Question

What is the pH of a solution when \(\left[\mathrm{H}^{+}\right]\) is \(9.04 \times 10^{-13} \mathrm{M}\) ?

Short answer

The pH of the solution is approximately 12.043.

Step-by-step solution

Understand the Formula

To find the pH of a solution, we use the formula \( ext{pH} = - ext{log}_{10} ext{[H}^+\text{]} \) where \( ext{[H}^+\text{]} \) is the concentration of hydrogen ions in moles per liter.

Substitute the Value

Substitute the given hydrogen ion concentration \( ext{[H}^+] = 9.04 \times 10^{-13} \mathrm{M} \) into the pH formula.

Apply the Logarithm

Calculate the logarithm of the concentration: \( \text{log}_{10}(9.04 \times 10^{-13}) \), which involves first using \( \text{log}_{10}(9.04) \) and \( \text{log}_{10}(10^{-13})\).

Calculate the Log Components

1. Find \( \text{log}_{10}(9.04) \approx 0.957 \).2. Calculate \( \text{log}_{10}(10^{-13}) = -13 \).

Finish the Calculation

Combine the results of the logs to get \( \text{pH} = -(0.957 - 13) = 12.043 \).

Interpret the Result

The pH value calculated is approximately \( 12.043 \), which indicates that the solution is basic since a pH greater than 7 is considered basic.

Key concepts

Hydrogen Ion Concentration

The hydrogen ion concentration \([ ext{H}^+]\) is a fundamental concept in chemistry, as it tells us how many hydrogen ions are present in a solution. This measurement is crucial because it directly affects the pH level of the solution. In this exercise, we have a hydrogen ion concentration of \(9.04 \times 10^{-13} \, ext{M}\), which means there are very few hydrogen ions per liter of the solution.
The concentration of hydrogen ions helps to determine the acidity or basicity of a solution. A lower concentration of hydrogen ions typically results in a basic solution, whereas a higher concentration indicates an acidic solution.
When calculating pH, it's essential to first understand the given \( ext{[H}^+]\) properly, as the entire calculation depends on it. Each level of hydrogen ion concentration corresponds to a step on the pH scale, where lower concentrations lead to higher pH values (more basic).

Logarithmic Function

Logarithmic functions are mathematical operations that make it easier to handle large or small numbers, like those often seen in chemistry. The pH calculation is a practical application of logarithms, specifically the base-10 logarithm, which is noted as \(\text{log}_{10}\).
When we calculate the pH of a solution, we essentially compute the negative logarithm of the hydrogen ion concentration: \(\text{pH} = -\text{log}_{10}[\text{H}^+]\).
Understanding how logarithms work is crucial:

• Logarithms convert multiplication into addition, simplifying calculations involving exponential numbers.
• When computing \(\text{log}_{10}(9.04 \times 10^{-13})\), we separately calculate \(\text{log}_{10}(9.04)\) and \(\text{log}_{10}(10^{-13})\). The second step involves recognizing that \(\text{log}_{10}(10^{-13}) = -13\).

By using logarithms, we avoid handling very tiny numbers directly and can interpret results more easily on a standard numerical scale like pH.

Basic Solution

A basic solution has a pH greater than 7. The pH measures the solution's acidity or basicity on a scale typically ranging from 0 to 14. In this example, we calculated a pH value of approximately 12.043, which clearly falls in the basic range.
Basic solutions have fewer hydrogen ions compared to hydroxide ions. This comparison means the solution is more alkaline and less acidic.
Characteristics of basic solutions include:

• They tend to feel slippery to the touch.
• They often taste bitter (though tasting chemical solutions is not advised).
• They turn litmus paper blue and phenolphthalein pink.

Understanding that our pH result signifies a basic solution helps determine its possible applications and necessary precautions when handling it. For example, some cleaning products are basic solutions, as well as some antacids used to neutralize stomach acid.