Question

The solubility of \(\mathrm{Cl}_{2}\) in 100 \(\mathrm{g}\) of water at STP is 310 \(\mathrm{cm}^{3}\) . Assume that this quantity of \(\mathrm{Cl}_{2}\) is dissolved and equilibrated as follows: $$\mathrm{Cl}_{2}(a q)+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{Cl}^{-}(a q)+\mathrm{HClO}(a q)+\mathrm{H}^{+}(a q)$$ (a) If the equilibrium constant for this reaction is \(4.7 \times 10^{-4}\) , calculate the equilibrium concentration of HClO formed. (b) What is the pH of the final solution?

Short answer

(a) To calculate the equilibrium concentration of HClO formed, first determine the moles of Cl2 dissolved in water using the equation PV/RT, with P = 1 atm, V = 0.31 L, R = 0.0821 L atm/K mol, and T = 273 K. Then calculate the initial concentration of Cl2 using the number of moles of solvent (100 g water / 18 g/mol). Use the equilibrium constant (4.7 x 10^-4) and stoichiometry relationships to find the equilibrium concentration of HClO through the equation K' = ([HClO]^3)/([Cl2]0 - [HClO]). (b) To calculate the pH of the final solution, find the equilibrium concentration of H+ using the relationships at equilibrium (as H+ = HClO) and then apply the pH formula: pH = -log([H+]).

Step-by-step solution

Determine the concentration of Cl2 in water

To find out the moles of Cl2 in water, we first need to convert the volume of Cl2 from cm³ to L and then use the equation: moles of Cl2 = PV/RT where P = pressure (1 atm) V = volume (Convert the volume of Cl2 from cm³ to L: 310 cm³ * (1 L/1000 cm³) = 0.31 L) R = gas constant (0.0821 L atm / K mol) T = temperature (273 K) Moles of Cl2 = (1 atm * 0.31 L) / (0.0821 L atm / K mol * 273 K)

Calculate the equilibrium concentration of HClO formed

Based on the stoichiometry of the balanced equation, the moles of Cl2 is equal to the moles of HClO formed. Total mass of solvent = 100 g water Molar mass of water = 18 g/mol Moles of solvent (water) = 100 g / 18 g/mol = 5.56 mol Now we calculate the initial concentration of Cl2: Initial concentration = Moles of Cl2 / Moles of solvent (5.56 mol) Using the calculated initial concentration of Cl2 and equilibrium constant (4.7 x 10^-4) given in the problem, we can now find out the equilibrium concentration of HClO formed. The reaction stoichiometry suggests the following relationships at equilibrium: [Cl2] = [Cl2]0 - [HClO] [Cl-] = [HClO] [HClO] = [H+] [Cl2] = Initial (Cl2) concentration - [HClO] Plugging these relationships into the equilibrium constant expression, we get: K = ([Cl-][HClO][H+])/([Cl2][H2O]) Since the concentration of H2O is large and relatively constant, we can simplify this expression to: K' = ([HClO]^3)/([Cl2]0 - [HClO]) Plug in K and initial concentration found earlier, then solve the cubic equation for [HClO].

Calculate the pH of the final solution

Use the equilibrium concentration of H+ ([H+]) found in step 2 to calculate the pH of the final solution using the pH formula: pH = -log([H+]) By calculating the pH, we get the final answer to this problem.

Key concepts

Understanding Solubility: The Basics

Solubility is a measure of how much of a substance can dissolve in a solvent at a given temperature and pressure. For instance, in the problem, the solubility of chlorine gas (\(\text{Cl}_2\)) in water at standard temperature and pressure (STP) is 310 cm³ per 100 grams of water. This tells us how much chlorine can dissolve in the water before reaching equilibrium.
To start with, remember that solubility depends on several factors:

• Nature of the Solute and Solvent: Polar solvents usually dissolve polar solutes, and non-polar solvents dissolve non-polar solutes.
• Temperature: Solubility can increase or decrease with temperature based on the substance.
• Pressure: Particularly affects gases; an increase in pressure usually increases solubility for gases.

At equilibrium, the rate of dissolution and the rate of precipitation of the solute balance each other. This helps establish a constant concentration of solute in the solution.

Equilibrium Constant: A Window into Reactions

The equilibrium constant (\(K\)) is a key concept in chemistry that quantifies the balance between reactants and products in a reversible reaction at equilibrium. In the example given, the reaction is:\[\mathrm{Cl}_{2}(aq) + \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{Cl}^{-}(aq) + \mathrm{HClO}(aq) + \mathrm{H}^{+}(aq)\]The equilibrium constant for this reaction is given as \(4.7 \times 10^{-4}\), reflecting the ratio of product concentrations to reactant concentrations at equilibrium.
Here's how we use the equilibrium constant:

• If \(K\) is large, the products are favored at equilibrium.
• If \(K\) is small, the reactants are favored.

To determine the equilibrium concentrations of the products, the initial concentration and \(K\) value are plugged into an equilibrium expression, often requiring solving for unknowns. Understanding \(K\) helps in predicting how changes in conditions (like concentration or temperature) might shift the equilibrium.

Exploring Acid-Base Reactions

Acid-base reactions involve the transfer of protons (\(\text{H}^+\)). In the equation from the problem, chlorine reacts with water to produce hydrochloric acid (\(\text{HCl}\)), which dissociates in water:
\[\mathrm{Cl}_{2}(aq) + \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{Cl}^{-}(aq) + \mathrm{HClO}(aq) + \mathrm{H}^{+}(aq)\] Here's what's happening:

• The chlorine reacts with water, producing hydrochloric acid and hypochlorous acid.
• The hydrochloric acid dissociates, increasing the concentration of hydrogen ions (\(\text{H}^+\)).
• This results in the acidic nature of the solution.

In equilibrium chemistry, understanding these reactions is crucial because they affect pH levels and the overall chemical behavior of the solution. Acid-base reactions are fundamental not only in chemistry but also in biological processes and industrial applications.

Calculating pH: The Power of Hydrogen

The pH scale measures the acidity or basicity of a solution. It's calculated using the concentration of hydrogen ions (\(\text{H}^+\)) in the solution. The formula is:
\[\text{pH} = -\log_{10}([\text{H}^+])\]This logarithmic scale means that a small change in hydrogen ion concentration significantly affects pH, which ranges from 0 (very acidic) to 14 (very basic).
Here's how we find the pH:

• Use the equilibrium \([\text{H}^+]\) concentration found from the reaction.
• Substitute \([\text{H}^+]\) into the pH formula.
• Calculate the logarithmic value to find the pH.

This calculation is vital for understanding the solution's acidity, which affects reactivity, solubility, and biological activity. In this exercise, calculating pH helped determine the condition of the solution at equilibrium. Understanding pH is crucial for tasks ranging from lab experiments to everyday applications like pool maintenance.