Question

What is the \(\mathrm{pH}\) of a solution that is \(2.5 \times 10^{-9} \mathrm{M}\) in \(\mathrm{NaOH}\) ? Does your answer make sense? What assumption do we normally make that is not valid in this case?

Short answer

The pH of the solution is approximately 5.4, which paradoxically suggests an acidic solution even though NaOH is a strong base. The result does not make sense because the low concentration of NaOH given makes the assumption of complete dissociation invalid in this case. In such dilute solutions, the self-ionization of water is significant and cannot be ignored, resulting in an acidic pH instead of the expected basic pH.

Step-by-step solution

Determine the concentration of hydroxide ions (OH⁻)

Given the concentration of NaOH is \(2.5\times10^{-9} \mathrm{M}\), and NaOH dissociates completely to yield OH⁻ ions, the concentration of OH⁻ ions in the solution is also \(2.5\times10^{-9} \mathrm{M}\).

Calculate the concentration of hydrogen ions (H⁺) using the ion product constant of water (\(K_w\))

Ion product constant of water, \(K_w\), is given by the relationship \(K_w = [\mathrm{H}⁺][\mathrm{OH}⁻]\). At 25°C, \(K_w = 1\times10^{-14}\). We can use this to calculate the H⁺ concentration: \[ [\mathrm{H}⁺] = \frac{K_w}{[\mathrm{OH}⁻]} \] Inserting the given values: \[ [\mathrm{H}⁺] = \frac{1 \times 10^{-14}}{2.5 \times 10^{-9}} \]

Solve for H⁺ ion concentration

Calculate the H⁺ ion concentration: \[ [\mathrm{H}⁺] = 4 \times 10^{-6} \mathrm{M} \]

Calculate the pH of the solution

Now that we have the H⁺ concentration, we can calculate the pH using the definition: \[ \mathrm{pH} = -\log_{10}[\mathrm{H}^+] \] Plug in the value of H⁺ concentration: \[ \mathrm{pH} = -\log_{10}(4 \times 10^{-6}) \]

Solve for pH

Calculate the pH of the solution: \[ \mathrm{pH} \approx 5.4 \]

Discuss the validity of the result and assumptions

The pH of the NaOH solution comes out to be around 5.4, which suggests an acidic solution. However, we know that NaOH is a strong base, so the answer does not make sense. This is because the concentration of NaOH given is very low, and the assumption that NaOH dissociates completely is not valid in this case. In such dilute solutions, the self-ionization of water becomes significant and cannot be ignored. The amount of H⁺ ions contributed by water is higher compared to the amount of OH⁻ ions contributed by the dissociation of NaOH in this dilute solution. This results in an acidic pH instead of the expected basic pH.

Key concepts

Hydroxide Ion Concentration

The hydroxide ion concentration ( OH⁻ ) is a crucial component when discussing the acidity or basicity of a solution. When a compound like sodium hydroxide ( NaOH ) dissolves in water, it dissociates completely. This means it breaks down to produce hydroxide ions ( OH⁻ ). In our original exercise, we were given that the sodium hydroxide concentration is very low at 2.5 imes 10^{-9} M. Due to complete dissociation, the concentration of hydroxide ions ( OH⁻ ) in the solution will also be 2.5 imes 10^{-9} M.

Understanding the OH⁻ concentration is important because it can help us measure the basicity of a solution. The more hydroxide ions present, the more basic the solution becomes.

• Hydroxide ions indicate a basic or alkaline environment.
• A higher concentration of OH⁻ results in a higher pH value.
• In very dilute solutions, contributions from water self-ionization must also be considered.

Ion Product Constant of Water

The Ion Product Constant of Water ( K_w ) is an essential concept in understanding the relationship between hydrogen ions ( H⁺ ) and hydroxide ions ( OH⁻ ) in water. At 25°C, K_w is consistently equal to 1 imes 10^{-14} . This constant describes how water self-ionizes and forms ions even in its pure state.

The formula K_w = [H⁺][OH⁻] expresses this relationship. When you have the concentration of one ion, you can calculate the concentration of the other. For example, in the exercise, we used the hydroxide ion concentration to find H⁺ by rearranging the formula. This highlights how K_w serves as a bridge between the acidic and basic parts of a solution.

• K_w helps maintain a balance between H⁺ and OH⁻ .
• It ensures that the product of [H⁺] and [OH⁻] remains constant at all times for water.
• This value can change with temperature, but remains fixed at 25°C.

Self-Ionization of Water

Water has a unique ability to partially ionize even when it's in its pure state. This phenomenon, known as the self-ionization of water, means that water molecules are continuously forming a small number of hydrogen ions ( H⁺ ) and hydroxide ions ( OH⁻ ) by interacting with each other. Despite being in minimal amounts, these ions play a significant role in solution chemistry.

In dilute solutions, like in our given exercise, the self-ionization becomes more critical. When the solute concentration is lower, the ions from water itself can influence the overall pH level more than the added solute does. For example, in a solution with an NaOH concentration of 2.5 imes 10^{-9} M, the hydrogen ion contribution from water self-ionization surpasses the hydroxide ions from NaOH , resulting in an unexpected pH level.

• Self-ionization occurs naturally in pure water at equilibrium.
• This effect becomes more obvious in extremely dilute or pure solutions.
• The process is significant in situations where expected contributions from solutes are minimal.