Problem 31 - Chemistry Matter and Change

Question

Challenge Calculate the \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in a sample of seawater with a \(\mathrm{pOH}=5.60\)

Step-by-Step Solution

Answer

In the given sample of seawater with \(\mathrm{pOH}=5.60\), we calculate the concentrations of hydroxide ions, \(\left[\mathrm{OH}^{-}\right]\), and hydrogen ions, \(\left[\mathrm{H}^{+}\right]\) as follows: 1. Find \(\left[\mathrm{OH}^{-}\right]\) using the equation \(\left[\mathrm{OH}^{-}\right] = 10^{-\mathrm{pOH}}\): $$\left[\mathrm{OH}^{-}\right] \approx 2.51 \times 10^{-6} \, M$$ 2. Find \(\left[\mathrm{H}^{+}\right]\) using the ion product constant of water, \(K_w\), and the equation \(\left[\mathrm{H}^{+}\right] = \frac{K_w}{\left[\mathrm{OH}^{-}\right]}\): $$\left[\mathrm{H}^{+}\right] \approx 3.98 \times 10^{-9} \, M$$

Step 1: Calculate the concentration of hydroxide ions using pOH

To find the concentration of hydroxide ions, we will use the formula $$\mathrm{pOH} = -\log{\left[\mathrm{OH}^{-}\right]}$$ We are given \(\mathrm{pOH} = 5.60\). We can find \(\left[\mathrm{OH}^{-}\right]\) by solving for it in the equation: $$\left[\mathrm{OH}^{-}\right] = 10^{-\mathrm{pOH}}$$

Step 2: Calculate the concentration of hydrogen ions using the ion product constant of water

The ion product constant of water, \(K_w\), is given by the following formula: $$K_w = \left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right]$$ The value of \(K_w\) at 25 degrees Celsius is \(1.0 \times 10^{-14}\). Using the calculated \(\left[\mathrm{OH}^{-}\right]\) in Step 1, we can find \(\left[\mathrm{H}^{+}\right]\) by solving for it in the equation: $$\left[\mathrm{H}^{+}\right] = \frac{K_w}{\left[\mathrm{OH}^{-}\right]}$$

Step 3: Calculate the concentrations for H+ and OH-

Now, let's calculate the concentrations using the equations gathered in the previous steps: 1. Find \(\left[\mathrm{OH}^{-}\right]\): $$\left[\mathrm{OH}^{-}\right] = 10^{-\mathrm{pOH}} = 10^{-5.60} \approx 2.51 \times 10^{-6} \, M$$ 2. Find \(\left[\mathrm{H}^{+}\right]\): $$\left[\mathrm{H}^{+}\right] = \frac{1.0 \times 10^{-14}}{2.51 \times 10^{-6}} \approx 3.98 \times 10^{-9} \, M$$ Thus, the concentration of hydrogen ions, \(\left[\mathrm{H}^{+}\right]\), in the seawater sample is approximately \(3.98 \times 10^{-9} \, M\), and the concentration of hydroxide ions, \(\left[\mathrm{OH}^{-}\right]\), is approximately \(2.51 \times 10^{-6} \, M\).

Key Concepts

pOH and pH Relationship

Understanding the balance between acidity and basicity in solutions such as seawater is essential, and the relationship between pOH and pH is central to this balance. The pH scale is a measure of the acidity or basicity of a solution, where lower pH values indicate higher acidity, and higher values indicate higher basicity. Conversely, pOH measures the hydroxide ion concentration and follows an inverse relationship to pH.

Mathematically, pH and pOH are connected through the equation:
\[ pH + pOH = 14 \]
at 25 degrees Celsius. This relationship stems from the ion product constant of water and allows us to calculate one value if the other is known. For instance, if a seawater sample has a pOH of 5.60, the pH can be easily figured out by subtracting this value from 14, which gives a pH of 8.40. This relatively high pH indicates that seawater is a basic (or alkaline) solution. This knowledge is crucial for marine biology, oceanography, and environmental science, as the pH of seawater affects marine life and ecosystem health.

When solving chemistry problems involving acidity and basicity, it is critical to keep this relationship in mind, as it can simplify calculations and enhance understanding of the chemical nature of a solution.

Ion Product Constant of Water

At the heart of understanding acid-base chemistry is the concept of the ion product constant of water, denoted as \(K_w\). This value is a constant at a given temperature, representing the product of the concentrations of hydrogen ions \(\left[\mathrm{H}^+\right]\) and hydroxide ions \(\left[\mathrm{OH}^-\right]\) in pure water. At the widely referenced temperature of 25 degrees Celsius, the \(K_w\) is equal to \(1.0 \times 10^{-14}\).

The significance of \(K_w\) is that it remains constant for all aqueous solutions, meaning the product of the concentrations of \(\left[\mathrm{H}^+\right]\) and \(\left[\mathrm{OH}^-\right]\) will always equate to \(1.0 \times 10^{-14}\) at 25 degrees Celsius, regardless of the solution's acidity or basicity. This fact allows us to calculate the missing ion concentration if one is known. For example, the exercise provided used the known value of \(K_w\) to determine the concentration of hydrogen ions after calculating the hydroxide ion concentration, thus showing a quantitative relation between \(\left[\mathrm{H}^+\right]\) and \(\left[\mathrm{OH}^-\right]\) in seawater.

Hydrogen and Hydroxide Ion Concentration

The concentrations of hydrogen ions \(\left[\mathrm{H}^+\right]\) and hydroxide ions \(\left[\mathrm{OH}^-\right]\) not only define the pH and pOH values but also give a complete picture of the solution's acidic or basic nature. These ion concentrations are usually expressed in molarity (M), which corresponds to moles per liter.

When we measure the pH of a solution, we are indirectly assessing the concentration of hydrogen ions present. A lower pH means a higher concentration of hydrogen ions, signifying an acidic solution. Conversely, the pOH reflects the concentration of hydroxide ions, with a lower pOH indicating a higher concentration and therefore a more basic solution. In the given seawater example, the concentration of hydroxide ions was calculated first, showing a relatively high concentration (2.51 \times 10^{-6} M), aligning with the basic nature of seawater.

These ion concentrations are inversely related due to the constant nature of \(K_w\). Therefore, a high \(\left[\mathrm{OH}^-\right]\) implies a low \(\left[\mathrm{H}^+\right]\), and vice versa. This inverse relationship is harnessed in various scientific fields to control the pH of solutions—for instance, by adding acids or bases in chemical processes or by buffering biological systems to maintain homeostasis.