Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A \(10 \mathrm{mg}\) effervescent tablet containing sodium bicarbonate and oxalic acid releases \(0.25 \mathrm{~mL}\) of \(\mathrm{CO}_{2}\) at \(\mathrm{T}=298.15 \mathrm{~K}\) and \(\mathrm{P}=1\) bar. If molar volume of \(\mathrm{CO}_{2}\) is \(25.0 \mathrm{~L}\) under such condition, what is the percentage of sodium bicarbonate in each tablet? [Main Jan. 11, 2019 (I)] \(\left[\right.\) Molar mass of \(\left.\mathrm{NaHCO}_{3}=84 \mathrm{~g} \mathrm{~mol}^{-1}\right]\) (a) \(0.84\) (b) \(33.6\) (c) \(16.8\) (d) \(8.4\)

Step-by-Step Solution

Answer

The percentage of sodium bicarbonate in the tablet is 8.4% (option d).

Step 1: Calculate moles of CO2 released

First, determine the moles of \( \mathrm{CO_{2}} \) released using the given volume and molar volume. The volume of \( \mathrm{CO_{2}} \) released is \( 0.25 \mathrm{~mL} = 0.25 \times 10^{-3} \mathrm{~L} \). The molar volume of \( \mathrm{CO}_{2} \) is given as \( 25.0 \mathrm{~L~mol}^{-1} \). Use the formula: \[ \text{moles of } \mathrm{CO}_{2} = \frac{\text{volume of } \mathrm{CO}_{2}}{\text{molar volume}} = \frac{0.25 \times 10^{-3}}{25.0} = 1.0 \times 10^{-5} \text{ mol} \].

Step 2: Relate CO2 to NaHCO3

Sodium bicarbonate (\( \mathrm{NaHCO}_3 \)) reacts according to the equation: \( \mathrm{NaHCO}_3 \rightarrow \mathrm{Na}^+ + \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2 \). This tells us that 1 mole of \( \mathrm{NaHCO}_3 \) produces 1 mole of \( \mathrm{CO}_2 \). Therefore, the moles of \( \mathrm{NaHCO}_{3} \) are also \( 1.0 \times 10^{-5} \) moles.

Step 3: Calculate mass of NaHCO3

Using the moles of \( \mathrm{NaHCO}_{3} \) and its molar mass \( 84 \mathrm{~g~mol}^{-1} \), calculate the mass: \[ \text{mass of } \mathrm{NaHCO}_3 = \text{moles of } \mathrm{NaHCO}_3 \times \text{molar mass} = 1.0 \times 10^{-5} \times 84 = 0.00084 \mathrm{~g} = 0.84 \mathrm{~mg} \].

Step 4: Calculate percentage of NaHCO3 in tablet

Finally, calculate the percentage of \( \mathrm{NaHCO}_3 \) in the tablet. The tablet's total mass is \(10~ \mathrm{mg}\). Thus, the percentage is: \[ \% \text{ of } \mathrm{NaHCO}_3 = \frac{0.84~ \mathrm{mg}}{10~ \mathrm{mg}} \times 100\% = 8.4\% \].

Key Concepts

Molar Volume

The concept of molar volume is essential in understanding gas reactions and conversions. Molar volume refers to the volume occupied by one mole of a substance, commonly a gas, at a specific temperature and pressure. For gases, it is usually measured in liters per mole (L/mol).
In the exercise, we have a molar volume of carbon dioxide (\(\mathrm{CO}_2\)) given as \(25.0 \mathrm{~L~mol}^{-1}\) at a temperature of \(298.15 \mathrm{~K}\) and a pressure of \(1\) bar.

• To find the moles of a gas, use the formula:\[\text{moles of gas} = \frac{\text{volume of gas}}{\text{molar volume}}\]
• This calculation bridges the gap between the macroscopic measure of volume and the microscopic concept of moles.

Understanding molar volume is key in converting between the physical amount of gas we observe and their corresponding chemical quantities.

Effervescent Tablet

Effervescent tablets are a popular form of medication and supplements, designed to dissolve in water and release carbon dioxide gas (\(\mathrm{CO}_2\)). This makes it fizzy and facilitates quicker absorption.
In the exercise scenario, sodium bicarbonate (\(\mathrm{NaHCO}_3\)) in the tablet reacts in water, resulting in the release of carbon dioxide gas.

Effervescent tablets offer several benefits:

• They provide a more palatable method of taking medications or vitamins.
• The effervescent action can enhance the bioavailability and rapid onset of the drug effect.

In a chemical sense, one mole of \(\mathrm{NaHCO}_3\) leads to one mole of \(\mathrm{CO}_2\) being produced, as shown through the decomposition equation:\[\mathrm{NaHCO}_3 \rightarrow \mathrm{Na}^+ + \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2\] Understanding this reaction is key to determining the amount of reactants involved in the process.

Percentage Composition

Percentage composition is a critical concept representing how much of a particular substance is contained in a mixture or compound. It's expressed as a percentage of the whole's mass or volume.
In this exercise, we calculate the percentage composition of sodium bicarbonate (\(\mathrm{NaHCO}_3\)) within an effervescent tablet.
The process involves:

• Calculation of the mass of \(\mathrm{NaHCO}_3\) using its moles and molar mass.
• Dividing this mass by the total mass of the tablet to find the proportion.
• Converting the proportion to a percentage by multiplying by 100.

For this problem, the calculation follows:\[\% \text{ of } \mathrm{NaHCO}_3 = \frac{0.84~ \mathrm{mg}}{10~ \mathrm{mg}} \times 100\% = 8.4\%\] This step is crucial in both scientific and industrial contexts where the precise composition is significant for quality control and regulatory compliance.