Question

You travel to a distant, cold planet where the ammonia flows like water. In fact, the inhabitants of this planet use ammonia (an abundant liquid on their planet) much as earthlings use water. Ammonia is also similar to water in that it is amphoteric and undergoes autoionization. The \(K\) value for the autoionization of ammonia is \(1.8 \times 10^{-12}\) at the standard temperature of the planet. What is the pH of ammonia at this temperature?

Short answer

The pH of ammonia at the standard temperature of the distant, cold planet is given by the expression: \[pH = -\log_{10} \sqrt{(1.8 \times 10^{-12})[NH_3]^2}\] where \([NH_3]\) represents the concentration of ammonia in the solution. Without knowing the exact ammonia concentration, it is not possible to provide a specific pH value. However, this expression allows anyone to find the pH as long as they have the ammonia concentration at the planet's standard temperature.

Step-by-step solution

Write the expression for the equilibrium constant, K

Based on the autoionization reaction of ammonia, we can write the expression for the equilibrium constant, K: \[K = \dfrac{[NH_4^+][NH_2^-]}{[NH_3]^2}\] Since the autoionization occurs in a 2:1 ratio, we can assume that the concentrations of \(NH_4^+\) and \(NH_2^-\) ions are equal, i.e., \([NH_4^+] = [NH_2^-]\). Then the expression becomes: \[K = \dfrac{[NH_4^+]^2}{[NH_3]^2}\]

Find the concentration of NH4+

We can rearrange the equation to find the concentration of \(NH_4^+\): \[K[NH_3]^2 = [NH_4^+]^2\] Taking the square root of both sides: \[[NH_4^+] = \sqrt{K[NH_3]^2}\] We are not given the concentration of \(NH_3\), but since we are interested in the pH, we can leave it as is for now.

Relate the concentration of NH4+ to the concentration of H+

We can now relate the concentration of \(NH_4^+\) to the concentration of \(H^+\) ions. Since \(NH_4^+\) is a weak acid, it forms \(H^+\) and \(NH_3\) in solution: \[NH_4^+ ⇌ H^+ + NH_3\] We can assume that the concentration of \(H^+\) ions is equal to the concentration of \(NH_4^+\) because of the 1:1 ratio in the reaction. So, we can write: \[[H^+] = [NH_4^+]\] Substituting the expression for \([NH_4^+]\) from Step 2: \[[H^+] = \sqrt{K[NH_3]^2}\]

Find the pH of ammonia solution

Now that we have an expression for the concentration of \(H^+\) ions, we can find the pH using the formula: \[pH = -\log_{10} [H^+]\] Substituting the expression for \([H^+]\) from Step 3: \[pH = -\log_{10} \sqrt{K[NH_3]^2}\] We are given the value of K, so we can plug it in: \[pH = -\log_{10} \sqrt{(1.8 \times 10^{-12})[NH_3]^2}\] Although we don't know the concentration of \(NH_3\), we can still conclude that the pH of ammonia at the standard temperature of the planet is calculated using this expression. As the concentration of ammonia is not specified, it is not possible to provide the exact pH value. The expression above will allow anyone to find the pH as long as they have the ammonia concentration in the solution at the standard temperature of the planet.