How many significant figures are there in the following num- bers: \(10.78,6.78,0.78 ?\) If these were 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.

To determine the number of significant figures in a number, follow these rules: 1. Non-zero digits are always significant. 2. Any zero between significant digits is also significant. 3. Leading zeros are not significant. 4. Trailing zeros are significant only if there is a decimal point in the number. Now, let's find out the significant figures for each given number: \(10.78\) - All the digits are non-zero, and there's a decimal point, so there are 4 significant figures. \(6.78\) - All the digits are non-zero, and there's a decimal point, so there are 3 significant figures. \(0.78\) - Ignore the leading zero, the other two digits are non-zero and there's a decimal point, so there are 2 significant figures.

The pH value is the negative base 10 logarithm of the H+ ion concentration, which can be expressed as: pH = -log10[H+] To calculate the H+ ion concentration, we need to use the inverse of this operation: [H+] = 10^(-pH) Now, let's find the H+ concentration for each given pH value and determine the number of significant figures: For \(10.78\): [H+] = \(10^{-10.78}\) ≈ \(1.62 \times 10^{-11}\) - There are 3 significant figures in the calculated H+ concentration. For \(6.78\): [H+] = \(10^{-6.78}\) ≈ \(1.66 \times 10^{-7}\) - There are 3 significant figures in the calculated H+ concentration. For \(0.78\): [H+] = \(10^{-0.78}\) ≈ \(1.66 \times 10^{-1}\) - There are 3 significant figures in the calculated H+ concentration.

We can observe that the number of significant figures in H+ concentrations is the same for all three values, which is 3. However, the number of significant figures is different among the given pH values. This discrepancy arises from the nature of the pH scale. The pH scale is logarithmic, not linear, and any small change in the pH value can result in a significant change in the concentrations of the hydrogen ions. Therefore, comparing significant figures between pH values and H+ concentrations doesn't always yield a direct correspondence.