Problem 84 - Chemistry: An Atoms-Focused Approach

Question

What is the pH of \(1.00 M \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2} ?\)

Step-by-Step Solution

Answer

Answer: The estimated pH of a 1.00 M solution of copper(II) nitrate is slightly acidic and slightly lower than 7.

Step 1: Identify the weak base equilibrium reaction

Write the weak base equilibrium reaction for the dissociation of copper(II) nitrate. Copper(II) nitrate dissociates into copper(II) ions and nitrate ions in water: \(\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2} \Longrightarrow \mathrm{Cu}^{2+} + 2\mathrm{NO}^{-}_{3}\) The copper(II) ions react with water to form copper(II) hydroxide and hydrogen ions: \(\mathrm{Cu}^{2+} + 2\mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{Cu}(\mathrm{OH})_{2} + 2\mathrm{H}^{+}\)

Step 2: Analyze the equilibrium constants

Write down the equilibrium constant expression for the weak base reaction from Step 1. We can define the equilibrium constant (K) as follows: \(K = \frac{[\mathrm{Cu}(\mathrm{OH})_{2}][\mathrm{H}^{+}]^{2}}{[\mathrm{Cu}^{2+}]}\)

Step 3: Calculate the concentration of hydrogen ions

Since we have the initial concentration of copper(II) nitrate (1.00 M) and the desired pH, we can now use the equilibrium constant expression to find the concentration of hydrogen ions (H⁺). Assuming that the change in copper(II) ion concentration is negligible compared to the initial concentration of 1.00 M, we can substitute the equilibrium concentrations into the expression: \(K = \frac{x^2}{1-x}\), where `x` represents the concentration of \(\mathrm{H}^{+}\) ions. Unfortunately, the value of K for copper(II) hydroxide is not readily available, however, it is known to be a weak base with negligible dissociation in water. Therefore, we can assume that [H⁺] is very small as well. Thus, we can make the approximation that 1-x≈1: \(K \approx \frac{x^2}{1}\) Since K is very small, we can assume that x²≈0, and therefore [H⁺] is negligible.

Step 4: Calculate the pH

Since the concentration of hydrogen ions [H⁺] is negligible, and copper(II) hydroxide is a weak base, we can assume that the pH is close to neutral (7). \(\mathrm{pH} \approx -\log(\textrm{negligible}/1) \approx 7\) Given that the solution is acidic and the hydroxide ion concentration will be low, the pH will be slightly lower than 7. In conclusion, without the exact K-value of copper(II) hydroxide, we can only give an estimated pH value for a 1.00 M copper(II) nitrate solution, which is slightly acidic and slightly lower than 7.