Problem 26 - Chemistry Matter and Change

Question

Challenge Calculate the \(\mathrm{pH}\) of a solution having \(\left[\mathrm{OH}^{-}\right]=8.2 \times 10^{-6} \mathrm{M}\).

Step-by-Step Solution

Answer

The pH of the solution is 8.91.

Step 1: Understand the relationship between pH and pOH

The pH of a solution is related to its hydrogen ion concentration \( [H^+] \), while the pOH is related to the hydroxide ion concentration \( [OH^-] \). They are connected by the equation \( pH + pOH = 14 \) at 25°C.

Step 2: Calculate the pOH from hydroxide ion concentration

To find the pOH of the solution, use the formula \( pOH = -\log([OH^-]) \). Plug in the given concentration: \( pOH = -\log(8.2 \times 10^{-6}) = 5.09 \).

Step 3: Use the relationship between pH and pOH to find pH

With the value of pOH calculated, find the pH by using the relationship \( pH + pOH = 14 \).So, \( pH = 14 - pOH = 14 - 5.09 = 8.91 \).

Key Concepts

pOH

Understanding pOH helps us link hydroxide ion concentration to the overall acidity or basicity of a solution. **pOH** is a measure of the concentration of hydroxide ions, [OH^-], in a solution. Just like pH, which measures the concentration of hydrogen ions, pOH helps indicate whether a solution is acidic, basic, or neutral.
To find pOH, you use the logarithmic function:

• Calculate pOH by taking the negative logarithm (base 10) of the hydroxide ion concentration: \( pOH = -\log([OH^-]) \).
• At 25°C, the sum of pH and pOH is always 14. This relationship helps us convert between pH and pOH when needed.

For instance, if you have a hydroxide concentration of \(8.2 \times 10^{-6}\) M, the pOH would be calculated using the formula above, resulting in a pOH of 5.09. This indicates a somewhat basic solution since pOH below 7 implies a higher concentration of [OH^-].
Understanding pOH without knowing pH can already reveal a lot about the nature of the solution. It serves as a crucial component in solving for pH efficiently.

hydroxide ion concentration

The **hydroxide ion concentration** represents the amount of hydroxide ions ([OH^-]) available in a solution. These ions are pivotal in determining a solution's basicity. A higher concentration of hydroxide ions generally indicates a more basic solution, while lower concentrations suggest a more acidic environment.
When given the hydroxide ion concentration, such as \(8.2 \times 10^{-6} \) M in our original exercise, chemists can use this value to calculate both the pOH and the eventual pH of the solution. The calculation begins by plugging the concentration into the pOH formula: \( pOH = -\log([OH^-]) \).

• Solutions with high hydroxide ion concentrations will have low pOH values, indicating a basic solution.
• Conversely, lower concentrations of hydroxide ions will result in higher pOH values, pointing to an acidic solution.

Overall, knowing the hydroxide ion concentration is invaluable because it sets the foundation for subsequent calculations that reveal the solution's full acidity or basicity. It provides a direct method to discern the nature of the solution.

logarithmic function

At the heart of pH and pOH calculations lies the **logarithmic function**. In chemistry, we often deal with very small or very large numbers. Logarithms help simplify these into more manageable numbers, making calculations easier.
The logarithm used in pOH and pH calculations is base 10, often written as \( \log_{10} \) or simply \( \log \). When calculating things like pOH, this designates the expression:

• For a hydroxide ion concentration \( [OH^-] \), pOH is calculated as \( pOH = -\log([OH^-]) \).
• This function effectively translates small concentrations into straightforward numeric values, such as changing \(8.2 \times 10^{-6} \) M to a pOH of 5.09.

Logarithmic functions bring clarity and utility to scientific calculations. They not only simplify handling very small and large quantities but also facilitate essential relationships, such as that between pH and pOH.In learning about these functions, recognize how they play a pivotal role in connecting different chemical properties for a comprehensive understanding of a solution's characteristics. They transform complex exponential and multiplicative relationships into linear and additive forms, bridging gaps in scientific analysis.