Using Beer's Law to find the Equilibrium Constant of a Complex

Background

The following reaction between Fe³⁺ and SCN^- to form the complex Fe(SCN)²⁺ occurs with an equilibrium formation constant, K_f. The equation for the reaction and for the calculation of K follow.

Fe³⁺(aq) + SCN^-(aq) ⇌ Fe(SCN)²⁺

` K=([Fe(SCN)^(2+)])/([Fe^(3+)][SCN^-]) `

Fe³⁺ and SCN^- in solution are colorless, while the Fe(SCN)²⁺ complex is a deep, “blood” red. Based on Le Chatelier’s principle, we can shift equilibrium to the left or right by adding product or reactants, respectively. This will, in turn, produce less or more Fe(SCN)²⁺ making the red color less or more prominent. The concentration of Fe(SCN)²⁺ can be related to the amount of light absorbed by the solution via the Beer-Lambert Law:

A = εcl,

where A is the absorbance, c is the concentration (in M), l is the pathlength of the cuvette through which the light passes (our cuvettes have l = 1.00 cm), and ε is the molar absorptivity of the colored compound.

There is, however, a complication with using Beer’s Law in connection with this reaction: because it exists at equilibrium, we can never make a solution that is 100% Fe(SCN)²⁺. In other words, it cannot be assumed that every Fe³⁺ and SCN^- ion will combine to form Fe(SCN)²⁺.

To address this complication, it is necessary to add either the Fe³⁺ or the SCN^- in great excess, such that the other ion will be nearly completely consumed. If we add Fe³⁺ in excess, then we can assume that all of the SCN^- is used to make Fe(SCN)²⁺. By looking at the reaction equation above, we can see that the total amount of SCN^- is contained within the lone ion and the complex, so

moles SCN^- initial = (moles SCN^- at equilibrium) + (moles Fe(SCN)²⁺ at equilibrium)

We have just said that we will drive moles SCN^- at equilibrium to ≈ 0 such that

moles SCN^- initial = (moles Fe(SCN)²⁺ at equilibrium)

Thus, if we know how much SCN^- was added at the start of the reaction, then we also know how much Fe(SCN)²⁺ exists at equilibrium.

The goal for this experiment is to calculate K for this reaction, which will require two parts:

Using the excess reactant principle above for five solutions to create a reference or calibration curve relating [Fe(SCN)²⁺] to absorbance and determine ε.
Create three more solutions with the same amount of Fe³⁺ and SCN^-, use Beer’s Law and ε to determine [Fe(SCN)²⁺], and calculate K.

Procedure

To start this activity, click this link for Equilibrium and Beer's Law The lab will load in a new tab. Click back to this tab to read further instructions and complete the questions below.

The lab is set up with a white light source (the super light bulb in the virtual lab) and a spectrometer. We will start by recording the absorption of five solutions for which we have added excess Fe³⁺.

Open the Stockroom tab on the right hand side of the screen and expand the Samples menu and then expand the Liquids menu, then the Beer’s Law menu. Click the Fe(SCN)²⁺ label to fill the cuvette with a solution of concentration 0.0001 M. Make sure that the Stoichiometric box is checked.

[As a simplification, checking the Stoichiometric box is what ensures you are adding excess Fe³⁺. The labeled concentration is that of SCN^- at the start and of Fe(SCN)²⁺ at equilibrium].

Click and drag the cuvette from the stockroom counter to the spot between the light source and the spectrometer. Open the Live Data tab and you will see the Intensity spectrum for the solution in the cuvette. Toggle to just the Visible spectrum with the button under the display, and from Intensity to Absorbance on the left hand side of the graph.
Drag your mouse over the peak around 454 nm and you will see the data values for the absorbance measurement.
Record the absorbance at the point closest to 454 nm in the table below.
Now you will measure the absorbance for a set of standard solutions. To do that, open the Stockroom tab and click the trashcan beside the Fe(SCN)²⁺ label to empty that solution. Type in the next concentration from the table below and make sure to leave the Stoichiometric box checked. Click Enter to set that concentration. Then click the Fe(SCN)²⁺ label to fill the cuvette with the new solution.
Open the Live Data tab and measure the absorbance of the new solution from the spectrum and record it in the table.
Repeat for all of the indicated concentrations.

Concentration Fe(SCN)²⁺ (M)	Absorbance
0.0001
0.0002
0.0003
0.0004
0.0005

Analysis Section 1

Open a spreadsheet program, like Excel or Desmos, and plot your data points, with Absorbance on the y-axis and Concentration on the x-axis.
Fit a linear trendline and make sure the equation is labeled on your graph, including the R² value.
Fill in your equation below.

Absorbance = ·Concentration +

The slope of your equation gives you the ε·l in the Beer's law equation: A = εcl. Since l = 1 cm, the slope is equal to the ε of Fe(SCN)²⁺. The units of ε are 1/(M*cm).

Now that we have the ε of Fe(SCN)²⁺, we can complete the second part of the experiment to calculate K, using Beer’s Law to relate solution absorbance to an unknown concentration of Fe(SCN)²⁺.

Procedure Section 2

Go to the Stockroom tab and dispose of your last standard solution using the trashcan. Now uncheck the Stoichiometric box and type in a concentration of 0.0006. Click Enter to set that concentration. Then click the Fe(SCN)²⁺ label to fill the cuvette with the new solution.

We are no longer adding one of the reagents in excess. Without the Stoichiometric box checked, the concentration you entered is the initial concentration of both Fe³⁺ and SCN^-.

Go to the Live Data tab and record the absorbance at 454 nm in the table below, as before.
Repeat for all of the indicated concentrations, and be sure that the Stoichiometric box is unchecked each time. Record the absorbances in the table.

Initial [Fe³⁺]=[SCN^-] (M)	Absorbance	Equilibrium [Fe(SCN)²⁺]
0.0006
0.0008
0.0010

Analysis Section 2

Now use the ε calculated in the first analysis section and the absorbances you just measured to calculate the equilibrium concentration of Fe(SCN)²⁺ using Beer’s Law for each solution. Record these values in the last column of the table.

[Fe(SCN)²⁺] = A/ε (as l is 1)

Fill out the tables below showing the concentrations of Fe³⁺, SCN^-, and Fe(SCN)²⁺ for each solution. Also record the equilibrium [Fe(SCN)²⁺] that you calculated in the last table for each solution. Because one mole of Fe³⁺ combines with one mole of SCN^- to create one mole of Fe(SCN)²⁺, the change for all three will be the same: the amount of Fe(SCN)²⁺ formed. This change will be an increase for Fe(SCN)²⁺ and a decrease for each of the reactants. Using the initial concentrations and the change for Fe³⁺ and SCN^- in each solution, you should now have the concentrations for Fe³⁺, SCN^-, and Fe(SCN)²⁺ at equilibrium for each one.

We have laid out the table for the first concentration. You can do the calculations for the second and third concentrations on your own. You will use the equilibrium concentrations for your K calculations below.

	[Fe³⁺] (M)	[SCN^-] (M)	[Fe(SCN)²⁺] (M)
Initial	0.0006	0.0006	0
Change	-x	-x	+x
Equilibrium	0.0006-x	0.0006-x

	[Fe³⁺] (M)	[SCN^-] (M)	[Fe(SCN)²⁺] (M)
Initial	0.0008	0.0008	0
Change
Equilibrium

	[Fe³⁺] (M)	[SCN^-] (M)	[Fe(SCN)²⁺] (M)
Initial	0.001	0.001	0
Change
Equilibrium

Calculate a K for each of the three solutions using the equilibrium constant equation.

[Fe³⁺] (M	K
0.0006
0.0008
0.001

What is your average K?

Submit your spreadsheet with your graph and other calculations to your instructor.