Problem 84 - Chemistry The Science in Context

Question

What is the pH of \(1.00 M \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2} ?\)

Step-by-Step Solution

Answer

Answer: The pH of a 1.00 M Cu(NO₃)₂ solution is 7, which is neutral.

Step 1: Understand the pH scale

The pH scale is a measure of the acidity or alkalinity of a solution, which is determined by the concentration of hydrogen ions (H⁺). A pH value of 7 is neutral, values less than 7 are acidic, and values greater than 7 are basic (alkaline). The pH of a solution can be calculated by using the formula: pH = −log₁₀[H⁺]

Step 2: Identify the main species in the solution

Copper(II) nitrate is a soluble salt and will dissociate in water to form Cu²⁺ and 2NO₃⁻ ions. None of these ions are acidic or basic, therefore, 1.00M Cu(NO₃)₂ solution will not introduce any new sources of H⁺ or OH⁻ ions. However, water is weakly ionizable and auto-ionizes to a very small extent as follows: 2H₂O (l) ⇌ H₃O⁺ (aq) + OH⁻ (aq) The equilibrium constant for auto-ionization of water is called the ion-product constant of water, denoted by Kw, and is given by: Kw = [H₃O⁺][OH⁻] = 1.0×10⁻¹⁴ (at 25°C)

Step 3: Calculate the concentration of H⁺ ions in the solution

Since the Cu(NO₃)₂ solution is neutral (does not affect the pH), the concentration of H⁺ ions comes only from the auto-ionization of water. In pure water, the concentration of H₃O⁺ and OH⁻ ions are equal: [H₃O⁺] = [OH⁻] Using the ion-product constant for water: Kw = [H₃O⁺][OH⁻] = 1.0×10⁻¹⁴ Since [H₃O⁺] = [OH⁻] in pure water, we can write the equation as: 1.0×10⁻¹⁴ = [H₃O⁺]² Now, we can find the concentration of H₃O⁺ ions: [H₃O⁺] = sqrt(1.0×10⁻¹⁴) = 1.0×10⁻⁷ M

Step 4: Determine the pH of the solution

Using the pH formula: pH = −log₁₀[H₃O⁺] = -log₁₀(1.0×10⁻⁷) = 7 The pH of the 1.00 M Cu(NO₃)₂ solution is 7, which is neutral.

Key Concepts

Auto-ionization of Water

Water is a fascinating liquid because it has the ability to ionize itself even though it is a neutral compound. This process is called auto-ionization or self-ionization. When water molecules interact, they can transfer a proton from one molecule to another. This results in the formation of one hydronium ion \[\text{H}_{3}\text{O}^{+}\] and one hydroxide ion \[\text{OH}^{-}\]. The resulting reaction is: \[2\text{H}_{2}\text{O} (l) \rightleftharpoons \text{H}_{3}\text{O}^{+} (aq) + \text{OH}^{-} (aq)\] The auto-ionization of water is an important concept because it demonstrates that even pure water contains a very small amount of ions. This process is in dynamic equilibrium, meaning that the rate at which water molecules dissociate equals the rate at which hydronium and hydroxide ions recombine back into water molecules. **Key Points of Auto-ionization:**

• Water auto-ionizes to form hydronium and hydroxide ions.
• It's a reversible and dynamic equilibrium process.
• This equilibrium is crucial for understanding acidity and basicity in solutions.

Neutral Solution

A neutral solution is one where the concentration of hydrogen ions \([\text{H}^{+}]\) is equal to the concentration of hydroxide ions \([\text{OH}^{-}]\). This balance leads to a pH of 7, which characterizes neutrality on the pH scale. The pH scale is broadly divided into three categories:

• Acidic: pH less than 7
• Neutral: pH equal to 7
• Basic or Alkaline: pH greater than 7

In the context of the exercise, the 1.00 M \(\text{Cu} \left(\text{NO}_{3}\right)_{2}\) solution does not change the neutral nature of pure water because it dissociates into ions that do not affect the \([\text{H}^{+}]\) or \([\text{OH}^{-}]\) concentrations. Thus, even as impurities are present, so long as they don't influence the ion concentrations, the solution remains neutral.

Ion-product Constant of Water

The ion-product constant of water, denoted as \(K_w\), is the equilibrium constant for the auto-ionization of water. It plays a vital role in understanding the pH of a solution, offering insight into the acidic or basic nature of any aqueous solution. At room temperature (25°C), \(K_w\) is given by: \[K_w = [\text{H}^{+}][\text{OH}^{-}] = 1.0 \times 10^{-14}\] This constantly holds true for pure water and dictates the neutral pH level.**Why is \(K_w\) Important?**

• It helps us calculate the concentrations of hydronium and hydroxide ions.
• It assists in calculating the pH of neutral solutions by establishing the baseline \( [\text{H}^{+}] \) and \( [\text{OH}^{-}] \) concentrations.
• It provides fundamental knowledge for both chemistry studies and practical applications in various fields.

Understanding \(K_w\) is essential for anyone working with chemical solutions, making it a cornerstone of acid-base chemistry.