Problem 120 - Chemistry: The Central Science in SI Units, Global Edition

Question

At \(50^{\circ} \mathrm{C}\), the ion-product constant for \(\mathrm{H}_{2} \mathrm{O}\) has the value \(K_{w}=5.48 \times 10^{-14} \cdot(\mathbf{a})\) What is the \(\mathrm{pH}\) of pure water at \(50^{\circ} \mathrm{C} ?\) (b) Based on the change in \(K_{w}\) with temperature, predict whether \(\Delta H\) is positive, negative, or zero for the autoionization reaction of water: $$ 2 \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) $$

Step-by-Step Solution

Answer

The pH of pure water at 50°C is 6.63, and based on the increase in the ion-product constant for water (Kw) with temperature, the enthalpy change (∆H) for the autoionization reaction of water is positive, indicating an endothermic reaction.

Step 1: Calculate the concentration of H3O+ ions

Be aware that in pure water, the concentration of H3O+ ions is equal to the concentration of OH- ions. Moreover, we know that Kw = [H3O+] * [OH-]. Since the concentrations are equal, we can rewrite the equation for Kw as follows: Kw = [H3O+]^2 We will use the given value for Kw at 50°C to find the concentration of H3O+ ions.

Step 2: Solve for H3O+ ion concentration

We need to determine the [H3O+] from the given Kw at 50°C: \(K_w = 5.48 \times 10^{-14} = [H_3O^+]^2\) \( [H_3O^+] = \sqrt{5.48 \times 10^{-14}}\) \[ [H_3O^+] = 2.34 \times 10^{-7} \text{ M}\]

Step 3: Calculate the pH of pure water

To find the pH of pure water, we use the relationship between pH and H3O+ ion concentration: \(pH = -\log ([H_3O^+])\) \(pH = -\log (2.34 \times 10^{-7})\) \[ pH = 6.63 \]

Step 4: Analyze the change in Kw with temperature

The value of Kw increases as the temperature increases. At 25°C, the Kw value is \(1.0 \times 10^{-14}\), and at 50°C it is \(5.48 \times 10^{-14}\). This means that the autoionization reaction becomes more favorable at higher temperatures.

Step 5: Predict the sign of ∆H for the autoionization reaction of water

Since the autoionization reaction of water becomes more favorable at higher temperatures, it indicates that the reaction is endothermic, absorbing heat from the surroundings. Therefore, the enthalpy change (∆H) for the autoionization reaction of water is positive. In summary, the pH of pure water at 50°C is 6.63, and the enthalpy change for the autoionization reaction of water is positive.

Key Concepts

Ion-product Constant

The ion-product constant of water, denoted as \( K_w \), is a crucial concept in understanding water chemistry. This constant expresses the balance between the concentration of hydronium ions \([H_3O^+]\) and hydroxide ions \([OH^-]\) in water. In pure water, these concentrations are equal.

For any given temperature, \( K_w \) is defined as the product of these concentrations:

• \( K_w = [H_3O^+] \times [OH^-] \)

This shows that at equilibrium, the multiplication of these ion concentrations gives a constant value. However, this equilibrium constant varies with temperature. For instance, at \( 50^{\circ} \mathrm{C} \), \( K_w \) is \( 5.48 \times 10^{-14} \).

As temperature increases, \( K_w \) usually increases as well, indicating a greater degree of ionization of water at higher temperatures.

Enthalpy Change

Enthalpy change, symbolized as \( \Delta H \), is a measure of heat absorbed or released during a reaction. In the context of the autoionization of water—the reaction where water molecules dissociate into hydronium and hydroxide ions—understanding \( \Delta H \) helps us understand the temperature dependency of \( K_w \).

The fact that \( K_w \) increases with rising temperature suggests an endothermic process. This means the reaction absorbs heat from its surroundings, leading to a larger concentration of ions as temperature rises.

In simple terms:

• If \( \Delta H \) is positive: the process is endothermic, absorbing energy.
• If \( \Delta H \) is negative: the process is exothermic, releasing energy.

Since the autoionization reaction of water absorbs heat (as indicated by the increase in \( K_w \) with temperature), \( \Delta H \) is positive.

pH Calculation

Calculating the pH is essential to understanding the acidity or basicity of a solution. The pH is linked to the concentration of hydronium ions in the solution. The formula to calculate pH from the \([H_3O^+]\) concentration is:

• \( pH = -\log([H_3O^+]) \)

In pure water at \( 50^{\circ} \mathrm{C} \), given \( K_w = 5.48 \times 10^{-14} \), the concentration of \( [H_3O^+] \) and \([OH^-]\) can be found using:

• \( [H_3O^+] = \sqrt{K_w} = \sqrt{5.48 \times 10^{-14}} \)

This results in \([H_3O^+] = 2.34 \times 10^{-7} \, \text{M}\). Substituting this value into the pH formula:

• \( pH = -\log(2.34 \times 10^{-7}) \)

We find that the pH of pure water at this temperature is approximately 6.63.

Interestingly, at \( 50^{\circ} \mathrm{C} \), the pH of water is less than 7, which usually indicates acidic. However, this is still considered neutral at this specific temperature due to the equal \([H_3O^+]\) and \([OH^-]\).