Problem 30 - Chemistry: The Central Science

Question

A buffer, consisting of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{HPO}_{4}^{2-}\), helps control the \(\mathrm{pH}\) of physiological fluids. Many carbonated soft drinks also use this buffer system. What is the \(\mathrm{pH}\) of a soft drink in which the major buffer ingredients are \(6.5 \mathrm{~g}\) of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) and \(8.0 \mathrm{~g}\) of \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) per \(355 \mathrm{~mL}\) of solution?

Step-by-Step Solution

Answer

The pH of the soft drink, which contains 6.5 g of NaH2PO4 and 8.0 g of Na2HPO4 per 355 mL of solution, can be calculated using the Henderson-Hasselbalch equation. After converting the given masses to moles and calculating the molarities of the buffer components, we find that the pH of the soft drink is 7.25.

Step 1: Convert grams to moles

We are given the mass of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) and \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\). We will first convert them to moles using their molar masses. For \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\): Molar mass = 1\(\times\)22.99 (Na) + 2\(\times\)1.01 (H) + 1\(\times\)30.97 (P) + 4\(\times\)16.00 (O)= 119.98 g/mol Moles of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) = (6.5 g) / (119.98 g/mol) = 0.0542 mol For \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\): Molar mass = 2\(\times\)22.99 (Na) + 1\(\times\)1.01 (H) + 1\(\times\)30.97 (P) + 4\(\times\)16.00 (O)= 141.96 g/mol Moles of \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) = (8.0 g) / (141.96 g/mol) = 0.0564 mol

Step 2: Calculate molarity

We will now find the molarity of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) and \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) in the solution. Volume of solution = 355 mL = 0.355 L Molarity of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) = (0.0542 mol) / (0.355 L) = 0.1526 M Molarity of \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) = (0.0564 mol) / (0.355 L) = 0.1589 M

Step 3: Use Henderson-Hasselbalch equation

We will now use the Henderson-Hasselbalch equation to find the pH of the solution: \(pH = pK_{a} + \log \frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\) \(pK_{a}\) for \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) is 7.21. A- represents the HPO42- ion, and HA represents the H2PO4- ion. \(pH = 7.21 + \log \frac{0.1589}{0.1526} = 7.21 + 0.0356 = 7.25\) The \(\mathrm{pH}\) of the soft drink is 7.25.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a formula that allows us to calculate the pH of a buffer solution based on the concentration of acid and its conjugate base. This equation is pivotal when dealing with buffer systems, such as those found in physiological fluids and carbonated soft drinks like in our exercise.

At its core, the equation is expressed as: \( pH = pK_{a} + \text{log} \frac{[A^-]}{[HA]} \), where \( pK_{a} \) is the acid dissociation constant, \( [A^-] \) is the concentration of the conjugate base, and \( [HA] \) is the concentration of the acid. Using the logarithm allows us to easily find the pH by knowing these concentrations.

When applying this to an exercise, we use it to find the pH of a particular buffer solution by plugging in the known molarity of the acid and its conjugate base. It's important for students to grasp how to interpret and manipulate this equation because it simplifies understanding acid-base properties in various chemical systems.

Molarity

Molarity, denoted as \( M \), is a measure of the concentration of a solute in a solution. It's defined as the number of moles of solute per liter of solution. The formula to calculate molarity is: \( M = \frac{\text{moles of solute}}{\text{liters of solution}} \).

Understanding molarity is crucial when solving chemistry problems because it directly relates to the number of particles available in a given volume of solution. This concept is particularly important in the context of our buffer solution exercise, where we need to assess the amount of acid and base present in the solution. Precise calculations of molarity ensure the accurate determination of the pH level of a solution using the Henderson-Hasselbalch equation.

Molar Mass

Molar mass is a property that represents the mass of one mole of a substance (usually in grams per mole). It's a bridge between the mass of a substance and the number of particles or moles. To find the molar mass, we sum the atomic masses of all the atoms in a molecule.

For example, in our buffer solution problem, calculating the molar mass of \( \text{NaH}_2 \text{PO}_4 \) and \( \text{Na}_2 \text{HPO}_4 \) allows us to convert the given mass of each substance in grams to the amount in moles. This conversion is fundamental to then finding the molarity of each component, which is an essential step towards pH calculation of the buffer solution.

Acid-Base Equilibrium

Acid-base equilibrium refers to the state in which the rates of the forward and reverse reactions of an acid-base pair are equal, resulting in a stable ratio of the concentrations of the products and reactants. This concept is deeply linked to the buffer capacity of solutions.

Buffers are solutions that resist changes in pH upon the addition of small amounts of acids or bases. They typically consist of a weak acid and its conjugate base. In our soft drink example, we have a buffer made of \( \text{H}_2 \text{PO}_4^- \) and \( \text{HPO}_4^{2-} \). The equilibrium between these two species is what allows the solution to maintain a relatively constant pH. Understanding acid-base equilibrium is vital to appreciating how buffered solutions function and how they can be manipulated to maintain a desired pH, which is a key aspect in a multitude of chemical and biological systems.