11: Thermodynamics of Solubility
 Page ID
 204135
Objectives
 To determine the thermodynamic variable for the solubility of \(\ce{KNO_{3}}\)
Introduction
This experiment will further develop your understanding of thermodynamics while relating it to two concepts you already have studied, equilibrium and solubility. You will determine three thermodynamic values associated with the solubility of potassium nitrate. The reaction we will be studying is the dissolution of potassium nitrate in water
\[\ce{KNO_{3}}(s) \ce{<=>} \ce{K^{+}}(aq) + \ce{NO_{3}^{}} (aq) \label{1}\]
Potassium nitrate is a highly soluble compound (large \(\ce{K_{sp}}\)) compared to other insoluble compounds that you have studied. The massaction expression for this chemical equation is shown in Eq 2.
\[\ce{K_{sp}} = \ce{[K^{+}]}\ce{[NO_{3}^{}]} \label {2}\]
As you have learned previously, the value of \(\ce{K_{sp}}\) (and all equilibrium constants) is temperature dependent. In this experiment we will identify several different temperatures when solid potassium nitrate is in chemical equilibrium with its respective ions, \(\ce{K^{+}}\) and \(\ce{NO_{3}^{+}}\) This will be the temperature when crystallization first occurs in a solution that is cooling after being heated to completely dissolve the compound. By measuring this temperature at various temperatures over a small temperature range you will calculate \(\ce{K_{sp}}\) at each temperature using Equation 2.
From the value of \(\ce{K_{sp}}\) obtained at any temperature, a variety of thermodynamic variables can be determined. One very useful thermodynamic variable is G, the Gibbs free energy. The mathematical relationship between the change in the Gibbs free energy, \(\Delta\)G and the solubility product, \(\ce{K_{sp}}\), as function of temperature is given in Equation 3
\[\Delta G =\; RT\; ln K_{sp} \label{3}\]
where \(\ce{R}\) is the ideal gas constant (8.314 \(\frac{J}{mol^{.}K}\)) and \(\ce{T}\) is the temperature in Kelvin. By substituting the value of \(\ce{K_{sp}}\) at each temperature into equation 3, \(\Delta\)G for each temperature can be calculated. The other two thermodynamic variables, change in enthalpy (\(\Delta\)H) and change entropy (\(\Delta\)S), for any reaction can be determined from \(\Delta\)G at any temperature, as shown in Equation 4
\[\Delta G=\; \Delta H\; \; T\Delta S \label{4}\]
These two variables, \(\Delta\)H and T\(\Delta\)S, can be substituted for \(\Delta\)G into Equation 3. With some rearrangement the mathematical relationship between \(\ce{K_{sp}}\), \(\Delta\)H, and \(\Delta\)S as a function of \(\ce{T}\) is given in a useful form for graphing in Equation 5 below. The equation is written in
\[ lnK_{sp}= \frac{\Delta H}{R} \bigg(\frac {1}{T}\bigg) + \frac {\Delta S}{R} \label{5}\]
\[y\;\;\; =\;\;\;\; m\;\;\;\; x\;\;+\;\;\;b\]
the form of a straight line, y = mx + b as shown above. We may assume that \(\Delta\)H and \(\Delta\)S are constant over these small temperature ranges. Equation 5 relates the overall change in Gibbs free energy, \(\Delta\)G, to the changes in \(\ce{K_{sp}}\) as given in Equation 3. It also allows for the other two thermodynamic variables to be estimated. In this experiment you will obtain the thermodynamic values \(\Delta\)G, \(\Delta\)H, and \(\Delta\)S, associated with the solubility of \(\ce{KNO_{3}}\)
One final note: Adding moderately large amounts of solids to moderately small amounts of water has a large affect on the final volume of the solution. When saturated solutions are prepared the volume of the resulting solution must be determined experimentally.
Pre Lab Video
Procedure
Safety and Waste Disposal

CAUTION: Potassium nitrate is a strong oxidizer and skin irritant. Avoid contact with skin, eyes, and mucous membranes. Avoid shock, heat and contact with combustible materials.

The solution must be disposed of in the hazardous waste container.
Step 1
Weigh 13.00 g of potassium nitrate quantitatively and transfer it to a large test tube (25 x 200 mm).
Step 2
Add 10.0 mL of water and heat the test tube in a boiling water bath with the temperature probe in the solution until all the potassium nitrate dissolves. You will want to carefully stir the solution to aid in dissolution.
Step 3
For each data point you will determine the volume in the hot solution by comparison to a cool solution. Once the solid is completely dissolved, fill a second test tube of the same size with water to the same level as the test tube containing the hot \(\ce{KNO_{3}}\) solution. Be sure to make sure temperature probe is not in the solution.
Step 4
Use a graduated cylinder to measure the volume of water in second test tube. Record the volume to the correct number of significant figures.
Step 5
Raise the test tube containing the hot \(\ce{KNO_{3}}\) solution out of the hot water bath and allow it to cool while gently stirring.
Step 6
When crystals first appear, record the temperature. We are assuming that this temperature is where the equilibrium between solid and solution occurs.
Step 7
Add 3.00 mL of water to the solution. Heat the solution again until the crystals just barely dissolve completely. If you heat the solution too high, it will take much longer for the solution to cool. Again, measure the volume of the solution once the crystals have dissolved.
Step 8
Cool slowly while stirring. Record the temperature at which the crystals first are visible in the solution.
Step 9
Repeat the cycle (steps 6 and 7) until you have obtained at least five good data points.
Observations
Volume  Temperature  

Trial 1  
Trial 2  
Trial 3  
Trial 4 

Trial 5 
Calculations
Using Excel or some spreadsheet application:
Step 1
Calculate the concentration of the ions at equilibrium; \(\ce{[K^{+}]}\) and \(\ce{[NO_{3}^{}]}\). We know the mass of potassium nitrate we added, so we need to calculate moles and then divide by the volume at the temperature we used. The two concentrations will be equal (Why?).
Step 2
Calculate the equilibrium constant \(\ce{K_{sp}}\) for the reaction at the temperature we measured. Remember that \(\ce{K_{sp}}=\ce{[K^{+}][NO_{3}^{}]}\).
Step 3
Calculate \(\ce{ln\;K_{sp}}\)
Step 4
Change the units of your temperature to Kelvin
Step 5
Calculate the value of \(\frac{1}{T}\)
Step 6
Plot \(\ce{ln\;K_{sp}}\)(y) versus \(\frac{1}{T}\) (x). Be sure to properly label your plot, including title. Be sure that the plot fills the graph space as well.
Step 7
Fit your data to a linear trend line and obtain the equation of the line and \(\ce{R^{2}}\).
Step 8
Calculate the value of \(\Delta\)H for this reaction using \(\ref {5}\).
Step 9
Calculate \(\Delta\)G at each temperature from your value of K using equation \ref{3}.
Step 10
Rearrange equation \ref{4} and calculate \(\Delta\)S at each temperature using your value of \(\Delta\)H and the value of \(\Delta\)G at the temperature used.