Problem 104 - Chemistry: An Atoms-Focused Approach

Question

The value of \(K_{w}\) increases as temperature increases. a. If the \(\mathrm{p} K_{\mathrm{w}}=13.017\) at \(60^{\circ} \mathrm{C},\) what is the \(\left[\mathrm{H}^{+}\right] ?\) b. What is the pH of water at \(60^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: The concentration of H+ ions is \(\sqrt{10^{-13.017}}\), and the pH of the water is \(-\log \sqrt{10^{-13.017}}\).

Step 1: Understand the pKw and Kw relationship

pKw is the negative logarithm of the ion product of water (Kw). We can find Kw by taking the inverse logarithm of its pKw value.

Step 2: Calculate Kw

Given \(\mathrm{p}K_{\mathrm{w}}= 13.017\), we can calculate Kw using the formula: $$K_w = 10^{-\mathrm{p}K_{\mathrm{w}}}$$ $$K_w = 10^{-13.017}$$

Step 3: Calculate H+ concentration

For water, the following relation holds: $$K_w = [\mathrm{H}^{+}] [\mathrm{OH}^{-}]$$ Since water is neutral, the concentrations of H+ ions and OH- ions are equal, that is: $$[\mathrm{H}^{+}] = [\mathrm{OH}^{-}]$$ Thus, we can write the equation as: $$K_w = [\mathrm{H}^{+}]^2$$ To find the concentration of H+ ions, we take the square root of Kw: $$[\mathrm{H}^{+}] = \sqrt{K_w}$$ $$[\mathrm{H}^{+}] = \sqrt{10^{-13.017}}$$ #b. Finding the pH of water at 60°C#

Step 4: Understand the pH and H+ concentration relationship

The pH of a solution is the negative logarithm of the concentration of H+ ions. We can use this relationship to find the pH using the H+ concentration calculated in part a.

Step 5: Calculate the pH

Using the formula for pH, we have: $$\mathrm{pH} = -\log [\mathrm{H}^{+}]$$ $$\mathrm{pH} = -\log \sqrt{10^{-13.017}}$$

Key Concepts

pH Calculation

In chemistry, pH is an essential measurement that tells us how acidic or basic a solution is. The formula for calculating pH is:
\[\mathrm{pH} = -\log [\mathrm{H}^+]\]Here, \([\mathrm{H}^+]\) is the concentration of hydrogen ions.
For example, in water at 60°C, we first found the hydrogen ion concentration using the ion product of water, \(K_w\). With \(K_w\) calculated, we determined \([\mathrm{H}^+]\) using:

• \([\mathrm{H}^+] = \sqrt{K_w}\)

Then, the pH is simply obtained by taking the negative logarithm of the \([\mathrm{H}^+]\) value. This results in the final pH measure.
Understanding pH helps us gauge the nature of solutions, predicting reaction behaviors or assessing environmental conditions.

Temperature Dependence

The ion product of water, \(K_w\), changes with temperature. This means that the acidity or alkalinity of water can shift as temperature varies. At higher temperatures:

• \(K_w\) increases.
• This leads to more hydrogen ions \([\mathrm{H}^+]\) and hydroxide ions \([\mathrm{OH}^-]\) present in the solution.

A higher \(K_w\) indicates that water is a better conductor of electricity because more ions are available. As a result, the pH of pure water at higher temperatures is not strictly 7, as often assumed at 25°C.
In our example, with \(pK_w = 13.017\) at 60°C, we see how pH differs from the standard room temperature expectations. Recognizing this adjustment helps in applications such as chemical reactions, which might be sensitive to changes in temperature.

pKw

\(pK_w\) is a measure of the strength of the ionization of water, found using:
\[\mathrm{p}K_w = -\log K_w\]This value helps scientists understand how water ionizes under different conditions. It equates the intricate balance between \([\mathrm{H}^+]\) and \([\mathrm{OH}^-]\) in water. By taking the negative logarithm of \(K_w\), you get \(pK_w\), specific to any temperature.
In our example, a \(pK_w\) of 13.017 at 60°C tells us that water has a lower pH than at room temperature. This insight is crucial when predicting the behavior of solutions and in processes such as neutralization reactions.
Understanding \(pK_w\) ensures you consider the temperature's effect, adjusting expectations in chemical analyses or laboratory settings.