Problem 51 - Chemistry The Molecular Nature of Matter

Question

Calculate the molar concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in solutions that have the following \(\mathrm{pH}\) values. (a) 8.14 (b) 2.56 (c) 11.25 (d) 13.28 (e) 6.70

Step-by-Step Solution

Answer

For each pH value: (a) \( [\mathrm{H}^{+}] = 7.24 \times 10^{-9} \mathrm{M}, [\mathrm{OH}^{-}] = 1.39 \times 10^{-6} \mathrm{M} \); (b) \( [\mathrm{H}^{+}] = 2.75 \times 10^{-3} \mathrm{M}, [\mathrm{OH}^{-}] = 3.63 \times 10^{-12} \mathrm{M} \); (c) \( [\mathrm{H}^{+}] = 5.62 \times 10^{-12} \mathrm{M}, [\mathrm{OH}^{-}] = 1.78 \times 10^{-3} \mathrm{M} \); (d) \( [\mathrm{H}^{+}] = 5.25 \times 10^{-14} \mathrm{M}, [\mathrm{OH}^{-}] = 1.90 \times 10^{-1} \mathrm{M} \); (e) \( [\mathrm{H}^{+}] = 2.00 \times 10^{-7} \mathrm{M}, [\mathrm{OH}^{-}] = 5.00 \times 10^{-8} \mathrm{M} \)

Step 1 - Understanding the relationship between pH, pOH, and concentration

Recall that the pH is the negative logarithm of the hydrogen ion concentration, which is expressed as \( \mathrm{pH} = -\log[\mathrm{H}^{+}] \). Similarly, the pOH is the negative logarithm of the hydroxide ion concentration, \( \mathrm{pOH} = -\log[\mathrm{OH}^{-}] \). The sum of pH and pOH is equal to 14 at 25 degrees Celsius (assuming the water is at 25 degrees Celsius unless otherwise stated) which is given by the equation \( \mathrm{pH} + \mathrm{pOH} = 14 \).

Step 2 - Calculating the concentration of \( \mathrm{H}^{+} \) from pH

Use the definition of pH to find the concentration of \( \mathrm{H}^{+} \). The formula to find the molar concentration is \( [\mathrm{H}^{+}] = 10^{-\mathrm{pH}} \). Apply this conversion to calculate the molar concentration of \( \mathrm{H}^{+} \) for each given pH value.

Step 3 - Calculating the concentration of \( \mathrm{OH}^{-} \) from pH

After finding the concentration of \( \mathrm{H}^{+} \) ions, calculate the pOH by subtracting the pH from 14, \( \mathrm{pOH} = 14 - \mathrm{pH} \). Then, convert pOH to the concentration of \( \mathrm{OH}^{-} \) using the formula \( [\mathrm{OH}^{-}] = 10^{-\mathrm{pOH}} \).

Step 4 - Calculating the concentrations for each given pH value

Repeat Steps 2 and 3 for each pH value given in the exercise to find the corresponding concentrations of \( \mathrm{H}^{+} \) and \( \mathrm{OH}^{-} \) for each solution.

Step 5 - Present the complete set of solutions

(a) For \( \mathrm{pH} = 8.14 \): Calculate \( [\mathrm{H}^{+}] \) and \( [\mathrm{OH}^{-}] \)(b) For \( \mathrm{pH} = 2.56 \): Calculate \( [\mathrm{H}^{+}] \) and \( [\mathrm{OH}^{-}] \)(c) For \( \mathrm{pH} = 11.25 \): Calculate \( [\mathrm{H}^{+}] \) and \( [\mathrm{OH}^{-}] \)(d) For \( \mathrm{pH} = 13.28 \): Calculate \( [\mathrm{H}^{+}] \) and \( [\mathrm{OH}^{-}] \)(e) For \( \mathrm{pH} = 6.70 \): Calculate \( [\mathrm{H}^{+}] \) and \( [\mathrm{OH}^{-}] \)Report the results in molar concentrations.

Key Concepts

pH and pOH Relationship

Understanding the pH and pOH relationship is crucial for grasping acid-base chemistry. pH stands for potential hydrogen and is a measure of the acidity or alkalinity of a solution. The pH scale ranges from 0 to 14, where lower values indicate high acidity, and higher values indicate high alkalinity (basicity). pH and pOH are related by the simple equation, \(\text{pH} + \text{pOH} = 14\). This relationship shows that if you know the pH of a solution, you can easily calculate the pOH, and vice versa. For instance, if a solution's pH is 7, which is neutral (like pure water), the pOH is also 7 because \(7 + 7 = 14\). This balance is due to the ion product constant for water, \(K_w\), at 25°C being \(1 \times 10^{-14}\), essentially the basis of the equation.

Hydrogen Ion Concentration

The hydrogen ion concentration, represented by \( [\mathrm{H}^{+}] \), is a measure of the number of hydrogen ions present in a solution. It is directly related to the acidity of the solution. pH is calculated using the negative logarithm of the hydrogen ion concentration: \(\text{pH} = -\log[\mathrm{H}^{+}]\). For example, a pH of 2 corresponds to a hydrogen ion concentration of \(10^{-2} M\), which is quite acidic. In comparison, a higher pH value means a lower \( [\mathrm{H}^{+}] \) and less acidity. With this logarithmic relationship, each one-unit change in pH corresponds to a tenfold change in hydrogen ion concentration, illustrating the power of the logarithmic scale in amplifying changes in ionic concentration.

Hydroxide Ion Concentration

On the other side of acid-base reactions is the hydroxide ion concentration, \( [\mathrm{OH}^{-}] \). The hydroxide ion concentration measures the number of hydroxide ions in a solution, and is indicative of its basicity. To calculate the pOH, we similarly take the negative logarithm of the hydroxide ion concentration: \(\text{pOH} = -\log[\mathrm{OH}^{-}]\). A solution with a high hydroxide ion concentration will have a low pOH and, based on the \(\text{pH} + \text{pOH} = 14\) relationship, a high pH. For example, if a solution has a pOH of 3, the corresponding \( [\mathrm{OH}^{-}] \) would be \(10^{-3} M\), which translates to a pH of 11, indicating a basic solution.

Logarithmic Scale in Chemistry

The logarithmic scale is an essential concept in chemistry, particularly in the context of pH and pOH. It converts multiplication and division of concentrations into simple addition and subtraction. This logarithmic scale expresses a wide range of concentration values in a more manageable form. Because it is a base-10 log scale, each step on the pH scale means a tenfold change in acidity or basicity. This scale allows chemists to work with the extremely large and small numbers associated with ion concentrations in an intuitive manner. Furthermore, due to the logarithmic scale's sensitivity, very small changes in hydrogen or hydroxide ion concentration result in more significant shifts in pH or pOH values, providing a clear indication of a solution's acid or base strength.

Acid-Base Chemistry

Acid-base chemistry revolves around understanding the behaviors of acids and bases in solution. Acids are substances that increase the hydrogen ion concentration \( [\mathrm{H}^{+}] \) in a solution, while bases increase the hydroxide ion concentration \( [\mathrm{OH}^{-}] \). The strength of acids and bases is quantified by their dissociation in water: strong acids and bases dissociate completely, whereas weak acids and bases do not. The pH scale serves as a tool to measure the relative strength of acids and bases. Through titration, a technique where a known concentration of acid reacts with a known concentration of base, the point of neutralization can be found, helping us determine unknown concentrations. Acid-base reactions are fundamental to a variety of chemical processes, including biological functions, industrial applications, and environmental systems.