Question 1 of 9intermediate🔧 ApplyNumerical3 marks

Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.

Correct Answer

The fraction of molecular volume to the actual volume occupied by oxygen gas at STP is approximately 3.8×103.8 \times 10⁻⁴.

Step 1: Calculate volume of one oxygen molecule Diameter (d) = 3 Å = 3×103 \times 10⁻¹⁰ m Radius (r) = d/2 = 1.5×101.5 \times 10⁻¹⁰ m Volume of one molecule = (4/3)πr³ = (4/3) × 3.14 × (1.5×101.5 \times 10⁻¹⁰)³ = 14.13×1014.13 \times 10⁻³⁰ m³

Step 2: Determine actual volume occupied by 1 mole of gas at STP From Exercise 12.2, molar volume at STP = 22.4 litres = 22.4×1022.4 \times 10⁻³ m³

Step 3: Calculate total molecular volume for 1 mole Number of molecules in 1 mole = Avogadro number = 6.022×106.022 \times 10²³ Total molecular volume = 6.022×106.022 \times 10²³ × 14.13×1014.13 \times 10⁻³⁰ = 8.5×108.5 \times 10⁻⁶ m³

Step 4: Calculate the required fraction Fraction = Molecular volume / Actual volume = (8.5×108.5 \times 10⁻⁶) / (22.4×1022.4 \times 10⁻³) = 3.8×103.8 \times 10⁻⁴

Thus, the molecular volume is only about 0.04% of the actual volume occupied by the gas at STP.

Exercise: EXERCISES | Q: 12.1
For More Understanding

Explanation

This question tests understanding of the difference between the actual volume a gas occupies and the volume actually taken up by the molecules themselves. At STP, gases have large intermolecular spaces, so the molecular volume is a tiny fraction of the total volume. The calculation follows the approach shown in Example 12.1 of the textbook, where a similar ratio is estimated for water vapour. The key steps involve finding the volume of a single spherical molecule, multiplying by Avogadro's number to get total molecular volume for one mole, and dividing by the molar volume (22.4 L at STP as established in Exercise 12.2).

Solution Steps

  1. Step 1: Calculate volume of one oxygen molecule using V = (4/3)πr³ with r = 1.5×101.5 \times 10⁻¹⁰ m

  2. Step 2: Use molar volume at STP = 22.4 L = 22.4×1022.4 \times 10⁻³ m³

  3. Step 3: Multiply volume of one molecule by Avogadro number (6.022×106.022 \times 10²³) to get total molecular volume

  4. Step 4: Divide total molecular volume by molar volume to get the fraction

Next