Problem 50 - Chemistry and Chemical Reactivity

Question

What is the molar concentration of \(\mathrm{Au}^{+}(\mathrm{aq})\) in a saturated solution of AuCl in pure water at \(25^{\circ} \mathrm{C} ?\) $$\mathrm{AuCl}(\mathrm{s}) \rightleftarrows \mathrm{Au}^{+}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})$$

Step-by-Step Solution

Answer

The molar concentration of \( \mathrm{Au}^{+} \) is approximately \( 1.41 \times 10^{-7} \) M.

Step 1: Write the Solubility Equilibrium Expression

The dissolution of AuCl in water is represented by the equation: \( \mathrm{AuCl(s)} \rightleftharpoons \mathrm{Au}^{+}(aq) + \mathrm{Cl}^{-}(aq) \). For this reaction, the equilibrium constant expression can be written as \( K_{sp} = [\mathrm{Au}^{+}][\mathrm{Cl}^{-}] \), where \([\mathrm{Au}^{+}]\) and \([\mathrm{Cl}^{-}]\) are the molar concentrations of the ions in solution.

Step 2: Establish the Relationship of Ion Concentrations

Since the dissociation of AuCl produces one mole of \( \mathrm{Au}^{+} \) and one mole of \( \mathrm{Cl}^{-} \), the molar concentrations of \( \mathrm{Au}^{+} \) and \( \mathrm{Cl}^{-} \) are equal at equilibrium. Let's assume \( s \) is the solubility of AuCl in mol/L, thus \( [\mathrm{Au}^{+}] = s \) and \( [\mathrm{Cl}^{-}] = s \).

Step 3: Relate Ion Concentrations to Solubility Product

Substitute the values from Step 2 into the solubility product equation: \( K_{sp} = s \cdot s = s^2 \). Therefore, \( s = \sqrt{K_{sp}} \).

Step 4: Use Provided or Lookup the Solubility Product Constant

Look up or use the given solubility product constant for AuCl, \( K_{sp} \). Let's assume \( K_{sp} = 2.0 \times 10^{-13} \) (the value should be checked in reference tables).

Step 5: Calculate the Molar Concentration of Au+

Substitute the \( K_{sp} \) value into the formula from Step 3: \( s = \sqrt{2.0 \times 10^{-13}} \). Calculate \( s \) to find \( s \approx 1.41 \times 10^{-7} \). This \( s \) value represents the molar concentration of \( \mathrm{Au}^{+} \) in the solution.

Key Concepts

Molar Concentration

Understanding molar concentration is key when discussing chemical solutions. It refers to the number of moles of a solute within a liter of solution, denoted as mol/L or \text{M}. In simpler terms, it tells us how many molecules or atoms of a substance are present in that solution.

When addressing the molar concentration of \( \mathrm{Au}^{+} \text{(aq)} \) in a saturated solution of AuCl, we use this measure to understand the concentration of gold ions present when equilibrium is achieved.

• The formula for calculating molar concentration is: \[ [\text{soluble ion}] = s \] where \( s \) represents solubility in mol/L.
• In a saturated solution, the molar concentration corresponds to the maximum amount of solute that can dissolve in a given volume at specific conditions.

It's all about determining how concentrated the solution is, making it possible to predict the behavior of ions when a compound like AuCl dissolves in water.

Solubility Product Constant

The solubility product constant, known as \( K_{sp} \), is a special type of equilibrium constant that applies to sparingly soluble compounds. It reflects how much of a solid solute dissolves to form a saturated solution.

For AuCl, we represent the equation of its dissolution as: \[ \mathrm{AuCl(s)} \rightleftharpoons \mathrm{Au}^{+}(aq) + \mathrm{Cl}^{-}(aq). \] Here, \( K_{sp} = [\mathrm{Au}^{+}][\mathrm{Cl}^{-}] \).

• Each ion concentration can be represented as \( s \) since they come from the same dissociating source. Thus, the equation becomes \( K_{sp} = s^2 \).
• The importance of \( K_{sp} \) lies in its ability to predict whether a precipitate will form under a given set of conditions.

By knowing the \( K_{sp} \) for a particular compound at a specific temperature, like \(25^\circ \text{C}\), you can calculate the molar solubility, which explains how much solute can dissolve in a solvent.

Chemical Equilibrium

Chemical equilibrium is a central concept in chemistry, especially when understanding reactions occurring in solution, such as the dissolving of solids like AuCl in water. At equilibrium, the rate of dissolution equals the rate of precipitation, meaning the concentration of ions in solution remains constant.

Consider the dissociation of AuCl: \[ \mathrm{AuCl(s)} \rightleftharpoons \mathrm{Au}^{+}(aq) + \mathrm{Cl}^{-}(aq) \].

• At this point, no net change is observed in the concentration of \( \mathrm{Au}^{+} \) or \( \mathrm{Cl}^{-} \).
• Equilibrium provides a stable state in which chemical processes proceed at the same rate in both directions.

Establishing chemical equilibrium helps us understand the behavior of saturated solutions and is crucial in calculating key values like molar concentration and the solubility product. Recognizing equilibrium conditions allows us to determine how variables such as concentration, pressure, or temperature affect reactions.