techniques. Nevertheless, for mixtures of compounds prevalent in the atmosphere, ZSR provides an excellent estimate of the water content providing the single solution molalities are known as a function of water activity. The ZSR relationship is used in several atmospheric chemistry models.
You should now review the conclusions on the main page for this lesson.