Question

Determine the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) to the correct number of significant figures in a solution with each pH. Describe how these calculations show the relationship between the number of digits to the right of the decimal place in \(\mathrm{pH}\) and the number of significant figures in concentration. \(\mathrm{pH}=2.50\) \(\mathrm{pH}=2.51\) \(\mathrm{pH}=2.52\)

Short answer

\([\mathrm{H}_3\mathrm{O}^+]\) for pH 2.50 is \(3.16 \times 10^{-3}\) M, for pH 2.51 is \(3.09 \times 10^{-3}\) M, and for pH 2.52 is \(3.02 \times 10^{-3}\) M, each with 2 significant figures.

Step-by-step solution

Understand the relationship between pH and \(\mathrm{H}_3\mathrm{O}^+\) concentration

The pH of a solution is the negative base-10 logarithm of the hydronium ion concentration. The formula that describes this relationship is \(\text{pH} = -\log[\mathrm{H}_3\mathrm{O}^+]\). To find the concentration of \(\mathrm{H}_3\mathrm{O}^+\), we need to take the antilog (also known as inverse logarithm) of the negative pH value.

Calculate \(\mathrm{H}_3\mathrm{O}^+\) concentration for \(\text{pH} = 2.50\)

To find the concentration of \(\mathrm{H}_3\mathrm{O}^+\) when \(\text{pH} = 2.50\), we use the formula: \[\mathrm{H}_3\mathrm{O}^+ = 10^{-\text{pH}} = 10^{-2.50} = 3.16 \times 10^{-3}\] M. The number of significant figures in the concentration is determined by the number of digits to the right of the decimal in the pH value, which is 2 in this case.

Calculate \(\mathrm{H}_3\mathrm{O}^+\) concentration for \(\text{pH} = 2.51\)

Following the same process for \(\text{pH} = 2.51\), the concentration of \(\mathrm{H}_3\mathrm{O}^+\) is: \[\mathrm{H}_3\mathrm{O}^+ = 10^{-\text{pH}} = 10^{-2.51} = 3.09 \times 10^{-3}\] M. Again, we must report the result with 2 significant figures, corresponding to the 2 digits after the decimal in the pH value.

Calculate \(\mathrm{H}_3\mathrm{O}^+\) concentration for \(\text{pH} = 2.52\)

Finally, for \(\text{pH} = 2.52\), the concentration of \(\mathrm{H}_3\mathrm{O}^+\) is: \[\mathrm{H}_3\mathrm{O}^+ = 10^{-\text{pH}} = 10^{-2.52} = 3.02 \times 10^{-3}\] M. The result is also reported with 2 significant figures due to the 2 digits after the decimal in the pH.

Explain the relationship of significant figures

The relationship between the number of digits to the right of the decimal place in \(\text{pH}\) value and the number of significant figures in the concentration of \(\mathrm{H}_3\mathrm{O}^+\) shows that the significant figures in the concentration directly correspond to the number of decimal places in the pH value.

Key concepts

pH Calculation

Understanding the pH calculation is crucial for chemistry students as it describes the acidity or basicity of a solution. The pH scale ranges from 0 to 14, where a pH of 7 is neutral, values below 7 are acidic, and values above 7 are basic or alkaline. The pH is calculated using the formula:
equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: When applying this with a given pH, for example, pH=2.50, the concentration of equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: Hydronium ion concentration can therefore be expressed in scientific notation to match the precision indicated by the pH value.

Significant Figures in Chemistry

In chemistry, reporting the correct number of significant figures in measurements and calculations is fundamental for precision and accuracy. Significant figures refer to all known digits plus one estimated digit in a number. The rules for identifying significant figures are designed to avoid overestimating the precision of calculations and measurements.

• Non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros are not significant.
• Trailing zeros are significant if there is a decimal point present.

Applying this to our earlier pH examples, the pH values have two digits to the right of the decimal point. This precision dictates that the hydronium ion concentrations must also be reported with two significant figures. Consequently, the scientific notation of these concentrations reflects the significant figures, providing a clear and accurate representation of the solution's concentration.

Hydronium Ion Concentration

Hydronium ion concentration is a measure of the amount of equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: present in a solution. It is a direct indicator of a solution's acidity. Understanding this concentration helps chemists make precise adjustments to pH when formulating or neutralizing solutions. Scientific notation is often used when expressing hydronium ion concentration because the values are frequently very small, as seen in our exercise with equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: equation: Moreover, as the pH changes even by a small decimal value, the hydronium ion concentration changes significantly due to the logarithmic scale reflecting a tenfold change for each unit change in pH. This sensitivity underscores the importance of significant figures in communicating the precise concentration of ions in the solution.