Question

Calculate the \(\mathrm{pH}\) of the following solutions. a. \(0.10 \mathrm{M} \mathrm{NaOH}\) b. \(1.0 \times 10^{-10} \mathrm{M} \mathrm{NaOH}\) c. \(2.0 \mathrm{M} \mathrm{NaOH}\)

Short answer

The pH values for the given NaOH solutions are: a. 0.10 M NaOH: pH = 13 b. \(1.0 \times 10^{-10} M\) NaOH: pH = 4 c. 2.0 M NaOH: pH > 14 or 'extremely basic'

Step-by-step solution

Identify the relationship between concentration and pH

To calculate the pH of a solution, we use the formula: pH = -log[H+] In this case, we are given the concentration of NaOH, which is a strong base. Since it dissociates completely in water, it will produce an equal concentration of OH⁻ ions. We need to calculate the pOH first and then use the relationship between pH and pOH to find the pH. The relationship between pOH and pH is as follows: pH + pOH = 14 Now the task is to find the pOH for each concentration and use this relationship to calculate pH.

Calculate pOH for 0.10 M NaOH

For the first solution, the NaOH concentration is 0.10 M. Since NaOH dissociates completely, the OH⁻ concentration will also be 0.10 M. To find the pOH, use the pOH = -log[OH⁻]: pOH = -log(0.10) pOH = 1

Calculate pH for 0.10 M NaOH

Now that we have the pOH, we can find the pH by using pH + pOH = 14: pH = 14 - pOH pH = 14 - 1 pH = 13 The pH of the 0.10 M NaOH solution is 13.

Calculate pOH for \(1.0 \times 10^{-10} M\) NaOH

For the second solution, the concentration of NaOH is \(1.0 \times 10^{-10} M\). The OH⁻ concentration will be the same: pOH = -log(\(1.0 \times 10^{-10}\)) pOH = 10

Calculate pH for \(1.0 \times 10^{-10} M\) NaOH

Now, find the pH by using pH + pOH = 14: pH = 14 - pOH pH = 14 - 10 pH = 4 The pH of the \(1.0 \times 10^{-10} M\) NaOH solution is 4.

Calculate pOH for 2.0 M NaOH

For the third solution, the concentration of NaOH is 2.0 M. pOH = -log(2.0) pOH ≈ -0.301

Calculate pH for 2.0 M NaOH

Now use pH + pOH = 14 to find the pH: pH = 14 - pOH pH = 14 - (-0.301) pH ≈ 14.301 However, since the pH scale typically ranges from 0 to 14, it's more appropriate to indicate that the solution is extremely basic and should be reported as pH > 14 or "extremely basic". Here are the final pH values for each NaOH solution: a. 0.10 M NaOH: pH = 13 b. \(1.0 \times 10^{-10} M\) NaOH: pH = 4 c. 2.0 M NaOH: pH > 14 or 'extremely basic'

Key concepts

strong bases

A strong base is a compound that dissociates completely in water, meaning it fully splits into its ions when dissolved. This results in a higher concentration of hydroxide ions (OH⁻) in the solution.
NaOH, or sodium hydroxide, is an excellent example of a strong base. When you dissolve NaOH in water, it breaks down entirely into sodium ions (Na⁺) and hydroxide ions (OH⁻).
The ability of strong bases to dissociate entirely makes them highly effective in increasing the basicity of a solution.

• This complete dissociation leads to a high pOH value (and correspondingly low or very high pH, depending on the concentration), which indicates the solution's basic nature.
• Understanding the behavior of strong bases, like NaOH, helps predict the pH of the solution and whether it's basic or highly basic.

NaOH dissociation

NaOH dissociation is a key process that occurs when sodium hydroxide is dissolved in water. This process involves breaking apart the compound into separate ions, specifically, Na⁺ and OH⁻ ions.
Since NaOH is a strong base, it dissociates completely in water, meaning each molecule of NaOH results in one sodium ion and one hydroxide ion:
\ \[ \text{NaOH} \rightarrow \text{Na}^+ + \text{OH}^- \]
This characteristic of complete dissociation allows us to reliably use the concentration of NaOH to determine the concentration of hydroxide ions.
This is crucial for calculating the pOH, and subsequently, the pH of the solution.

• Complete dissociation of NaOH implies that the concentration of OH⁻ ions in solution is equal to the initial concentration of NaOH.
• Knowing the concentration of OH⁻ enables direct calculation of the pOH.

pH and pOH relationship

pH and pOH are two interrelated measures that help describe the acidity or basicity of a solution. They are linked by a simple mathematical relationship: \ \[ \text{pH} + \text{pOH} = 14 \]
This relationship is derived from the ion product constant of water (\(\text{K}_w \)), which is always \(1.0 \times 10^{-14} \text{ at 25°C}\).

This key formula allows you to determine one value if you know the other, making pH and pOH calculations interdependent:

• To calculate pH, subtract pOH from 14. This tells you how acidic or basic a solution is.
• A low pH indicates acidity, while a high pH (like those > 7) signifies basic solutions.
• The relationship helps convert between the concentration of hydrogen ions ([H⁺]) and hydroxide ions ([OH⁻]), cementing the dual understanding of a solution's behavior.

basic solutions

Basic solutions, also known as alkaline solutions, are those with a pH greater than 7.
Such solutions have a higher concentration of hydroxide ions (OH⁻) than hydrogen ions (H⁺).
The pH scale, which ranges from 0 to 14, is a convenient way to express the acidity or basicity of a solution:

• In basic solutions, pH values typically range from just above 7 to 14.
• If the pH is equal to 7, the solution is considered neutral, as is the case with pure water.
• Values above 7 indicate a basic solution, and the closer the pH approaches 14, the stronger the basicity.

Understanding these principles of basic solutions allows students to predict the behavior of substances like NaOH when dissolved in water and to comfortably navigate the broader concepts of pH and solution chemistry.