How to Calculate the pH of a Solution Given [OH-]

Understanding the relationship between pH and pOH is essential for chemists and anyone working with solutions. In this article, we will explore how to calculate the pH of a solution given the hydroxide ion concentration [OH-].

Introduction to pH and pOH

In aqueous solutions, the pH and pOH are related to the concentration of hydrogen ions [H ] and hydroxide ions [OH-], respectively. The pH is defined as the negative logarithm of the hydrogen ion concentration, and the pOH is defined as the negative logarithm of the hydroxide ion concentration. These values are interconnected through the following equation:

pH pOH 14

Casestudy: [OH-] 1 x 10^-11 M

Let's go through a detailed example. If the concentration of hydroxide ions [OH-] is 1 x 10^-11 M, we need to determine the pH of the solution.

Step 1: Calculate the pOH

The pOH is calculated using the formula:

pOH -log[OH-]

For [OH-] 1 x 10^-11 M, the calculation is as follows:

pOH -log(1 x 10^-11) 11.0

Using the relationship between pH and pOH:

pH pOH 14

Substitute the value of pOH:

pH 14 - 11.0 3.0

Additional Example: [OH-] 10^-10 M

Now, let’s consider another scenario where the concentration of hydroxide ions [OH-] is 10^-10 M.

Step 1: Calculate the pOH

Again, using the formula:

pOH -log[OH-]

For [OH-] 10^-10 M, the calculation is:

pOH -log(10^-10) 10.0

Using the relationship between pH and pOH:

pH pOH 14

Substitute the value of pOH:

pH 14 - 10.0 4.0

Further Example: [OH-] 10^-7 M 10^-10 M

Now, let's consider the case where the concentration of hydroxide ions [OH-] is a sum of 10^-7 M and 10^-10 M.

Step 1: Calculate the total [OH-] concentration

First, add the concentrations:

[OH-] 10^-7 10^-10

Since 10^-10 is much smaller than 10^-7, we can approximate this as:

[OH-] ≈ 10^-7 M

Step 2: Calculate the pOH

Using the formula:

pOH -log[OH-]

For [OH-] ≈ 10^-7 M, the calculation is:

pOH -log(10^-7) 7

Using the relationship between pH and pOH:

pH pOH 14

Substitute the value of pOH:

pH 14 - 7 7.0

Step 3: Detailed Calculation for Accuracy

For more precise calculation, use the exact values:

[OH-] 1 x 10^-7 1 x 10^-10 M

Since 1 x 10^-10 is very small compared to 1 x 10^-7, we can express it as:

[OH-] ≈ 1 x 10^-7 0.001 x 10^-7 1.001 x 10^-7 M

Step 4: Calculate pOH with more precision

pOH -log(1.001 x 10^-7)

Using logarithm properties:

pOH ≈ 6.9996

Step 5: Calculate pH

Using the relationship:

pH pOH 14

Substitute the value of pOH:

pH 14 - 6.9996 7.0004

This precision ensures that the concentration of hydrogen ions [H ] in the solution is accurate.

Conclusion

In summary, understanding how to calculate the pH of a solution given the concentration of hydroxide ions [OH-] is crucial for many applications. By applying the relationship between pH and pOH and using the fundamental logarithmic properties, we can determine the pH of any aqueous solution with high precision.

References

1. Gilmann, PM. Mackowiak, PA. (2002). Principles of Physiological Chemistry. McGraw-Hill.

2. Casiday, R, Goldman, R. Kotz, JC. (2008). Chemistry: Structure and Properties. Pearson Prentice Hall.

3. Muller, K. (2007). Chemistry for the Biosciences: A Molecular Approach. Oxford University Press.