Calculations:

To evaluate the equilibrium expression, you need to determine the equilibrium concentration of the reactants and products.

[H⁺]_eq:
Since the solution is buffered at a constant pH, the [H⁺] does not change during the reaction. The value of the [H⁺] _eq will equal [H⁺] and can be calculated from the pH by

               (eqn. 7)

[AlQ^-]_eq:
The product, AlQ^-, absorbs light at 550 nm, and its concentration is determine from Beer's law. The reactant, H₄Q, does not absorb strongly at this wavelength, and the "blank" will correct for any absorbance from the H₄Q. Beer's law gives a relationship between the absorbance and concetration. Explicitly stated Beer's law is

               (eqn. 8)

where ABS = absorbance of the solution
= extinction coefficient. For AlQ^- = 25,000 L/mol cm
l = path length of the cell. In this case l = 0.90 cm
c = equilibrium concetration in moles/L of AlQ^-

so [AlQ^-]_eq is calculated by

               (eqn. 9)

However you recorded the % transmission for your solutions, not the absorbance. Recording the % transmittance is more accurate then recording the absorbance, since the % transmittance scale is linear while the absorbance scale is logrithmic. Therefore the % transmittance is converted to ABS by

               (eqn. 10)

[Al³⁺]_eq: To calculate this value you first need to determine the [Al³⁺]_initial value. In this experiment you placed 2 mL of stock [Al³⁺] solution in a test tube with 2 mL of stock [H₄Q] solution. The net result is that the Al³⁺ solution was diluted by1/2. The initial concetration is not equal to the value on the lable of the stock reagent bottle. The concetration after a dilution can be calculated by

               (eqn. 11)
where Vol_initial = 2.0 mL and Vol_final = 4.0 mL.( Remember that during a dilution process, the number of moles of solute does not change.)
Now [Al³⁺]_eq = [Al³⁺]_initial - [AlQ^-]_eq

[H₄Q]_eq:
This is calculated in a similar method as [Al³⁺]_eq is calculated, since the same process was use to prepare the solution.

               (eqn 12)

Vol_initial and Vol_final are the same as above. And [H₄Q]_eq = [H₄Q]_initial - [AlQ^-]_eq

You will need to calculate the equilibrium constant at each temperature. You only need to show one sample calculations, and summarize your results in a table.

Gibbs Free Energy:

For the graph you need values for 1/T and ln K at each temperture reading. Remember that temperature is in kelvins (°C + 273.15). Summarize this data in a table.

Draw a graph of ln K vs 1/T following these basic guidelines:

1. use a graphing program (Excel, KaleidaGraph, Cricket graph) or use good graphing paper.
     (see http://www.ncsu.edu/labwrite/res/gt/graphtut-home.html for a brief into to graphing in Excel)
2. axes should be labeled. Plot the dependent varialble on the y-axis, and the independent variable on the x-axis.
3. a caption or title explaining what the graph means.

The values of H° and S° are obtained from the slope and y-intercept of your graph. One method for determining the slope and y-intercept is to perform a Least Squared Analysis. This is also refered to as a Linear Regression Analysis or a Linear Curve Fit. Most graphical calculators or graphing programs are capable of this analysis. Otherwise you will determine these values from the graph. The slope is

               (eqn. 13)

and

               (eqn. 14)

where R is the gas constant. R = 8.3145 J/mol K or 1.987 cal/mol K

The y-intercept, the point of intersection of the line on the y-axis when x=0, is

               (eqn. 15)
and

               (eqn. 16)

There is no uncertainity analysis for this experiment.