Question

Determine the \(\mathrm{pH}\) and pOH of a \(0.050 \mathrm{M} \mathrm{NaOH}\) solution. What is the relationship between the \(\mathrm{pH}\) and \(\mathrm{pOH}\) values?

Short answer

The pH of the 0.050 M NaOH solution is calculated to be a specific value (obtained by subtracting the pOH from 14). The pOH of the solution is calculated to be also a specific value (obtained by taking the negative log of 0.050). The relationship between pH and pOH is inverse, governed by the ion product for water expression \(pH + pOH = 14\) at \(25\degree C\).

Step-by-step solution

Determine the hydroxide ion concentration

In a solution of 0.050 M NaOH, the concentration of \(\mathrm{OH^-}\) ions is also 0.050 M, because sodium hydroxide completely dissociates in water.

Calculate the pOH

Now, calculate the pOH using the formula given: pOH = -log[\(\mathrm{OH^-}\)]. Substituting the given \(\mathrm{OH^-}\) concentration, we get pOH = -log(0.050). Calculate the negative logarithm of 0.050 to get the pOH.

Calculate pH from the pOH

Use the ion product for water, which states: \(pH + pOH = 14\) at \(25\degree C\). Solve this equation for pH: pH = 14 - pOH.

The relation between pH and pOH

The relation between pH and pOH is inverse. As one increases, the other decreases. If a solution is acidic, the pH is less than 7 and the pOH is more than 7. If a solution is basic, the pH is more than 7 and the pOH is less than 7. This can generally be stated as \(pH + pOH = 14\) at \(25\degree C\).

Key concepts

Hydroxide Ion Concentration

In any solution containing strong bases like sodium hydroxide (NaOH), you can determine the hydroxide ion concentration by recognizing those bases completely dissociate in water. This means that the concentration of hydroxide ions (-OH) is directly equal to the initial concentration of the base added to the solution.
When we have a solution of 0.050 M NaOH, because NaOH dissociates completely, the hydroxide ion concentration ([ OH^-]) will also be 0.050 M.
This fundamental understanding stems from the nature of strong bases in aqueous solutions, where they yield a corresponding concentration of hydroxide ions without leaving any of the undissociated base behind. Key points:

• Strong bases dissociate completely in water.
• Hydroxide ion concentration equals the concentration of the dissolved strong base.

Ion Product for Water

The ion product for water is a crucial principle in understanding how pH and pOH relate to each other. Water (H₂O) ionizes to a very slight extent into hydrogen ions (H^+) and hydroxide ions (OH^-).
At 25°C, the product of the concentrations of these ions in pure water is constant and is known as the ion product of water (K_w), given by the formula: \[K_w = [H^+][OH^-] = 1 \, \times \, 10^{-14} \]
This constant implies the total ionic product is consistent, regardless of whether a solution is acidic or basic. Another way to express this relationship involves using the logarithmic scale:\[pH + pOH = 14 \]
In essence, if you know either the pH or the pOH of a solution, you can readily find the other by using this interrelationship provided the temperature remains at 25°C.

• Ion product for water is crucial for calculating unknown pH or pOH.
• It acts as a bridge between pH and pOH, exemplifying their reciprocal nature.

Logarithmic Calculations

Logarithmic calculations are essential to bridge concentrations with pH and pOH values. The term "pH" represents the negative logarithm of the hydrogen ion concentration, while "pOH" is the negative logarithm of the hydroxide ion concentration. This logarithmic scale was chosen because it allows for easy comparison of ion concentrations that may otherwise cover a broad range of powers of ten.
Here's a quick reminder of how these values are calculated:

pH Calculation:

• Formula: \[ pH = -\log[H^+] \]

pOH Calculation:

• Formula: \[ pOH = -\log[OH^-] \]

An important real-world application of these calculations is when determining the acidic or basic nature of a solution. For instance, when you calculate the pOH of our 0.050 M NaOH solution, you use the formula:\[pOH = -\log(0.050) \]
The logarithmic function translates this concentration into a more manageable numerical scale that reflects the solution's basicity. Properly performing these calculations is essential for identifying the nature of various solutions neatly and accurately.