Caffeine (C₈H₁₀N₄O₂) is a weak base with a pK_b of 10.4. Calculate the pH of a solution containing a caffeine concentration of 455 mg/L.

Verified step by step guidance

1. Convert the concentration of caffeine from mg/L to moles/L (molarity). To do this, first find the molar mass of caffeine (C8H10N4O2). Calculate the molar mass by summing the atomic masses of all the atoms in the formula: (8 \times 12.01) + (10 \times 1.008) + (4 \times 14.01) + (2 \times 16.00) grams/mol.
2. Use the molar mass to convert 455 mg/L of caffeine to moles/L. Divide the mass of caffeine by its molar mass: \( \frac{455 \text{ mg}}{1000 \text{ mg/g}} \div \text{molar mass of caffeine} \).
3. Calculate the Kb value using the pKb given. The relationship between Kb and pKb is given by: \( \text{Kb} = 10^{-\text{pKb}} \).
4. Use the Kb and the concentration of caffeine to find the concentration of OH- ions in the solution using the formula for the base ionization constant, Kb: \( \text{Kb} = \frac{[\text{OH}^-][\text{BH}^+]}{[\text{B}]} \), where [B] is the initial concentration of the base.
5. Calculate the pOH of the solution using the concentration of OH- ions: \( \text{pOH} = -\log[\text{OH}^-] \). Finally, find the pH from the pOH using the relationship: \( \text{pH} = 14 - \text{pOH} \).

Key Concepts

Weak Bases and pKb

A weak base is a substance that partially ionizes in solution, establishing an equilibrium between the base and its conjugate acid. The pKb value indicates the strength of the base; a higher pKb means a weaker base. In this case, caffeine has a pKb of 10.4, which suggests it does not ionize significantly in water, affecting the pH calculation.

pH and pOH Relationship

pH is a measure of the hydrogen ion concentration in a solution, while pOH measures the hydroxide ion concentration. The relationship between pH and pOH is given by the equation pH + pOH = 14 at 25°C. Understanding this relationship is crucial for calculating the pH of a solution when dealing with weak bases, as it allows for the determination of hydroxide ion concentration from the base's ionization.

Concentration and Molarity

Concentration refers to the amount of solute present in a given volume of solution, often expressed in molarity (moles per liter). To calculate the pH of the caffeine solution, it is necessary to convert the given mass (455 mg) into moles and then into molarity. This conversion is essential for applying the equilibrium expression related to the weak base and determining the resulting pH.