Question

For a hydrogen ion concentration of \(2.33 \times 10^{-6} M\), how many decimal places should we give when expressing the \(\mathrm{pH}\) of the solution?

Short answer

The pH of the solution with a hydrogen ion concentration of \(2.33 \times 10^{-6} M\) should be expressed as \(5.63\), with two decimal places.

Step-by-step solution

Calculate the pH of the solution

To calculate the pH of the solution, we will use the formula: \[ \mathrm{pH} = -\log[\ce{H^+}] \] where \([\ce{H^+}]\) is the hydrogen ion concentration. We have a hydrogen ion concentration of \( 2.33 \times 10^{-6} M\). Plug this concentration into the formula: \[ \mathrm{pH} = -\log (2.33 \times 10^{-6}) \]

Solve for pH

To solve for pH, we simply compute the logarithm: \[ \mathrm{pH} = -(-5.632) \] \[ \mathrm{pH} = 5.632 \]

Determine the appropriate number of decimal places

The pH value is 5.632, and it has three decimal places. We usually report the pH value to two decimal places, but it is important to remember that the number of decimal places should be related to the precision of the given hydrogen ion concentration. Since the hydrogen ion concentration was given with two significant figures (\(2.33 \times 10^{-6} \)), it is appropriate to report the pH value with two decimal places as well. Therefore, the pH of the solution should be expressed as \(5.63\).

Key concepts

Understanding Hydrogen Ion Concentration

Hydrogen ion concentration is a measure of the number of hydrogen ions, \(\ce{H^+}\), present in a solution. It's crucial in determining the acidity or basicity of a solution. \[\ [\ce{H^+}] = 2.33 \times 10^{-6} \space M \]This value tells us there's a very small amount of hydrogen ions in the solution, indicating it's slightly acidic. The smaller the concentration, the less acidic the solution. Hydrogen ion concentrations are usually expressed in scientific notation, which allows for the precise handling of very small or large numbers.Why it Matters:

• The concentration directly influences pH levels.
• Small changes in \(\ce{H^+}\) concentration can result in significant changes in pH.

In chemistry, understanding hydrogen ion concentration is essential for predicting how a substance will behave in different chemical reactions.

The Importance of Significant Figures

Significant figures are all about precision in measurement. They tell you how many meaningful digits there are in a number. For hydrogen ion concentration like \(2.33 \times 10^{-6} \) M, the three numbers "2.33" represent significant figures.Key Points About Significant Figures:

• They help ensure accuracy and reliability in scientific calculations.
• The number of significant figures indicates the certainty of the measurement.
• In calculations, the result should generally have the same number of significant figures as the quantity with the fewest significant figures.

When determining the pH, it's important to maintain consistency with significant figures to ensure the calculated pH reflects the precision of the initial concentration.Therefore, reporting the pH with the same number of significant figures as those in the initial concentration (2 significant figures in this case) keeps our results accurate and precise.

Understanding Decimal Places in pH

When reporting pH, the number of decimal places is crucial for ensuring that the value represents the concentration accurately. For a pH derived from \(2.33 \times 10^{-6} \) M, the decimal places tie directly to the significant figures of the hydrogen ion concentration.Here's Why Decimal Places Matter:

• The number of decimal places in a pH value should match the number of significant figures in the \(\ce{H^+}\) concentration.
• More decimal places indicate greater precision in how closely the pH value matches the actual hydrogen ion concentration.
• For the concentration given, rounding the pH to 5.63 ensures that the number of decimal places aligns with the precision of the input data.

By maintaining this consistency, you can trust that the pH value accurately reflects the acidity level of the solution, minimizing errors in interpretations and ensuing calculations.