Problem 43 - Chemistry: Principles and Reactions

Question

The equilibrium constant at \(25^{\circ} \mathrm{C}\) for the reaction $$\mathrm{Br}_{2}(l)+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{Br}^{-}(a q)+\mathrm{HBrO}(a q)$$ is \(1.2 \times 10^{-9} .\) This is the system present in a bottle of "bromine water." Assuming that HBrO does not ionize appreciably, what is the pH of the bromine water?

Step-by-Step Solution

Answer

Answer: The pH of the bromine water is approximately 2.98.

Step 1: Write the balanced equation and equilibrium expression

The balanced equation is: $$\mathrm{Br}_{2}(l)+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{Br}^{-}(a q)+\mathrm{HBrO}(a q)$$ The equilibrium expression is: $$K = \frac{[\mathrm{H}^{+}][\mathrm{Br}^{-}][\mathrm{HBrO}]}{[\mathrm{Br}_{2}][\mathrm{H}_{2}\mathrm{O}]}$$ However, since the concentration of liquid substances (like \(\mathrm{Br}_{2}\) and \(\mathrm{H}_{2}\mathrm{O}\)) do not affect the equilibrium constant, we can rewrite the expression as: $$K = [\mathrm{H}^{+}][\mathrm{Br}^{-}][\mathrm{HBrO}]$$

Step 2: Set up the ICE table

We begin by setting up an ICE table to find the equilibrium concentrations of each species: |Species |Initial |Change |Equilibrium| |-----------|--------|-------|-----------| |H+ | 0 |+x |x | |Br− | 0 |+x |x | |HBrO | 0 |+x |x | Since we are assuming that HBrO does not ionize significantly, the equilibrium concentrations of H+, Br- and HBrO are all equal to x.

Step 3: Substitute the equilibrium concentrations into the equilibrium expression

Substituting the equilibrium concentrations into the equilibrium expression, we get: $$K = (x)(x)(x) = x^3$$ As we know, the value of K is \(1.2 \times 10^{-9}\). So, we have: $$x^3 = 1.2 \times 10^{-9}$$

Step 4: Solve for x (concentration of H+ ions)

To find the concentration of H+ ions, we need to solve for x: $$x = \sqrt[3]{1.2 \times 10^{-9}}$$ Now we can use a calculator to find the value of x: $$x \approx 1.06 \times 10^{-3}$$ This represents the concentration of H+ ions at equilibrium.

Step 5: Calculate the pH of the solution

Now that we know the concentration of H+ ions, we can use the pH formula to find the pH of the bromine water: $$pH = -\log_{10}([\mathrm{H}^{+}])$$ Substitute the value of x into the formula: $$pH = -\log_{10}(1.06 \times 10^{-3})$$ Now we can use a calculator to find the pH: $$pH \approx 2.98$$ The pH of the bromine water is approximately 2.98.

Key Concepts

Equilibrium Constant

In chemical reactions, the equilibrium constant is a numerical value that represents the ratio of product concentrations to reactant concentrations when the system is in equilibrium. For a given reaction: \[ \text{a}A + \text{b}B \rightleftharpoons \text{c}C + \text{d}D \]The equilibrium constant, \(K\), is given as:\[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]It is important to note that

• Only reactants and products with variable concentrations, such as aqueous or gaseous substances, are included in the equilibrium constant expression.
• Solids and liquids are excluded because their concentrations do not change.

In the context of bromine water, the equilibrium constant helps in understanding the ratio of the concentrations of hydrogen ions, bromide ions, and hypobromous acid to the initial amounts of bromine and water.

ICE Table

An ICE (Initial, Change, Equilibrium) table is an excellent tool used in chemistry to help find the concentrations of reactants and products at equilibrium. Here's how it works:

• **Initial:** List the initial concentrations of reactants and products before the reaction starts. If not provided, assume it zero for products.
• **Change:** Determine the change that occurs in each concentration as the system moves towards equilibrium.
• **Equilibrium:** Write down the equilibrium concentrations, which are the initial concentration plus or minus the change.

In the bromine water example, because the reaction starts with no products, the equilibrium concentrations of all products are represented equally as "x". This method simplifies the calculation and makes solving the equilibrium expression straightforward.

pH Calculation

The pH of a solution indicates its acidity or alkalinity and is determined using the concentration of hydrogen ions (\([ ext{H}^+]\)). The formula for pH is:\[pH = -\log_{10}([ ext{H}^+])\]In the bromine water example, once we have determined the equilibrium concentration of \([ ext{H}^+]\) using the ICE table, we can insert this value into the pH formula. A low pH value, such as 2.98, indicates a highly acidic solution, which is coherent with bromine water being acidic due to the presence of hydrogen ions.

Bromine Water

Bromine water is a solution of bromine gas (\( ext{Br}_2\)) dissolved in water (\( ext{H}_2 ext{O}\)). It is known for its reactivity and is used in various chemical tests, such as detecting unsaturation in organic compounds. In the discussed chemical equilibrium, bromine reacts with water to form hydrogen and bromide ions along with hypobromous acid (HBrO). Although HBrO is a weak acid and only ionizes slightly, it plays a crucial role in determining the acidity of the solution. Since bromine and water do not change much during the reaction, their concentrations are not used directly in equilibrium constant calculations. Understanding the behavior of bromine water and its applications can be valuable in both academic and practical chemistry contexts.