Question

Calculate the molar concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in solutions that have the following \(\mathrm{pH}\) values. (a) 12.67 (b) 5.18 (c) 11.55 (d) 4.22 (e) 6.06

Short answer

The molar concentrations of \(\mathrm{H}^+\)\) and \(\mathrm{OH}^-\)\) for the given pH values are:(a) \(\mathrm{H}^+ = 2.14 \times 10^{-13}\) M, \(\mathrm{OH}^- = 4.68 \times 10^{-2}\) M(b) \(\mathrm{H}^+ = 6.61 \times 10^{-6}\) M, \(\mathrm{OH}^- = 1.51 \times 10^{-9}\) M(c) \(\mathrm{H}^+ = 2.82 \times 10^{-12}\) M, \(\mathrm{OH}^- = 3.55 \times 10^{-3}\) M(d) \(\mathrm{H}^+ = 6.03 \times 10^{-5}\) M, \(\mathrm{OH}^- = 1.66 \times 10^{-10}\) M(e) \(\mathrm{H}^+ = 8.71 \times 10^{-7}\) M, \(\mathrm{OH}^- = 1.15 \times 10^{-8}\) M.

Step-by-step solution

Understand the Relationship Between pH and H+ Concentration

The pH of a solution is defined as the negative base-10 logarithm of the hydrogen ion concentration. This relationship can be expressed by the equation \( \mathrm{pH} = -\log_{10}(\mathrm{H}^+) \). To find \( \mathrm{H}^+ \) concentration from pH, we can reverse this relationship by taking the inverse logarithm (antilogarithm) of the negative pH value, which gives \( \mathrm{H}^+ = 10^{-\mathrm{pH}} \).

Calculate the Hydrogen Ion Concentration

Use the formula \( \mathrm{H}^+ = 10^{-\mathrm{pH}} \) to find the molar concentrations of \( \mathrm{H}^+ \):(a) \( \mathrm{H}^+ = 10^{-12.67} \) molar(b) \( \mathrm{H}^+ = 10^{-5.18} \) molar(c) \( \mathrm{H}^+ = 10^{-11.55} \) molar(d) \( \mathrm{H}^+ = 10^{-4.22} \) molar(e) \( \mathrm{H}^+ = 10^{-6.06} \) molar

Calculate the Hydroxide Ion Concentration Using the Ionic Product of Water

At 25°C, the ionic product of water (\( K_w \) ) is \( 1.0 \times 10^{-14} \) M². The concentration of hydroxide ions can be found using the formula \( \mathrm{OH}^- = \frac{K_w}{\mathrm{H}^+} \). Calculate \( \mathrm{OH}^- \):(a) \( \mathrm{OH}^- = \frac{1.0 \times 10^{-14}}{10^{-12.67}} \) molar(b) \( \mathrm{OH}^- = \frac{1.0 \times 10^{-14}}{10^{-5.18}} \) molar(c) \( \mathrm{OH}^- = \frac{1.0 \times 10^{-14}}{10^{-11.55}} \) molar(d) \( \mathrm{OH}^- = \frac{1.0 \times 10^{-14}}{10^{-4.22}} \) molar(e) \( \mathrm{OH}^- = \frac{1.0 \times 10^{-14}}{10^{-6.06}} \) molar

Key concepts

Hydrogen Ion Concentration

Understanding the hydrogen ion concentration in a solution is crucial in chemistry, as it determines the solution's acidity. The concentration of hydrogen ions, denoted as \( [H^+] \), affects the pH level of the solution. As shown in the step-by-step solution, the pH value is the negative logarithm to the base 10 of the hydrogen ion concentration: \[ pH = -\log_{10}([H^+]) \].

For students, it's important to grasp that the lower the pH, the higher the hydrogen ion concentration and vice versa. This inverse relationship helps to calculate the concentration from a given pH. For example, a pH of 7 corresponds to a neutral solution where \( [H^+] = 10^{-7} \) M, while a pH of 3 would indicate a more acidic solution with \( [H^+] = 10^{-3} \) M. These values demonstrate how even small changes in pH can indicate large differences in acidity due to the logarithmic scale.

Hydroxide Ion Concentration

In relation to hydrogen ion concentration, hydroxide ion concentration, represented as \( [OH^-] \), is another fundamental aspect of aqueous solutions. The concentration of \( OH^- \) ions is particularly relevant in the context of basicity. For example, in a solution with a lower hydrogen ion concentration (indicating basic conditions), \( [OH^-] \) concentrations are higher.

The exercise solution showcases how to find the hydroxide ion concentration using the ionic product of water, which remains constant at \( 1.0 \times 10^{-14} \) M² at 25°C. The formula \( [OH^-] = K_w / [H^+] \) allows the calculation of \( [OH^-] \) once the \( [H^+] \) is known. This relationship between \( [H^+] \) and \( [OH^-] \) emphasizes the balance that exists within a water-based solution, directly affecting the solution's pH.

Ionic Product of Water

The ionic product of water (\( K_w \)) is a constant value that represents the product of the molar concentrations of hydrogen ions and hydroxide ions in pure water. At room temperature (25°C), it is always \( 1.0 \times 10^{-14} \) M². This value is critical because it signifies that the concentrations of \( H^+ \) and \( OH^- \) are inversely proportional in a given solution.

When solving pH-related problems, \( K_w \) provides a pivotal reference point. It allows us to deduce one ion’s concentration knowing the other. The exercise solutions utilize \( K_w \) to find the \( [OH^-] \) from the calculated \( [H^+] \), hence illustrating the direct application of \( K_w \) in such calculations. Students should internalize that even in solutions that are not pure water, the product of \( [H^+] \) times \( [OH^-] \) remains \( K_w \), thereby linking the two ion concentrations in any aqueous system.

pH and pOH Relationship

The relationship between pH and pOH is a foundational concept in acid-base chemistry that helps in understanding the degree of acidity or basicity of a solution. pH represents the acidity level, while pOH represents the basicity level. The two are related by the following equation: \[ pH + pOH = 14 \] at 25°C, which is derived from the ionic product of water (\( K_w \)).

This equation shows that as the pH of the solution decreases (becoming more acidic), the pOH value must increase (becoming less basic), and vice versa, ensuring that their sum always equals 14. Hence, if a solution has a pH of 10, its pOH will be 4. Such understanding helps students to convert between the two measures of hydrogen and hydroxide ion concentrations, providing a complete picture of the solution's characteristics.