Question

Calculate the \(\mathrm{pH}\) of each acid solution. Explain how the resulting pH values demonstrate that the \(\mathrm{pH}\) of an acid solution should carry as many digits to the right of the decimal place as the number of significant figures in the concentration of the solution. $$ \begin{aligned} \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] &=0.044 \mathrm{M} \\ \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] &=0.045 \mathrm{M} \\ \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] &=0.046 \mathrm{M} \end{aligned} $$

Short answer

pH values: 1.356 for [H3O+]=0.044 M, 1.347 for [H3O+]=0.045 M, 1.337 for [H3O+]=0.046 M. The pH values carry as many digits to the right of the decimal as there are significant figures in the concentrations.

Step-by-step solution

Understand the concept of pH

The pH of a solution is calculated using the formula pH = -log[H3O+], where [H3O+] represents the concentration of hydronium ions in moles per liter (M). The number of significant figures in the hydronium ion concentration should be reflected in the number of decimal places in the pH value.

Calculate the pH for the first concentration

Use the first concentration [H3O+] = 0.044 M and apply the pH formula: pH = -log(0.044). Make sure to use a scientific calculator and keep the significant figures in mind for the final pH value.

Calculate the pH for the second concentration

Use the second concentration [H3O+] = 0.045 M and apply the pH formula: pH = -log(0.045). The result should have the same number of decimal places as the significant figures in the concentration.

Calculate the pH for the third concentration

Use the third concentration [H3O+] = 0.046 M and apply the pH formula: pH = -log(0.046). The number of significant figures in the concentration should dictate the number of decimal places in the pH value.

Round the results according to significant figures

After obtaining the pH values from the previous steps, round them if necessary to match the number of significant figures from the respective hydronium ion concentrations.

Key concepts

pH of Acid Solution

Understanding the pH of an acid solution is crucial for students in chemistry. The pH scale is a measure of how acidic or basic a solution is, ranging from 0 to 14, with 0 being the most acidic, 7 neutral, and 14 the most basic.

For an acid solution, the pH is calculated with the formula \(\text{pH} = -\log[\text{H}_3\text{O}^{+}]\), where \[\text{H}_3\text{O}^{+}\] represents the concentration of hydronium ions. A lower pH value indicates a higher acidity. To calculate this, one must be capable of using a scientific calculator to find the logarithm of the hydronium ion concentration. It's important to note that the pH is a logarithmic scale, meaning each whole number on the scale represents a tenfold change in acidity.

Understanding the Logarithmic Nature of pH

Because pH is based on a logarithmic function, it is not measured in units but rather as a dimensionless number. This is why the pH of a solution is often misunderstood. To clarify, each step on the pH scale represents a tenfold increase or decrease in acidity or basicity.

Significant Figures in Chemistry

Significant figures play an integral role in ensuring the precision of measured values in chemistry. They are the digits in a measurement that carry meaning contributing to its precision. This includes all non-zero digits, zeros between non-zero digits, and zeros that are placeholders for significant decimals.

For pH calculations, the significant figures in the hydronium ion concentration dictate the number of decimal places in the calculated pH value. This is because the significant figures represent the precision of our knowledge about the concentration's measurement, which carries through to the accuracy of the calculated pH.

Applying Significant Figures to pH Calculation

When performing the pH calculation, you must ensure that the final answer reflects the significant figures from the initial concentration. If the concentration has two significant figures, as in the example of \(0.044 \text{M}\), the resulting pH value should have two digits after the decimal point to maintain the consistent level of precision.

Hydronium Ion Concentration

The hydronium ion concentration in a solution is a key component in determining the acidity of the solution. In chemistry, the concentration of these ions is usually expressed in moles per liter (M), and this concentration is what we use to calculate the pH.

As we saw in the given exercise, varying concentrations of hydronium ions, such as \(0.044 \text{M}\), \(0.045 \text{M}\), and \(0.046 \text{M}\), will each yield a slightly different pH value. This is reflective of the sensitivity of the pH to the precise amount of hydronium ions present. It is critical to measure and record these concentrations accurately, as they significantly impact the pH calculations and the corresponding significant figures.

Impact on Chemical Reactions

The concentration of hydronium ions not only affects the pH but also influences the rates of many chemical reactions. For instance, a small difference in hydronium ion concentration can significantly alter the outcome of reactions that are sensitive to pH, such as those in biological systems or chemical equilibria.