Redox titration, complex formation effect on lg'K?

Question

We have performed the following redox titration

(1): $\ce{MnO_4^- + 8H^+ + 5Fe^2+ <=> Mn^2+ + 4H_2O + 5Fe^3+}$

where we have used a iron(II)solution (the analyte) and titrated it with a permanganate solution (titrant). The iron(II) was added by measuring up $\ce{FeSO_4*7H_2O}$. There is also $\ce{H_2SO_4}$ and $\ce{NaNO_3}$ in the analyte solution. The permanganate solution was prepared by an instructor. When the equivalence point was reached we could calculate the original concentration of $\ce{Fe^2+}$ in the analyte solution, we have pe-values, E-values, the volume of titrant needed to reach the equivalence point and a lot more different values.

The equilibrium constant lgK for

(2): $\ce{Fe^3+ + e- <=> Fe^2+}$

is 13,02 (from a data collection). Our calculated value for lg'K based of the concentrations that was calculated in the titration in reaction (2) is 11,67.

We know that $\ce{Fe^3+}$ forms a complex with $\ce{SO4^2-}$

(3): $\ce{Fe^3+ + nSO_4^2- <=> Fe(SO_4)_n^{3-2n}}$

where n=1 or 2. This is supposed to be a reason for the difference in the theoretical value of the equilibrium constant and our calculated one(the difference between 13,02 and 11,67). Why does this complex formation make our calculated value lower than the theoretical one? How does this affect the equilibrium constant?

Another reason for this difference is the ionic strength in the solution, but in which way? What is influencing the ionic strength in this system?

(there have been no previous calculations with ionic strength or activities in this laboration and there is no other mentioning of ionic strength anywhere else in our instructions)

The calculation of our lg'K has been calculated as following:

$\ce{'K = \frac{[Fe^2+]}{[Fe^3+]*({e^-)}}}$

$\ce{<=>}$

$\ce{lg'K = lg[Fe^2+] - lg[Fe^3+] + pe}$

Where [] are the concentrations in Molar and () is the activity (we can not write $\ce{e^-}$ as a concentration but we have pe-values).

Answer 1

I suspect the primary source for the error (≈10%) you are encountering between your calculated $K'$ value and the theoretical $K$ value is the deviation of the molar equilibrium concentrations of $\ce{Fe^2+}$ and $\ce{Fe^3+}$ from the activities of both ions.

The presence of many ions in a solution, especially when some possess multiple charges (like $\ce{Fe^3+}$ and $\ce{Fe^2+}$, not to mention the sulphate iron complexes noted) or are highly concentrated, contribute to a non-negligible ionic strength $I_c$. The reason for this is the following:

$I_c$ for a solution with $n$ charged species can be calculated with the formula:

$$I_c=0.5\sum_{i=1}^n c_i\;z_i^2$$

Where:

$z_i$ represents the charge of species $i$

$c_i$ represents the concentration of species $i$

When $I_c$ is relatively high ($I_c>0.01M$), the activity coefficient $\gamma$ of multi-charged ions deviate significantly from unity (are reduced significantly), and the ones with a higher charge tend to experience a higher reduction than those with a lower charge.

The activity of a species can be calculated using its activity coefficient and concentration in solution:

$$a_i=\gamma_i\;c_i$$

This means that the relationship between $K$ and $K'$ in our case is:

$$K=\frac{a_{\ce{Fe^2+}}}{a_{\ce{Fe^3+}}}=\frac{\gamma_{\ce{Fe^2+}}}{\gamma_{\ce{Fe^3+}}}\frac{c_{\ce{Fe^2+}}}{c_{\ce{Fe^3+}}}=K_\gamma\;K'$$

Or simply:

$$K=K_\gamma\;K'$$

We can observe that as both activity coefficients approach unity, $K$ approaches the value of $K'$:

$$\gamma_{\ce{Fe^3+}}≈\gamma_{\ce{Fe^2+}}≈1\implies K_\gamma ≈1\implies K≈K'$$

However, if $I_c$ is high enough to cause significant deviation, such deviation will be more impactful over $\gamma_{\ce{Fe^3+}}$ than it would over $\gamma_{\ce{Fe^2+}}$, since the former has a higher charge than the latter.

Just to give you an idea of the magnitude of error that arises when assuming ionic strength is negligible in cases when it shouldn't be, we will consider the theoretical activity coefficients for both ions at 25°C and a solution with an ionic strength of 0.01M:

$$\gamma_{\ce{Fe^3+}}=0.443$$

$$\gamma_{\ce{Fe^2+}}=0.676$$

$$K_\gamma=1.53$$

This means that the thermodynamic equilibrium constant $K$ would be more than 150% higher than the calculated equilibrium constant $K'$ with molar concentrations.

If you want to estimate the value of both activity coefficients in accordance with your experiment, you can use the Debye-Hückel equation.

After independently substituting all the parameters for $\ce{Fe^3+}$ and $\ce{Fe^2+}$, the resulting expressions are:

$$\pu{-log}\;\gamma_{\ce{Fe^3+}}=\frac{4.59\sqrt{I_c}}{1+2.97\sqrt{I_c}}$$

$$\pu{-log}\;\gamma_{\ce{Fe^2+}}=\frac{2.04\sqrt{I_c}}{1+1.98\sqrt{I_c}}$$

Or you can combine them and get a function of $K_\gamma$ in terms of $I_c$:

$$\pu{-log}\;K_\gamma=\frac{2.04\sqrt{I_c}}{1+1.98\sqrt{I_c}}-\frac{4.59\sqrt{I_c}}{1+2.97\sqrt{I_c}}$$

Plotting $K_\gamma$ vs $I_c$ would give:

I suspect the ionic strength of your particular solution at equilibrium has a value of approximately $4\times10^{-4}M$ since:

$$I_c=4\times10^{-4}M\implies K_\gamma=1.12=\frac{K}{K'}=\frac{13.02}{11.67}$$

However, if there were a significant difference between the experimental temperature and the theoretical temperature, that would also factor in the error, since both equilibrium constants $K$ and $K'$ are temperature-dependent. Measurement errors and impurities can also contribute to the error.