Question

Nicotinic acid, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NO}_{2}\), is found in minute amounts in all living cells, but appreciable amounts occur in liver, yeast, milk, adrenal glands, white meat, and corn. Whole-wheat flour contains about \(60 . \mu g\) per gram of flour. One gram (1.00 g) of the acid dissolves in water to give \(60 .\) mL of solution having a pH of \(2.70 .\) What is the approximate value of \(K_{\mathrm{a}}\) for the acid?

Short answer

The approximate value of \(K_a\) is \(5.0 \times 10^{-5}\).

Step-by-step solution

Calculate the Molarity of the Solution

First, find the molarity (M) of the solution. Given that 1.00 g of nicotinic acid is dissolved in 60.0 mL of solution, convert the mass to moles using its molar mass (C₆H₅NO₂ has a molar mass of approximately 123.11 g/mol). The formula for moles is: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{1.00}{123.11}\). Then, convert 60.0 mL to liters (0.060 L) and use the formula: \(\text{Molarity} = \frac{\text{moles}}{\text{liters}}\).

Determine the Concentration of Hydrogen Ions

Using the pH value of 2.70, calculate the concentration of hydrogen ions \([H^+]\) with the formula: \([H^+] = 10^{-\text{pH}} = 10^{-2.70}\).

Write the Acid Dissociation Expression

The dissociation of nicotinic acid (HA) in water can be represented as: \( HA \rightleftharpoons H^+ + A^-\). The equilibrium expression for the acid dissociation constant is \( K_a = \frac{[H^+][A^-]}{[HA]}\).

Determine Initial and Equilibrium Concentrations

Initially, consider the initial concentration of nicotinic acid as the molarity calculated in Step 1. At equilibrium, the concentration of \([H^+]\) is what you calculated in Step 2. Assume \([A^-] = [H^+]\). The concentration of \([HA]\) starts at its initial concentration and decreases by \([H^+]\).

Calculate \(K_a\)

Substitute the equilibrium concentrations into the expression for \(K_a\): \(K_a = \frac{([H^+])^2}{[HA]_{\text{initial}} - [H^+]}\). Plug in the values from Steps 1 and 2, and simplify to find the approximate \(K_a\).

Execute the Calculation

Substitute \([H^+] = 10^{-2.70} \approx 0.002 \) M into the formula from Step 5. If the initial concentration from Step 1 is \(\approx 0.081\) M (as calculated from 1 g in 60 mL), then: \(K_a \approx \frac{(0.002)^2}{0.081 - 0.002}\). Calculate to find \(K_a\).

Key concepts

Nicotinic Acid

Nicotinic acid, also known as niacin or vitamin B3, plays a vital role in many important biological processes. It's an organic compound that appears naturally in small quantities in all living cells. However, notable amounts can be found in certain foods like liver, yeast, and white meat.

This compound, with the formula \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{NO}_2\), is one of the essential vitamins that the human body requires for proper conversion of food into energy. In our bodies, nicotinic acid assists in maintaining healthy skin and nerves, and it also supports liver function. A deficiency in this nutrient can lead to several health issues, such as pellagra, which is characterized by diarrhea, dermatological problems, and dementia.

Outside of our physiological needs, nicotinic acid has chemical properties that are important in various reactions. It is a weak acid that dissociates in water, releasing hydrogen ions \((H^+)\). This means it can be used to study equilibrium in chemistry, as it helps to understand the behavior of weak acids and their dissociation constants in solution.

Molarity Calculation

Calculating molarity is an essential skill when working with solutions in chemistry. It tells us how much solute is present in a specific volume of solvent, which is measured in liters. Molarity \((M)\) is defined as the number of moles of solute per liter of solution.

To determine the molarity of nicotinic acid dissolved in water, first find the number of moles of the acid. Start by using its molar mass, which is calculated based on the atomic masses of its constituent atoms: carbon, hydrogen, nitrogen, and oxygen. With a molar mass of approximately 123.11 g/mol, and given that 1.00 g of nicotinic acid dissolves in the solution, the moles are calculated using the formula:

\[ \text{Moles of nicotinic acid} = \frac{1.00 \, \text{g}}{123.11 \, \text{g/mol}} \approx 0.0081 \, \text{mol} \]

Since 60.0 mL of solution is equivalent to 0.060 L, the molarity is:

\[ \text{Molarity} = \frac{0.0081 \, \text{mol}}{0.060 \, \text{L}} \approx 0.135 \, M \]

Equilibrium Concentration

When studying equilibrium in the context of chemical reactions, particularly those involving acids, it's crucial to understand how concentrations change.

Nicotinic acid dissociates in water, forming hydrogen ions \((H^+)\) and acetate ions \((A^-)\). The initial concentration of nicotinic acid, calculated through its molarity, provides a baseline. At equilibrium, some of the nicotinic acid remains undissociated, while a portion dissociates to release hydrogen ions and corresponding anions.

We use the concept of equilibrium concentration to describe these changes. The equilibrium concentrations are then plugged into the expression for the acid dissociation constant \(K_a\). The equation is defined as:

\[ K_a = \frac{[H^+][A^-]}{[HA]} \]

This represents the ratio of the product of the concentrations of the ions over the concentration of the undissociated acid. Understanding these dynamics in weak acids like nicotinic acid is fundamental for predicting how acids will behave in different chemical environments and under varying conditions.

pH and Hydrogen Ion Concentration

The pH of a solution is a measure of its acidity or basicity. For any given acid, the pH can tell how strong the acid is and how completely it dissociates in water. The pH is calculated from the hydrogen ion concentration \([H^+]\), which represents the number of hydrogen ions present in a solution. The relationship is given by the logarithmic formula:

\[ \text{pH} = -\log_{10} [H^+] \]

In the example of nicotinic acid dissolved in water with a pH of 2.70, the concentration of hydrogen ions can be found using the equational relationship that includes logarithms:

\[ [H^+] = 10^{-2.70} \approx 0.002 \, M \]

This concentration represents the level of dissociation in the solution, providing information on how much of the nicotinic acid molecules have released hydrogen ions. Knowing the pH and subsequently the \([H^+]\) helps in estimating the strength and behavior of the acid in equilibrium, facilitating the calculation of the acid dissociation constant \(K_a\), which indicates the acid's dissociation tendency in water.