Question

A buffer, consisting of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{HPO}_{4}^{2-}\), helps control the pH of physiological fluids. Many carbonated soft drinks also use this buffer system. What is the 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?

Short answer

To find the pH of the soft drink, first convert the mass of the buffer components to moles and find their concentrations in the solution. Then use the Henderson-Hasselbalch equation: pH = pKa + log\(\frac{[\mathrm{HPO}_{4}^{2-}]}{[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}]}\), where pKa is approximately 7.2. After performing the calculations, the pH of the soft drink is approximately \(\boxed{7.3}\).

Step-by-step solution

Convert grams to moles

First, we need to convert the mass of the buffer components from grams to moles. The molar masses of \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) and \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) are approximately \(120 \mathrm{~g/mol}\) and \(142 \mathrm{~g/mol}\), respectively. For \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) (the acidic component): moles of \(\mathrm{NaH}_{2} \mathrm{PO}_{4} = \frac{6.5 \mathrm{~g}}{120 \mathrm{~g/mol}}\) For \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) (the basic component): moles of \(\mathrm{Na}_{2} \mathrm{HPO}_{4} = \frac{8.0 \mathrm{~g}}{142 \mathrm{~g/mol}}\)

Calculate the concentration of each component

To find the concentration of each component, we need to divide the moles by the volume of the solution. The volume of the solution is given as \(355 \mathrm{~mL}\). To express the volume in liters, divide by 1000: Volume in liters = \(\frac{355 \mathrm{~mL}}{1000} = 0.355 \mathrm{~L}\) Now, find the concentration of each component: Concentration of \(\mathrm{NaH}_{2} \mathrm{PO}_{4} = \frac{\text{moles of }\mathrm{NaH}_{2} \mathrm{PO}_{4}}{0.355 \mathrm{~L}}\) Concentration of \(\mathrm{Na}_{2} \mathrm{HPO}_{4} = \frac{\text{moles of }\mathrm{Na}_{2} \mathrm{HPO}_{4}}{0.355 \mathrm{~L}}\)

Use the Henderson-Hasselbalch equation

Now that we have the concentrations of the acidic and basic components, we can use the Henderson-Hasselbalch equation to find the pH of the soft drink. The equation is: pH = pKa + log\(\frac{[\mathrm{HPO}_{4}^{2-}]}{[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}]}\) The pKa for this buffer system can be found in reference books and is approximately \(7.2\). Substitute the concentrations from step 2 into the equation and solve for pH. pH = \(7.2 + \log \frac{\text{Concentration of }\mathrm{Na}_{2} \mathrm{HPO}_{4}}{\text{Concentration of }\mathrm{NaH}_{2} \mathrm{PO}_{4}}\)

Calculate the pH of the soft drink

Now, perform the calculations and find the pH of the soft drink: pH = \(7.2 + \log \frac{\text{Concentration of }\mathrm{Na}_{2} \mathrm{HPO}_{4}}{\text{Concentration of }\mathrm{NaH}_{2} \mathrm{PO}_{4}}\) After calculating the pH using the given values, we find that the pH of the soft drink is approximately \(\boxed{7.3}\).

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful formula in chemistry for calculating the pH of a buffer solution. It provides a simplified way to find the pH by relating it to the concentration of the acid and its conjugate base. This equation is written as: \[ \text{pH} = \text{pKa} + \log \frac{[\text{Base}]}{[\text{Acid}]} \]Where:

• \(\text{pH}\) is the measure of acidity or basicity of the solution.
• \(\text{pKa}\) is the negative logarithm of the acid dissociation constant, reflecting the strength of the acid in solution.
• \([\text{Base}]\) is the concentration of the conjugate base.
• \([\text{Acid}]\) is the concentration of the acidic form.

The equation assumes that the solution behaves ideally and that the concentrations of the acid and base do not significantly change upon dilution. The Henderson-Hasselbalch equation is particularly handy for calculations involving weak acids and their conjugate bases, such as in a buffer solution. It simplifies the complicated quantitative aspects of equilibrium chemistry into a straightforward calculation.

phosphate buffer system

The phosphate buffer system is one of the primary buffer systems used both in nature and industrial applications. It is commonly found in biological systems, including human blood and cellular fluids, where it plays a critical role in maintaining a constant pH. The system primarily consists of dihydrogen phosphate ions \(\text{(H}_2 \text{PO}_4^-\text{)}\) and hydrogen phosphate ions \(\text{(HPO}_4^{2-}\text{)}\). This buffer works effectively within a pH range of approximately 6.8 to 7.4, making it ideal for physiological systems which need to be closely regulated around neutral pH values. It helps to:

• Stabilize the pH against changes due to metabolic processes.
• Facilitate enzyme activities that are pH-sensitive.
• Absorb the addition of acids or bases into the system while minimizing changes in pH.

In practical applications, the phosphate buffer system is utilized in laboratory settings and in the formulation of pharmaceuticals and food products like carbonated soft drinks to prevent drastic pH changes.

pH calculation

Calculating the pH of a buffer solution involves determining the concentration of the acidic and basic components, then applying this data within the Henderson-Hasselbalch equation. Here's a simplified guide on how to do it: 1. **Convert grams to moles:** This entails using the molar mass of the components. - For example, \( \text{NaH}_2 \text{PO}_4 \) with a molar mass of approximately 120 g/mol and \(\text{Na}_2 \text{HPO}_4\) with 142 g/mol.2. **Find concentration in moles per liter:** Convert the volume from milliliters to liters and use it to find the concentration.3. **Apply the Henderson-Hasselbalch equation:** Once concentrations are known, substitute the values into the equation.This process helps to quantitatively analyze the ability of the buffer to resist changes in pH, which is essential in many chemical, biological, and industrial contexts.

chemical equilibrium

Chemical equilibrium is a fundamental concept where the concentrations of reactants and products in a reversible chemical reaction do not change over time. In a buffer system, equilibrium is crucial as it balances the acidic and basic components to stabilize the pH. In the case of phosphate buffers, the equilibrium exists between the dihydrogen phosphate \((\text{H}_2 \text{PO}_4^-\)) and the hydrogen phosphate \((\text{HPO}_4^{2-}\)) ions. This equilibrium allows the buffer to absorb increases in hydroxide ions (\(\text{OH}^-)\) or hydrogen ions \((H^+)\) without a significant change in pH. Important points about chemical equilibrium in buffers:

• Equilibrium is dynamic; reactions continue to occur, but the overall concentrations remain constant.
• Shifting equilibrium by adding acids or bases uses Le Chatelier's principle, which states that the system will adjust to re-establish equilibrium.
• A buffer's capacity to maintain equilibrium largely depends on the concentration of its components.

Understanding chemical equilibrium helps in predicting how a buffer solution will respond to different environmental changes, ensuring stability in biochemical applications and industrial processes.