Question

a. What is the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) of pure water at \(25^{\circ} \mathrm{C} ?\) b. Is this true at all temperatures? Why or why not?

Short answer

a. \[ [\text{H}_3\text{O}^+] = 10^{-7} \text{M} \] b. No, \[ [\text{H}_3\text{O}^+] \] changes with temperature because \[ \text{K}_w \] varies.

Step-by-step solution

- Understanding Pure Water at 25°C

At 25°C, pure water is neutral, meaning its pH is 7. This implies that the concentration of \(\text{H}_3\text{O}^+\) ions is equal to the concentration of OH\text{^−} ions.

- Using the Definition of pH

Recall the definition of pH: \[ \text{pH} = -\text{log}[\text{H}_3\text{O}^+] \] Given that the pH of pure water at 25°C is 7, we can solve for \([\text{H}_3\text{O}^+]\).

- Solving for \([\text{H}_3\text{O}^+]\)

Since \[ 7 = -\text{log}[\text{H}_3\text{O}^+] \], converting the equation gives us: \[ [\text{H}_3\text{O}^+] = 10^{-7} \text{M} \]

- Addressing All Temperatures

The value \[ [\text{H}_3\text{O}^+] = 10^{-7} \text{M} \] is specific to 25°C because the ion product of water (\text{K}_w) changes with temperature. Therefore, \[ [\text{H}_3\text{O}^+] \] is not always \[ 10^{-7} \text{M} \], as the pH of pure water varies at different temperatures.

- Conclusion on Temperature Dependence

As temperature increases or decreases, the equilibrium constant for the dissociation of water changes, which alters the concentration of \[ \text{H}_3\text{O}^+ \]. This means the concentration is true only at 25°C and not all temperatures.

Key concepts

Hydronium Ion Concentration

In pure water at 25°C, the pH is exactly 7, indicating a balance between hydronium ions \(\text{H}_3\text{O}^+\) and hydroxide ions \( \text{OH}^- \). This balance is expressed through the relationship: \([ \text{H}_3\text{O}^+ ] = [ \text{OH}^- ]\). To find the hydronium ion concentration, we use the definition of pH: \[ \text{pH} = -\text{log}[\text{H}_3\text{O}^+] \] For pure water at 25°C, where pH = 7, the equation becomes: \[ 7 = -\text{log}[\text{H}_3\text{O}^+ ] \] Solving for \[ [\text{H}_3\text{O}^+] \], we get: \[ [\text{H}_3\text{O}^+ ] = 10^{-7} \text{M} \] This shows the concentration of hydronium ions in pure water at 25°C is \(\text{10}^{-7} \text{mol/L} \).
The hydronium ion concentration is crucial because it grounds our understanding of the pH scale, which measures acidity or basicity.

Temperature Dependence of pH

The pH of pure water at 25°C is 7, but this does not hold at all temperatures. The ion product of water, \(\text{K}_w \), varies with temperature, affecting the concentrations of \(\text{H}_3\text{O}^+ \) and \(\text{OH}^- \) ions. For example, at higher temperatures, \( \text{K}_w \) increases. This means:

• The product \[ [ \text{H}_3\text{O}^+ ] \times [ \text{OH}^- ] \] becomes greater than \(\text{10}^{-14} \).
• To maintain neutrality, \[ [ \text{H}_3\text{O}^+ ] \] increases beyond \(\text{10}^{-7} \text{M} \).
• Consequently, pH drops below 7, even though the solution remains neutral.

Conversely, at lower temperatures, \( \text{K}_w \) decreases, meaning:

• The product \[ [ \text{H}_3\text{O}^+ ] \times [ \text{OH}^- ] \] is less than \(\text{10}^{-14} \).
• \[ [ \text{H}_3\text{O}^+ ] \] falls below \(\text{10}^{-7} \text{M} \).
• The pH rises above 7, while the solution remains neutral.

This shows why pH varies with temperature and why the neutral pH of water changes.

Ion Product of Water

The ion product of water, \(\text{K}_w \), is a fundamental concept in chemistry. It is defined as the product of the concentrations of \( \text{H}_3\text{O}^+ \) and \( \text{OH}^- \) ions in water: \[ \text{K}_w = [ \text{H}_3\text{O}^+ ] \times [ \text{OH}^- ] \] At 25°C, \( \text{K}_w \) is equal to \(\text{10}^{-14} \text{mol}^2/\text{L}^2 \). This expresses the equilibrium state of water's autoionization:

• When water ( \(\text{H}_2\text{O} \)) dissociates, it forms \( \text{H}_3\text{O}^+ \) and \( \text{OH}^- \) ions.
• \( \text{H}_2\text{O} \) \rightarrow \( \text{H}_3\text{O}^+ + \text{OH}^- \)
• For pure water, the concentrations of \( \text{H}_3\text{O}^+ \) and \( \text{OH}^- \) are equal.

Temperature changes affect \( \text{K}_w \) significantly:

• As temperature increases, \[ \text{K}_w \] increases, altering \( [ \text{H}_3\text{O}^+ ] \) and \( [ \text{OH}^- ] \).
• As temperature decreases, \[ \text{K}_w \] decreases, impacting the equilibrium.

Understanding \( \text{K}_w \) is essential for grasping how temperature influences pH and the balance of ion concentrations in water.