Lesson 1d: properties over a range of RH

### Content

The graph plotting capabilities of the E-AIM model are used here to illustrate the variation of thermodynamic properties of aqueous NH4NO3 and HNO3 particles over a wide range of relative humidity.

### Part 1

Earlier parts of this lesson have shown that as relative humidity decreases the concentration of the dissolved electrolyte increases and the amount of water in the particles decreases. We will now extend what we have learned about aqueous NH4NO3 HNO3 particles at individual relative humidities with calculations that show how water uptake and activity coefficients vary continuously.

Select this link to open the data input page for "variable relative humidity, or total water" calculations using Model III (http://www.aim.env.uea.ac.uk/aim/model3/mod3rhw.php). This page allows you to vary the relative humidity over a range and produce results in a number of formats, including graphs. Fill in the input form as follows:

 1st Calculation Select "Graph" as the form of output, and then enter the values and options under the following headings: Ambient Conditions Select "Relative humidity" as the variable. Start Value = 0.65 (i.e., 65%), End Value = 0.99 (i.e., 99%), Number of points = 60. Ionic Composition in Moles Ammonium = 1.0, Nitrate = 1.0. Trace Gases Click on "HNO3" and "NH3" to prevent these gases being partitioned into the vapour phase. Solid Phases There are no entries under this heading. Click on the "Run" button at the end of the page to do the calculation. Note:  the above should be entered on the "variable relative humidity or total water" parametric calculations page of Model III (http://www.aim.env.uea.ac.uk/aim/model3/mod3rhw.php).

#### Viewing and Interpreting the Results

A page will appear which enables you to plot various quantities against each other, by choosing the X and Y variables, their ranges, and scales (log10 or linear) from the lists on the left hand side. Instructions and details of the variables are given in the right frame.

First, plot the liquid water content of the particle as a function of relative humidity by following the instructions in the table below.

 1st Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "moles of H2O(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

The water content of the particle rises smoothly, and the slope becomes very steep as the relative humidity approaches 1.0 (100%). This is because a vapour phase at 100% relative humidity is in equilibrium with pure water or, put another way, a solution at infinite dilution. Thus the moles of liquid water associated with an amount of soluble electrolyte tends to infinity as relative humidity tends to unity. Notice that the liquid water content at 95% relative humidity is about 30 moles.

Now plot the molality of the NH4+(aq) ion in the aqueous phase by making the selections below:

 2nd Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "mNH4+(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

The molality is, as we saw before, highest at low relative humidity (22.49 mol kg-1 at 65%), and declines smoothly to zero at a 1.0 relative humidity (100%).

Next, plot the activity coefficient of NH4+(aq) against relative humidity:

 3rd Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "fNH4+(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

The value of the activity coefficient declines with decreasing relative humidity (or increasing NH4NO3 concentration). We know that the value of fNH4+ is unity in an infinitely dilute solution, but clearly this value is only approached at relative humidities close to 100% and at very high dilution.

### Part 2

Next, we do a similar calculation for an HNO3 particle as a comparison. Use the back button on the other browser window to return to the data input page, or select this link. Fill in the input form as follows:

 2nd Calculation Select "Graph" as the form of output, and then enter the values and options under the following headings: Ambient Conditions Select "Relative humidity" as the variable. Start Value = 0.65 (i.e., 65%), End Value = 0.995 (i.e., 99.5%), Number of points = 60. Ionic Composition in Moles Hydrogen = 1.0, Nitrate = 1.0. Trace Gases Click on "HNO3" to prevent this gas being partitioned into the vapour phase. Solid Phases There are no entries under this heading. Click on the "Run" button at the end of the page to do the calculation. Note:  the above should be entered on the "variable relative humidity, or total water" parametric calculations page of Model III (http://www.aim.env.uea.ac.uk/aim/model3/mod3rhw.php).

#### Viewing and Interpreting the Results

First, plot the liquid water content of the particle against relative humidity:

 4th Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "moles of H2O(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

The water content of the particle rises smoothly with relative humidity, just as it did for NH4NO3. At 95% relative humidity the water content of the particle is about 40 moles, compared to the 30 found for NH4NO3 above. This reflects what we saw before in part (c) of this lesson: that water uptake, per mole of solute, is greater for HNO3 than it is for NH4NO3.

Next, plot the molality of the H+(aq) ion in the aqueous phase:

 5th Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "mH+(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

For HNO3 we see that a maximum molality of just over 8 mol kg-1 is reached at 65% relative humidity (compared to 22.49 mol kg-1 for NH4NO3). Also, the molality declines almost linearly with relative humidity which was not the case for NH4NO3. This reflects the different non-ideal (activity coefficient) effects in the two solutions, as we will see next.

Now plot the activity coefficient of H+(aq) against relative humidity:

 6th Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: leave blank Scale: linear (the default) Y Variable: "fH+(aq)" Range: leave blank Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

The value of the activity coefficient in this case generally increases with decreasing relative humidity - the opposite of what we found for the NH4NO3 solution. Is fH+ really equal to unity in an infinitely dilute solution at 1.0 relative humidity? We will now replot the graph above, but focus on the behaviour close to 1.0 relative humidity:

 7th Graph:  select the variables and enter the options as given below. X Variable: "relative humidity" Range: Min = 0.95;  Max = 1.0 Scale: linear (the default) Y Variable: "fH+(aq)" Range: Min = 0.70;  Max = 1.0 Scale: linear (the default) Click on the "Draw the Graph" button at the end of the page, and the plot will appear in the right frame.

It is clear from this magnified plot that at relative humidities greater than 0.99 the activity coefficient does indeed increase, but will approach unity only very close to 100% relative humidity.

Both the water uptake and ion activity coefficients of the two solutes differ. In fact, if either one of these two quantities is different then it implies that the other one will be too, as water and solute activity coefficients are related by the Gibbs-Duhem equation. In laboratory experiments on salts such as NH4NO3 it is usually the water activity that is determined as a function of molality, and the mean activity coefficient of the ions is then calculated from this. Conversely, for volatile electrolytes such as HNO3 or HCl the mean ion activity coefficients are usually determined from partial pressure or electromotive force measurements. The water activities can then be calculated from the Gibbs-Duhem equation. These relationships are explained in most thermodynamics text books.

You should now review the conclusions on the main page for this lesson.