Lesson 4b: the Zdanovskii-Stokes-Robinson relationship


The Zdanovskii-Stokes-Robinson relationship, which relates the water content of a mixture to that of the individual solutes at the same water activity, is demonstrated.

Entering the Data

Calculations for three different ionic compositons at 80% RH are now required, using the same data input page as for the previous part of this lesson. Record the total water content of the solutions, and the ionic molalities, in each case. The calculations are specified below:

1st, 2nd and 3rd Calculations
  1. Enter the values and select the options under the following headings:

    Ambient Conditions
    Relative humidity = 0.80 (i.e., 80%).

    Ionic Composition in Moles
    1.   Ammonium = 1.0, Nitrate = 1.0.
    2.   Sodium = 1.0, Nitrate = 1.0.
    3.   Ammonium = 1.0, Sodium = 1.0, Nitrate = 2.0.

    Solid Phases
    There are no entries under this heading.

  2. Click on the "Run" button at the end of the page to do the calculation.
Note:  the above should be entered on the "simple" calculations page of Model III (http://www.aim.env.uea.ac.uk/aim/model3/model3a.php).

Interpreting the Results

You should now have the following set of calculated quantities:

 No.  moles NH4+   moles Na+   moles NO3   moles H2  mNO3 

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 non-salt solutions, so this is called the Zdanovskii-Stokes-Robinson or "ZSR" relation. For an arbitrary number of solutes it is written as follows:

(m1 / m0,1) + (m2 / m0,2) + (m3 / m0,3) + ... = 1.0

where each mi is the molality of solute "i" in the mixture, and m0,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 mi is equal to the number of moles of solute "i" (ni) in the mixture, divided by the number of kg of water present. Thus, replacing all mi by ni 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 E-AIM 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.