Question

Salts containing the phosphate ion are added to municipal water supplies to prevent the corrosion oflead pipes. (a) Based on the \(\mathrm{pK}_{a}\) values for phosphoric acid \(\left(\mathrm{pK}_{a 1}=7.5 \times 10^{-3} , \right.\) \(\mathrm{p} K_{a 2}=6.2 \times 10^{-8}, \mathrm{p} K_{a 3}=4.2 \times 10^{-13} )\) what is the \(\mathrm{K}_{b}\) value for the \(\mathrm{PO}_{4}^{3-}\) ion? (b) What is the pH of a \(1 \times 10^{-3}\) \(M\) solution of \(\mathrm{Na}_{3} \mathrm{PO}_{4}\) (you can ignore the formation of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{H}_{3} \mathrm{PO}_{4} ) ?\)

Short answer

To find the $\mathrm{K}_{b}$ value for the $\mathrm{PO}_{4}^{3-}$ ion, we first determine the $\mathrm{K}_{a}$ values of $\mathrm{H}_{2} \mathrm{PO}_{4}^{-}$ and $\mathrm{H}\mathrm{PO}_{4}^{2-}$ using the given $\mathrm{pK}_{a}$ values. Then, we use the relationship $\mathrm{K}_{w}=\mathrm{K}_{a} * \mathrm{K}_{b}$ with the $\mathrm{K}_{a}$ value for $\mathrm{H}\mathrm{PO}_{4}^{2-}$ to find the $\mathrm{K}_{b}$ value. To calculate the pH of a $1 \times 10^{-3} M$ solution of $\mathrm{Na}_{3} \mathrm{PO}_{4}$, we assume that the $\mathrm{PO}_{4}^{3-}$ ion hydrolyzes only to $\mathrm{H}\mathrm{PO}_{4}^{2-}$ and use the $\mathrm{K}_{b}$ value to find the $\mathrm{OH}^{-}$ concentration. We then calculate the solution's $\mathrm{pOH}$ and find the pH using the relationship $\mathrm{pH} = 14 - \mathrm{pOH}$.

Step-by-step solution

Find the Ka of HPO4^2-

We are given pKa1 for the first ionization of phosphoric acid: H3PO4 -> H2PO4^- + H+. Use the first given pKa value to find the Ka of H_PO4^(-). pKa1 = 7.5 x 10^(-3) Ka1 = 10^(-pKa1) = 10^(-7.5 x 10^(-3))

Find the Ka of PO4^3-

Now, we need to find the Ka for the second ionization of phosphoric acid: H2PO4^- -> HPO4^2- + H+. Use the second given pKa2 value to find the Ka of PO4^3-. pKa2 = 6.2 x 10^(-8) Ka2 = 10^(-pKa2) = 10^(-6.2 x 10^(-8))

Calculate the Kb

Now we can find the Kb value for PO4^3-. To do this, we will use the relationship between Ka and Kb: Kw = Ka * Kb where Kw is the ion product of water, which is equal to 1x10^(-14). We will use the Ka value of HPO4^2- (from step 2) to find the Kb value for PO4^3-. Kb = Kw / Ka2 = (1x10^(-14)) / (10^(-6.2 x 10^(-8)))

Calculate the OH- concentration of Na3PO4 solution

Now that we have the Kb value for PO4^3-, we can find the OH- concentration (C_OH) for a 1x10^-3 M solution of Na3PO4. Using the equation Kb = ([OH-] * [HPO4^2-]) / [PO4^3-], we can find the OH- concentration by assuming that [HPO4^2-] = [OH-] and [PO4^3-] = 1x10^-3 M. Let x = [OH-]. Then, Kb = (x * x) / (1x10^-3 - x)

Solve for x (OH- concentration)

Solve the Kb equation for x: x^2 = Kb * (1x10^-3 - x) For very small Kb values, we can approximate x^2 ≈ Kb * 1x10^-3. Then x= √(Kb * 1x10^-3).

Calculate the pOH of the Na3PO4 solution

We can now find the pOH of the solution using the OH- concentration (x): pOH = -log10(x)

Calculate the pH of the Na3PO4 solution

Finally, we can find the pH of the solution using the pOH value: pH = 14 - pOH

Key concepts

Phosphoric Acid pKa

Understanding the concept of acid dissociation constants, denoted as pKa, is vital in buffer chemistry. The pKa values are a measure of the strength of an acid in solution, with lower values indicating a stronger acid. Phosphoric acid, a triprotic acid, has three hydrogens it can lose, having three distinct dissociation steps and hence three pKa values. The first dissociation gives off a hydrogen ion and forms dihydrogen phosphate (H₂PO₄^-), associated with pKa1. Each pKa value corresponds to the dissociation of one proton, moving from a less charged to a more charged species of phosphate. Having these pKa values, we can predict the behavior of phosphoric acid and its derivatives in buffer solutions and determine concentrations of various species at equilibrium.

For example, a lower pKa1 means that the first hydrogen is relatively easy to lose, and H₃PO₄ readily donates a proton to become H₂PO₄^-. In contrast, the significantly higher pKa3 means losing the third hydrogen from HPO₄^2- to form PO₄^3- is much less likely under most conditions.

Kb Value Calculation

The Kb value is the base dissociation constant, representing the strength of a base. Calculating the Kb for a polyatomic ion like PO₄^3- involves an understanding of the chemical equilibrium in aqueous solutions. Since PO₄^3- is the conjugate base of HPO₄^2-, which in turn is derived from H₂PO₄^-, you can calculate the Kb by using the relationship between the ion product of water (Kw) and the Ka of its conjugate acid.

The formula used is Kw = Ka * Kb, where Kw is a constant at a given temperature (1 x 10^-14 at 25°C). Therefore, to find the Kb of PO₄^3-, one must divide Kw by the given Ka value (Ka2 in the exercise). An important note for students - when you perform these calculations, ensure that you maintain consistency in units and pay attention to the correct scientific notation. Small mistakes can lead to a significant error. It's also essential to remember that a strong acid will have a weak conjugate base, and so, a high Ka value will correspond to a low Kb value.

pH Calculation of Buffer Solutions

The pH of a buffer solution can be a tricky aspect to grasp, but it's crucial for understanding the acid-base chemistry. A buffer can resist changes in pH upon the addition of acid or base, mostly because of the presence of a weak acid and its conjugate base (or vice versa). The pH of a buffer is governed by the Henderson-Hasselbalch equation:

pH = pKa + log([A^-]/[HA]),

where [A^-] is the concentration of the conjugate base and [HA] is the concentration of the acid. However, in the case of phosphate buffers, we often work with solutions containing salts that provide these ions. For example, with sodium phosphate, Na₃PO₄, there's an assumption that it fully dissociates in water to give PO₄^3- ions. To calculate pH, one also needs to consider the hydroxide ions produced and their impact on the solution's pH. Typically, this involves calculating the concentration of OH^- ions, then the pOH, and finally, using the relationship pH + pOH = 14 to find the pH. It's imperative to solve the dissociation reactions methodically, considering assumptions like negligible change in concentration for weak bases when confronted with very small Kb values, to achieve accurate results.