Problem 23 - Chemistry An Atoms First Approach

Question

How many significant figures are there in the following numbers: \(10.78,6.78,0.78 ?\) If these were \(\mathrm{pH}\) values, to how many significant figures can you express the \(\left[\mathrm{H}^{+}\right] ?\) Explain any discrepancies between your answers to the two questions.

Step-by-Step Solution

Answer

There are 4 significant figures in \(10.78\), 3 in \(6.78\), and 2 in \(0.78\). If these were pH values, the corresponding [H+] concentrations would be \(1.62 \times 10^{-11}\), \(1.66 \times 10^{-7}\), and \(1.66\), having 2 significant figures each. The discrepancy arises because the significant figures of [H+] are determined only by the logarithm part of the pH value, not the overall number of significant figures in the pH value.

Step 1: Identify the significant figures in the given numbers

To identify the significant figures, we'll apply the following rules: 1. Non-zero digits are always significant. 2. Any zeros between significant figures are significant. 3. Leading zeros are not significant. 4. Trailing zeros are significant only if they appear after the decimal point. Applying these rules, we find the number of significant figures in each number: \(10.78\): has four significant figures (1, 0, 7, 8) \(6.78\): has three significant figures (6, 7, 8) \(0.78\): has two significant figures (7, 8)

Step 2: Calculate the [H+] concentration using the pH formula

To calculate the concentration of hydrogen ions [H+] from pH values, we use the following formula: pH = -log[H+] To find [H+], we need to take the antilogarithm (inverse of the logarithm) of the pH values: [H+] = 10^(-pH) Using this formula, we calculate the [H+] values for the given pH values: [H+] for pH \(10.78 = 10^{-10.78} ≈ 1.62 \times 10^{-11}\) [H+] for pH \(6.78 = 10^{-6.78} ≈ 1.66 \times 10^{-7}\) [H+] for pH \(0.78 = 10^{-0.78} ≈ 1.66\)

Step 3: Determine the number of significant figures in the [H+] values

Now we'll find the number of significant figures in the [H+] values: - [H+] for pH \(10.78 : 1.62 \times 10^{-11}\) - has two significant figures (1, 6) - [H+] for pH \(6.78 : 1.66 \times 10^{-7}\) - has two significant figures (1, 6) - [H+] for pH \(0.78 : 1.66\) - has two significant figures (1, 6)

Step 4: Compare and explain the discrepancies

We'll now compare the significant figures of the given numbers (as pH values) and their corresponding [H+] values: | pH value | Significant figures | [H+] value | Significant figures | |:--------:|:-------------------:|:----------------:|:-------------------:| | 10.78 | 4 | 1.62 x 10^{-11} | 2 | | 6.78 | 3 | 1.66 x 10^{-7} | 2 | | 0.78 | 2 | 1.66 | 2 | We observe that the significant figures of the pH values and the [H+] values differ for pH 10.78 and 6.78 consistently. This is because the number of significant figures of [H+] is determined only by the logarithm part of the pH value and not the overall number of significant figures in the pH value. The calculation involving logarithm results in a new count of significant digits, which is not affected by the initial count of significant digits in the pH values.