Problem 40 - Chemistry Matter and Change

Question

Calculate The pH of a tomato is approximately \(4.50 .\) What are \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in a tomato?

Step-by-Step Solution

Answer

The concentration of hydrogen ions \(\left[\text{H}^+\right]\) in a tomato is approximately \(3.16 \times 10^{-5}\,\text{M}\), and the concentration of hydroxide ions \(\left[\text{OH}^-\right]\) is approximately \(3.17 \times 10^{-10}\,\text{M}\).

Step 1: Calculate the concentration of hydrogen ions

To convert the pH to the concentration of hydrogen ions, we can use the following formula: \[ \text{pH} = -\log_{10}\left[\text{H}^+\right] \] Rearranging to isolate \(\left[\text{H}^+\right]\): \[ \left[\text{H}^+\right] = 10^{-\text{pH}} \] Now, we can plug in the given pH value (4.50): \[ \left[\text{H}^+\right] = 10^{-4.50} \] Calculating this value gives: \[ \left[\text{H}^+\right] = 3.16 \times 10^{-5}\,\text{M} \]

Step 2: Calculate the concentration of hydroxide ions

To find the concentration of hydroxide ions, we will use the ion product constant of water (Kw), which is: \[ K_\text{w} = \left[\text{H}^+\right] \cdot \left[\text{OH}^-\right] = 1.0 \times 10^{-14} \] We already found the concentration of hydrogen ions, so we can use this value to find the concentration of hydroxide ions: \[ \left[\text{OH}^-\right] = \frac{K_\text{w}}{\left[\text{H}^+\right]} = \frac{1.0 \times 10^{-14}}{3.16 \times 10^{-5}} \] Calculating this value gives: \[ \left[\text{OH}^-\right] = 3.17 \times 10^{-10}\,\text{M} \] So, the concentration of hydrogen ions \(\left[\text{H}^+\right]\) in a tomato is approximately \(3.16 \times 10^{-5}\,\text{M}\), and the concentration of hydroxide ions \(\left[\text{OH}^-\right]\) is approximately \(3.17 \times 10^{-10}\,\text{M}\).

Key Concepts

Hydrogen Ion Concentration

Understanding the concentration of hydrogen ions, denoted as \(\left[\text{H}^+\right]\), is crucial for calculating pH, a measure of acidity or basicity of a solution. The pH scale is logarithmic, which means each whole pH value below 7 (which is neutral) is ten times more acidic than the next higher value. For instance, a pH of 3 is ten times more acidic than a pH of 4.

To determine the \(\left[\text{H}^+\right]\) from the pH, one can rearrange the basic pH formula:\[\text{pH} = -\log_{10}\left[\text{H}^+\right]\]\to:\[\left[\text{H}^+\right] = 10^{-\text{pH}}\]\For a tomato with a pH of approximately 4.50, this calculation gives us an \(\left[\text{H}^+\right]\) of \(3.16 \times 10^{-5}\) M (molarity), which indicates a relatively acidic environment within the tomato.

Hydroxide Ion Concentration

In a similar vein, the hydroxide ion concentration \(\left[\text{OH}^-\right]\) represents the amount of hydroxide ions present in a solution. It plays an integral role in defining the solution's basicity. Just as hydrogen ions are crucial for acidity, hydroxide ions are for basicity. The higher the concentration of \(\left[\text{OH}^-\right]\), the more basic the solution is.

To find this concentration when the pH or \(\left[\text{H}^+\right]\) is known, we use the ion product constant of water (Kw) as follows:\[\left[\text{OH}^-\right] = \frac{K_{\text{w}}}{\left[\text{H}^+\right]}\]\With the value of Kw being \(1.0 \times 10^{-14}\), for our example of a tomato with a hydrogen ion concentration \(3.16 \times 10^{-5}\) M, we find \(\left[\text{OH}^-\right]\) to be \(3.17 \times 10^{-10}\) M.

Ion Product Constant of Water

The ion product constant of water, often symbolized as Kw, is a fundamental constant of water chemistry. It reflects the equilibrium concentration of hydrogen and hydroxide ions in pure water at a certain temperature, typically 25°C. At this temperature, Kw has a value of \(1.0 \times 10^{-14}\).

This relationship is expressed as:\[K_{\text{w}} = \left[\text{H}^+\right] \cdot \left[\text{OH}^-\right]\]\This constant is crucial in determining the relationship between hydrogen and hydroxide ion concentrations in a solution. Whenever one of these concentrations changes, the other adjusts to maintain the constant value of Kw.

pH and pOH Relationship

pH and pOH are two sides of the same coin; they are inversely related to each other and together they provide a full picture of a solution's acidity and basicity. While pH focuses on the concentration of hydrogen ions, pOH looks at the concentration of hydroxide ions. The sum of pH and pOH always equals 14, which is derived from the ion product constant of water (Kw) at standard temperature.

To find pOH given the pH (or vice versa), you can use this relationship:\[\text{pH} + \text{pOH} = 14\]\So for a tomato with a pH of 4.50, its pOH would be 14 - 4.50, equating to 9.50. This reveals that the tomato is acidic since its pH is below 7, and accordingly, its pOH is above 7, indicating a low concentration of hydroxide ions.