Problem 154 - Chemistry An Atoms First Approach

Question

Quinine \(\left(\mathrm{C}_{20} \mathrm{H}_{24} \mathrm{N}_{2} \mathrm{O}_{2}\right)\) is the most important alkaloid derived from cinchona bark. It is used as an antimalarial drug. For quinine, \(\mathrm{p} K_{\mathrm{b}_{1}}=5.1\) and \(\mathrm{p} K_{\mathrm{b}_{2}}=9.7\left(\mathrm{p} K_{\mathrm{b}}=-\log K_{\mathrm{b}}\right) .\) Only 1 g quinine will dissolve in \(1900.0 \mathrm{mL}\) of solution. Calculate the \(\mathrm{pH}\) of a saturated aqueous solution of quinine. Consider only the reaction \(\mathrm{Q}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{QH}^{+}+\mathrm{OH}^{-}\) described by \(\mathrm{p} K_{\mathrm{b}_{1}},\) where \(\mathrm{Q}=\) quinine.

Step-by-Step Solution

Answer

In summary, to calculate the pH of a saturated aqueous solution of quinine, follow these steps: 1. Calculate the concentration of quinine: \(\text{Quinine concentration (M)} = \frac{\text{1 g}}{\text{1900 mL} \times \text{324.43 g/mol}}\) 2. Convert pKb1 to Kb: \(K_{b1} = 10^{-5.1}\) 3. Set up and solve an equilibrium expression: \(K_{b1} = \frac{[\text{OH}^-]^2}{[\text{Q}]-[\text{OH}^-]}\) 4. Calculate pOH: pOH = \(-\log{[\text{OH}^-]}\) 5. Determine the pH: pH = 14 - pOH

Step 1: Calculate the concentration of quinine

Using the given dissolution information, we can calculate the concentration of quinine in a saturated solution. Since 1 gram of quinine dissolves in 1900 mL of solution, the concentration can be calculated as: Quinine concentration (M) = \(\frac{\text{mass of quinine}}{\text{volume of solution}\times \text{molar mass of quinine}}\) where the molar mass of Quinine = \(C_{20}H_{24}N_{2}O_{2} = 20\times 12.01 + 24\times 1.01 + 2\times 14.01 + 2\times 16.00 = 324.43 \,g/mol\)

Step 2: Convert pKb1 to Kb

We are given the pKb1 value for quinine, but we need to convert it to Kb in order to use it in further calculations. The relation between pKb and Kb is: \(K_{b} = 10^{-\text{p}K_{b}}\) Now, plug in the given pKb1 value to obtain Kb: \(K_{b1} = 10^{-5.1}\)

Step 3: Set up and solve an equilibrium expression

Consider the reaction equation: Q + H2O \(\rightleftharpoons\) QH+ + OH- Since we are interested in the OH- concentration, set up an equilibrium expression: \(K_{b1} = \frac{[\text{QH}^+][\text{OH}^-]}{[\text{Q}]}\) Notice that the concentration of [OH-] will be equal to [QH+] at equilibrium due to the stoichiometry of the reaction. So we can simplify the expression: \(K_{b1} = \frac{[\text{OH}^-]^2}{[\text{Q}]-[\text{OH}^-]}\) Using the calculated quinine concentration from step 1 and Kb1 from step 2, we can now solve for the OH- concentration.

Step 4: Calculate the pOH and pH of the solution

Now that we have the concentration of OH-, we can easily find pOH by: pOH = \(-\log{[\text{OH}^-]}\) Next, we can determine the pH of the solution using the relationship between pH and pOH: pH + pOH = 14 Therefore, pH = 14 - pOH By calculating the pH, we will have the answer to the problem.

Key Concepts

Understanding Quinine in Acid-Base Equilibrium

Quinine is an organic compound that acts as a weak base in water. It's known for its antimalarial properties and is derived from the bark of the cinchona tree. In acid-base chemistry, such substances can accept hydrogen ions (H+) from water (H2O) to form their conjugate acid (QH+) and hydroxide ions (OH-). This reversible process can be represented with an equilibrium constant expression, where the base dissociation constant, or Kb, quantifies the position of the equilibrium. The larger the Kb, the stronger the base, as it implies a greater tendency to accept H+ from water.

A saturated solution of quinine has reached its maximum solubility, meaning no more quinine will dissolve at that temperature and pressure. The equilibrium between undissolved quinine and the ions in solution is crucial for calculating the pH. The pH is a measure of the acidity or basicity of a solution. With quinine acting as a base, its interaction with water and subsequent release of OH- is directly related to the solution's pH level.

Molar Mass Calculation for Quinine

The molar mass is a fundamental property of a substance, representing the mass of one mole of its molecules. It's critical in converting between the mass of a substance and the number of moles. To calculate the molar mass of quinine ((C_{20}H_{24}N_{2}O_{2})), you’ll need to consider the atomic masses of carbon (C), hydrogen (H), nitrogen (N), and oxygen (O), and sum them up. Each element’s atomic mass is multiplied by the number of atoms in one molecule of quinine.

For quinine, this calculation involves adding together the mass contributions from 20 carbons, 24 hydrogens, 2 nitrogens, and 2 oxygens. Molar mass calculations are crucial in chemistry because they allow for the conversion of mass to moles, helping to determine concentration, which is a vital step in predicting the behavior of substances in solution.

Acid and Base Constants: Understanding Kb and pKb

In the context of acid-base reactions, Kb and pKb are numbers that express the strength of a base. Kb is the base dissociation constant, a numerical value that measures the ability of a base to form hydroxide ions (OH^-) when dissolved in water. The pKb is the negative base-10 logarithm of the base dissociation constant (K_{b}) and offers a more convenient way to express base strength, especially for very small Kb values.

The relationship between Kb and pKb is crucial for understanding how they influence the pH of a solution. A lower pKb indicates a stronger base, which tends to create more OH- in solution, leading to a higher pH (more basic). In quinine's case, it's given to have certain pKb values. By converting these to their respective Kb values, we can calculate how much the quinine will impact the solution’s pH.

Calculating the pH of a Solution

The pH scale is a measure of the hydrogen ion concentration in a solution and is inherently related to the properties of acids and bases. Calculating the pH of a solution is a multiple-step process involving determining the concentration of hydrogen ions or hydroxide ions. For a base like quinine, we're more interested in the hydroxide ion concentration since its presence indicates basicity.

To calculate the pH of a quinine solution, you first determine the quinine concentration, account for its base properties through the base constant (Kb), and set up an equilibrium expression for the reaction with water. Once the hydroxide ion (OH-) concentration is known, the pOH can be calculated using the negative logarithm, then used to find the pH by utilizing the relationship pH + pOH = 14. This step-by-step approach can help students understand the link between solubility, base strength, ion concentration, and pH.

pH and pOH Relationship in Chemistry

pH and pOH are mathematical tools that chemists use to describe the acidity or basicity of a solution. pH is a measure of the concentration of hydrogen ions (H^+), while pOH measures the concentration of hydroxide ions (OH^-). Both scales range typically from 0 to 14, with lower pH values indicating higher acidity and lower pOH values indicating higher basicity.

The sum of pH and pOH is always 14 in aqueous solutions at 25°C, which comes from the water self-ionization equilibrium constant (K_{w}). This relationship is central to solving acid-base equilibrium problems. By calculating the pOH value from the hydroxide ion concentration in a basic solution, you can deduce the pH by subtracting the pOH from 14, thus finding how basic or acidic the solution is. This concept is a vital part of understanding the acid-base behavior of substances in solution.