Question

Calculate the \(\mathrm{pH}\) of water at \(40^{\circ} \mathrm{C}\), given that \(K_{\mathrm{w}}\) is \(3.8 \times 10^{-14}\) at this temperature.

Short answer

The pH of water at \(40^{\circ} \mathrm{C}\) is \(7.21\).

Step-by-step solution

Understand the Problem and Recognize the Formula

The \(K_{\mathrm{w}}\) indicates product of the molar concentration of H+ and OH- ions in water. For pure water, [H+] = [OH-]. So we can express the \(K_{\mathrm{w}}\) as \(K_{\mathrm{w}}\ =\ [H+]^2\). The task requires finding the pH, which is the negative logarithm of the [H+] concentration. Therefore it includes recognizing and using these formulas.

Solve for [H+]

First, solve the \(K_{\mathrm{w}}\ =\ [H+]^2\) for [H+]. This will require square rooting both sides of the equation. Thus, we get [H+] = \(\sqrt{K_{\mathrm{w}}}\), [H+] = \(\sqrt{3.8 \times 10^{-14}}\) and using the given value of \(K_{\mathrm{w}}\).

Calculate [H+]

Using the square root that we calculated in step 2, we calculate the [H+] concentration as: [H+] = \(6.1644 \times 10^{-8}\) M.

Calculate pH

Once we have the [H+], we can calculate the pH using the formula: pH = -log[H+]. Therefore, pH = -log(\(6.1644 \times 10^{-8}\)), and pH = \(7.21\) at \(40^{\circ} \mathrm{C}\).

Key concepts

Kw (ion product of water)

The ion product of water, denoted as \( K_{\mathrm{w}} \), is a crucial concept when calculating pH and understanding solution chemistry. At any given temperature, \( K_{\mathrm{w}} \) is the equilibrium constant for the self-ionization of water, which can be represented by this chemical reaction:

• \( \mathrm{H_2O} \leftrightharpoons \mathrm{H^+} + \mathrm{OH^-} \)

In pure water, the concentrations of hydrogen ions \([\mathrm{H^+}]\) and hydroxide ions \([\mathrm{OH^-}]\) are equal. Thus, \( K_{\mathrm{w}} = [\mathrm{H^+}][\mathrm{OH^-}] = [\mathrm{H^+}]^2 \). Remember, \( K_{\mathrm{w}} \) is temperature-dependent, which means its value varies with changes in temperature. For example, at room temperature (25°C), \( K_{\mathrm{w}} \) is \( 1.0 \times 10^{-14} \), but at 40°C, \( K_{\mathrm{w}} \) increases to \( 3.8 \times 10^{-14} \).
Knowing \( K_{\mathrm{w}} \) allows us to determine the concentrations of \( \mathrm{H^+} \) and subsequently, the pH of the solution.

temperature effect on pH

Temperature has a significant impact on pH levels, particularly in aqueous solutions like water. The pH of pure water is neutral and generally around 7.0 at 25°C. However, this neutrality changes with temperature shifts due to the temperature dependence of \( K_{\mathrm{w}} \).

At higher temperatures, self-ionization of water increases, leading to a higher \( K_{\mathrm{w}} \). For example, at 40°C, \( K_{\mathrm{w}} \) is \( 3.8 \times 10^{-14} \). This increase means that the concentration of \( \mathrm{H^+} \) ions is higher, resulting in a pH lower than 7.0 despite the solution remaining neutral.

• Greater \( K_{\mathrm{w}} \) values indicate more ionization at higher temperatures.
• Neutral pH decreases as temperature increases.

This understanding helps explain why pH isn't a fixed number and is pivotal in fields like biology and environmental science, where temperature changes can affect biochemical processes.

concentration of ions

In the context of water ionization, the concentration of ions like \( \mathrm{H^+} \) and \( \mathrm{OH^-} \) directly influences pH calculation. For pure water, these ion concentrations are equal, and their product is \( K_{\mathrm{w}} \). Solving for the \( \mathrm{H^+} \) concentration requires knowing \( K_{\mathrm{w}} \), using the equation:

• \( [\mathrm{H^+}] = \sqrt{K_{\mathrm{w}}} \)

This illustrates how the concentration of \( \mathrm{H^+} \) can be derived easily, providing the necessary value to find the pH.

Calculated concentrations of ions are crucial in determining the pH of solutions. For example, at a temperature of 40°C with \( K_{\mathrm{w}} = 3.8 \times 10^{-14} \), the \( \mathrm{H^+} \) concentration becomes \( 6.1644 \times 10^{-8} \text{ M} \).

• Knowing the \( \mathrm{H^+} \) concentration, the pH is calculated as \(-\log(\mathrm{H^+})\).

For this example, you compute the pH to be 7.21, reflecting the neutrality at that particular temperature. Understanding ion concentrations thus is fundamental to grasping how pH behaves under different conditions.