Question

Autoionization of Water and pH Find the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+},\) to the correct number of significant figures, in a solution with each pH. Describe how your 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. $$\begin{array}{l}{\mathrm{pH}=2.50} \\ {\mathrm{pH}=2.51} \\\ {\mathrm{pH}=2.52}\end{array} $$

Short answer

The concentrations of H_3O^+] are 3.16 x 10^{-3} M, 3.09 x 10^{-3} M, and 3.02 x 10^{-3} M for pH values of 2.50, 2.51, and 2.52 respectively, each with three significant figures.

Step-by-step solution

Understand the concept of pH

pH is a measure of the hydrogen ion concentration in a solution. The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration. The equation is given by pH = -log(H_3O^+]). A higher pH value corresponds to a lower hydrogen ion concentration, and vice versa.

Converting pH to H_3O^+] concentration

To find the concentration of H_3O^+] from the pH, use the equation H_3O^+] = 10^{-pH}. Simply plug in the pH value given into this equation to calculate the concentration of hydrogen ions.

Calculate the concentration for pH = 2.50

For a pH of 2.50, the concentration of H_3O^+] is H_3O^+] = 10^{-2.50}. Calculate this value to find the concentration.

Calculate the concentration for pH = 2.51

For a pH of 2.51, use the same formula: H_3O^+] = 10^{-2.51}. Calculate to find the concentration.

Calculate the concentration for pH = 2.52

Repeat the calculation for a pH of 2.52: H_3O^+] = 10^{-2.52}.

Determine the significance of figures

The number of digits to the right of the decimal place in the pH value corresponds to the number of significant figures in the H_3O^+] concentration. Therefore, for pH = 2.50, the H_3O^+] concentration should have three significant figures. The same applies to pH = 2.51 and pH = 2.52.

Compile the findings

List the calculated H_3O^+] concentrations for each pH value, ensuring that the number of significant figures reflects the number of decimal places in each provided pH value.

Key concepts

Autoionization of Water

Water, H₂O, can undergo a process called autoionization or self-ionization, where two water molecules react with each other to produce a hydronium ion, H₃O⁺, and a hydroxide ion, OH^–. The reaction can be represented as:
2 H₂O ⇌ H₃O⁺ + OH^–.
The equilibrium constant for this reaction, Kw, is a vital constant in chemistry and is given by the product of the concentrations of the hydronium and hydroxide ions. At room temperature (25°C), Kw is typically 1.0 × 10^–14. Understanding this reaction is essential when discussing pH and hydrogen ion concentration since it establishes the basis of the pH scale.

Significance in pH and pOH

The pH of pure water is 7, which is considered neutral. This neutrality comes from the equal concentrations of H₃O⁺ and OH^– in pure water. If either concentration increases, the other decreases, resulting in a change in pH. Thus, the autoionization concept is pivotal in comprehending how solutions can be acidic or basic.

Significant Figures in Chemistry

Significant figures are crucial in chemistry as they indicate the precision of measurements and calculations. In the context of pH and concentration calculations, significant figures become particularly important.
The number of significant figures is guided by the precision of the measurement. For example, when we express a pH as 2.50, it implies that the measurement is precise to two decimal places, and thus the resulting H₃O⁺ concentration calculated from this pH should also be reflected with three significant figures.

Why Precision Matters

Chemists must use the correct number of significant figures to accurately convey the certainty of their results. When solving for concentrations from pH, every decimal place in the pH translates to a significant digit in the concentration, since the pH is based on a logarithmic scale. This reflects the exponential relationship between H₃O⁺ concentration and pH values.

Calculating H3O+ Concentration

To calculate the hydronium ion concentration, H₃O⁺, from a given pH, we use the formula related to the definition of pH:
H₃O⁺ concentration = 10^–pH.
Engaging in an example, a pH value of 2.50 would mean a H₃O⁺ concentration of 10^–2.50, or 3.16 × 10^–3 M, considering three significant figures reflecting those in the pH value.

Relating pH and H₃O⁺ Concentration

In a step-by-step procedure, you take the antilog (inverse logarithm) of the negative pH value to get the concentration of H₃O⁺. It's also imperative to meticulously adhere to the significant figures throughout this calculation to maintain the integrity of the data precision. As observed from the exercise, with pH values 2.50, 2.51, and 2.52, the amount of decimal places directly influences the number of significant figures in the calculated hydronium ion concentration, underscoring the relationship between these two measures.