Problem 56 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A \(20.0 \mathrm{~cm}^{3}\) mixture of \(\mathrm{CO}, \mathrm{CH}_{4}\) and He gases is exploded by an electric discharge at room temperature with excess of oxygen. The volume contraction is found to be \(13.0 \mathrm{~cm}^{3}\). A further contraction of \(14.0 \mathrm{~cm}^{3}\) occurs when the residual gas is treated with KOH solution. Find out the composition of the gaseous mixture in terms of volume percentage.

Step-by-Step Solution

Answer

The mixture consists of 60% CO, 10% CH4, and 30% He.

Step 1: Understanding Volume Contraction

When the gases are exploded with excess oxygen, an initial contraction of 13.0 cm e^{3} occurs. This contraction is due to the chemical reactions between the gases and oxygen, primarily forming CO_{2} and H_{2}O . The initial contraction indicates the moles of reacting gases other than helium, which does not participate in reactions.

Step 2: Identify Gases Involved

The explosion involving CO and CH_{4} will lead to the following reactions:\[ 2CO + O_2 \rightarrow 2CO_2 \]\[ CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O \]The contraction is due to the reduction in gas moles when reactants turn into products. Helium (He), being inert, does not cause any volume change during the reactions.

Step 3: Calculate Volume Due to First Contraction

The first contraction by 13.0 cm e^{3} implies that this volume was attributed to the reactions of CO and CH_{4} with O_{2} . Assuming complete reactions, every 1 volume of CO gives 1 volume of CO _{2} and every 1 volume of CH _{4} gives 1.5 volumes of contraction with excess oxygen.

Step 4: Calculate Volume Due to Second Contraction

The second contraction of 14.0 cm e^{3} upon treatment with KOH suggests the absorption of carbon dioxide formed from both reactions, as KOH absorbs CO_{2} .

Step 5: Calculate Volumes of CO and CH4

From the first contraction:\[ x + 1.5y = 13 \]From the second contraction:\[ x + y = 14 \]where, x = volume of CO and y = volume of CH_4.Solving these simultaneous equations:Subtract the second from the first:\[ 0.5y = -1 \]\[ y = 2 \]} ,Substitute y = 2 in x + y = 14:\[ x + 2 = 14 \]\[ x = 12 \]Thus, CO = 12 cme^{3}, CH_{4} = 2 cm^{3}.

Step 6: Determine Volume of He

The initial volume of the mixture is 20.0 cm^{3}. The sum of CO and CH_{4} is 14 cm^{3}, so the volume of He is:\[ 20 - (12 + 2) = 6 \]Thus, He = 6 cm^{3}.

Step 7: Calculate Volume Percentages

Now, calculate the volume percentages of each component:- CO: \[ \frac{12}{20} \times 100 = 60\% \] - CH_4: \[ \frac{2}{20} \times 100 = 10\% \] - He: \[ \frac{6}{20} \times 100 = 30\% \]

Key Concepts

Gas Mixture Analysis

When analyzing a gas mixture, it is important to understand the components involved and how they behave during chemical reactions. In this exercise, a mixture of carbon monoxide (CO), methane (CH₄), and helium (He) is analyzed. Helium is an inert gas, meaning it won't react during the explosion with oxygen.
The process begins with an electric discharge in the presence of excess oxygen, causing CO and CH₄ to react. These reactions lead to the formation of carbon dioxide (CO₂) and water (H₂O), which can be accounted for by analyzing the resulting volume changes.
It is crucial to note that the volume contraction observed is due to these reactions because the products have fewer moles of gas compared to the reactants. By measuring these contractions, one can determine the composition of the original gas mixture.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. In our scenario, stoichiometry helps in determining how much of each gas reacts with oxygen and the subsequent changes in volume.
The reaction equations provided are:

• CO reaction: \[2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2\]
• CH₄ reaction: \[\text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O}\]

Each mole of CO produces a mole of CO₂, while CH₄ not only forms CO₂ but also water, causing a different volume contraction. Using stoichiometry helps relate the observed volume contractions to the original amounts of CO and CH₄ in the mixture, negating the effect of helium.
By setting up equations based on these reactions, we solve for the volumes of the individual gases in the mixture, bringing clarity to the overall composition.

Percentage Composition Calculation

Calculating the percentage composition of each gas in the mixture involves determining the volume each gas occupies relative to the total mixture volume.
From our calculations, we find that CO and CH₄ react, while He remains unchanged. By determining their respective volumes, such as CO being 12 cm³ and CH₄ being 2 cm³, we can compute their percentage by dividing by the total volume and multiplying by 100.
Therefore, the formula to calculate volume percentage is:

• For CO: \[\frac{(\text{volume of CO})}{\text{total volume}} \times 100\]\[\frac{12}{20} \times 100 = 60\%\]
• For CH₄: \[\frac{2}{20} \times 100 = 10\%\]
• For He: \[\frac{6}{20} \times 100 = 30\%\]

This calculation helps us identify how much of each gas was present in the initial mixture by percentage, which is essential for understanding the mixture's characteristics.