Question

Apply Concepts Like all equilibrium constants, the value of \(K_{w}\) varies with temperature. \(K_{w}\) equals \(2.92 \times 10^{-15}\) at \(10^{\circ} \mathrm{C}, 1.00 \times 10^{-14}\) at \(25^{\circ} \mathrm{C},\) and \(2.92 \times\) \(10^{-14}\) at \(40^{\circ} \mathrm{C}\) . In light of this information, calculate and compare the pH values for pure water at these three temperatures. Based on your calculations, is it correct to say that the pH of pure water is always 7.0\(?\) Explain.

Short answer

The pH values for pure water at the given temperatures are found to be different from each other. Specifically, the pH is 7.0 only at \(25^\circ\mathrm{C}\), while it deviates from 7.0 at \(10^\circ\mathrm{C}\) and \(40^\circ\mathrm{C}\). Therefore, it is not correct to say that the pH of pure water is always 7.0. The pH varies with temperature due to the change in the autoionization constant of water (\(K_w\)) with temperature.

Step-by-step solution

Calculate the concentration of \(H_3O^+\) at each temperature

Since in pure water, the concentration of \(H_3O^+\) and \(OH^-\) ions is equal, we have: At \(10^\circ\mathrm{C}: K_{w} = 2.92 \times 10^{-15} = x^2\) At \(25^\circ\mathrm{C}: K_{w} = 1.00 \times 10^{-14} = x^2\) At \(40^\circ\mathrm{C}: K_{w} = 2.92 \times 10^{-14} = x^2\) Step 2: Solve for \(x\) (concentration of \(H_3O^+\) ions)

Calculate the square root to find \(x\)

Take the square root of both sides of the equations obtained in the previous step: At \(10^\circ\mathrm{C}: x = \sqrt{2.92 \times 10^{-15}}\) At \(25^\circ\mathrm{C}: x = \sqrt{1.00 \times 10^{-14}}\) At \(40^\circ\mathrm{C}: x = \sqrt{2.92 \times 10^{-14}}\) Step 3: Calculate the pH of water at each temperature

Apply the pH formula

By using the pH equation, calculate the pH values at each temperature: At \(10^\circ\mathrm{C}: pH = -\log(\sqrt{2.92 \times 10^{-15}})\) At \(25^\circ\mathrm{C}: pH = -\log(\sqrt{1.00 \times 10^{-14}})\) At \(40^\circ\mathrm{C}: pH = -\log(\sqrt{2.92 \times 10^{-14}})\) Step 4: Compare the pH values

Evaluate if the pH of pure water is always 7.0

By evaluating the pH values for each temperature, we can now compare the results and analyze if the pH of pure water is always 7.0: At \(10^\circ\mathrm{C}: pH \neq 7.0\) At \(25^\circ\mathrm{C}: pH = 7.0\) At \(40^\circ\mathrm{C}: pH \neq 7.0\) Based on this comparison, pure water has a pH of 7.0 only at \(25^\circ\mathrm{C}\). The pH values change with different temperatures and the pH of pure water is not always equal to 7.

Key concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K_{eq} \), is a number that expresses the ratio of the concentration of products to reactants at equilibrium for a given reaction. It tells us the extent to which a reaction proceeds. For the ionization of water, the equilibrium constant is referred to as \( K_{w} \) and is specific to this reaction:

• \( K_{w} = [H_3O^+][OH^-] \)

In pure water, because the concentration of hydronium ions (\( H_3O^+ \)) is equal to the concentration of hydroxide ions (\( OH^- \)), \( K_{w} \) can be expressed as \( x^2 \), where \( x \) is the concentration of each ion.
The value of \( K_{w} \) depends on certain conditions such as temperature, and through this exercise, we see that it varies significantly across different temperatures. Understanding how equilibrium constants shift is crucial for predicting how the position of equilibrium responds to external changes like temperature variations.

Temperature Dependence

Temperature plays a crucial role in the behavior of chemical equilibria. For water, as the temperature changes, so does the \( K_{w} \), which indicates the extent of water ionization.

• At \( 10^\circ \mathrm{C}, \; K_{w} = 2.92 \times 10^{-15} \)
• At \( 25^\circ \mathrm{C}, \; K_{w} = 1.00 \times 10^{-14} \)
• At \( 40^\circ \mathrm{C}, \; K_{w} = 2.92 \times 10^{-14} \)

These values show that as the temperature increases, \( K_{w} \) increases, meaning that water ionizes more at higher temperatures.
Because of this, the concentration of \( H_3O^+ \) and \( OH^- \) in water also increases with temperature, which affects the resulting pH. The pH of pure water is therefore not always 7; it varies with temperature. Recognizing these dependencies helps in understanding how chemical reactions can be controlled or predicted under varying environmental conditions.

Kw (Ionic Product of Water)

The ionic product of water, \( K_{w} \), is a special equilibrium constant that applies to the auto-ionization of water:

• \( H_2O \rightleftharpoons H_3O^+ + OH^- \)

This expression essentially defines the product of the molar concentrations of hydronium and hydroxide ions at equilibrium. At standard conditions (25°C), \( K_{w} \) is typically \( 1.00 \times 10^{-14} \).
As temperature affects \( K_{w} \), by breaking and forming more water molecules into ions, it is clear that chemical equilibria are dynamic and can be influenced by external conditions. This equilibrium also underscores why water can self-ionize and why pure water is neutral only under standard conditions; it maintains a balance where the product of the concentrations of \( H_3O^+ \) and \( OH^- \) equals \( K_{w} \).

Hydronium Ion Concentration

The concentration of hydronium ions \( (H_3O^+) \) is a key factor in determining the acidity of a solution. In the case of pure water at different temperatures, \( H_3O^+ \) concentration is found via the expression \( K_{w} = x^2 \), where x represents the concentration of \( H_3O^+ \):

• \( x = \sqrt{K_{w}} \)

For example, at 25°C, \( x = \sqrt{1.00 \times 10^{-14}} \approx 1.00 \times 10^{-7} \; M \), which corresponds to a neutral pH of 7.0.
At temperatures other than 25°C, the hydronium ion concentration changes, leading to pH values that are not exactly 7.0. Thus, understanding hydronium ion concentrations helps in appreciating why water might have a slightly acidic or basic nature under non-standard conditions and assists in comprehending how changes in the environment can alter the pH of a solution.