Question

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.

Short answer

The given numbers \(10.78\), \(6.78\), and \(0.78\) have 4, 3, and 2 significant figures, respectively. If these numbers are pH values, the corresponding H+ concentrations are \(1.62 \times 10^{-11}\), \(1.66 \times 10^{-7}\), and \(1.66 \times 10^{-1}\), all with 3 significant figures. The discrepancy between the significant figures arises from the logarithmic nature of the pH scale, which means that small changes in pH values can result in significant changes in H+ concentrations.

Step-by-step solution

Find the number of significant figures in each number

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.

Calculate H+ concentration and its significant figures if these numbers are pH values

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.

Explain any discrepancies between the significant figures

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.

Key concepts

pH Calculation

The concept of pH is fundamental in chemistry, acting as a measure of the acidity or basicity of a solution. It is calculated using the equation:

This means the pH value represents the negative logarithm of the hydrogen ion concentration \([H^+]\). A low pH indicates a high concentration of hydrogen ions, meaning the solution is more acidic. Conversely, a higher pH suggests a lower concentration, indicating a more basic solution. Understanding the pH calculation helps in analyzing the acidic or basic nature of different substances.

H+ Ion Concentration

The concentration of hydrogen ions \(H^+\) in a solution is crucial in determining its pH level. This concentration can be derived from the pH value using the inverse logarithmic function:

• [H^+] = 10^{-pH}

This formula highlights that for every unit change in pH, the hydrogen ion concentration changes tenfold. Knowing \(H^+\) concentration is essential for understanding reactions in acidic or basic environments. For example, in a solution with a pH of 3, the \(H^+\) concentration is \(10^{-3}\) mol/L, indicating a higher acidity compared to a solution with a pH of 7.

Logarithmic Scale

The pH scale is logarithmic, which means it is based on powers of ten. Unlike linear scales where equal differences indicate equal changes, the logarithmic scale represents exponential changes. Every unit of pH change corresponds to a tenfold change in \(H^+\) concentration.

• A small pH change can lead to a significant change in acidity or basicity.
• This scale compresses wide-ranging \(H^+\) concentrations into manageable numbers between 0 and 14.

This scale is particularly useful in chemistry because it allows easy comparison of vastly different concentrations, making analysis and interpretation straightforward.

Chemical Measurement

In chemical measurement, accuracy and precision are vital, and significant figures play a crucial role in this process. Significant figures in a given measurement convey the level of precision. When calculating properties like pH, it's essential to maintain consistency in significant figures to ensure reliable results.

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant, but trailing zeros can be if a decimal is present.

This concept ensures that data is represented consistently and measurements reflect true scientific accuracy. Understanding how significant figures apply to pH calculations and \(H^+\) concentrations is key in the precise measurement of chemical properties.