Question 15 of 17advanced🔧 ApplyNumerical3 marks

One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large?

Correct Answer

Step 1: Calculate Molar Volume in m³ Molar volume = 22.4 L = 22.4×1022.4 \times 10⁻³ m³

Step 2: Calculate Atomic Volume of one hydrogen molecule Size of hydrogen molecule = 1 Å = 10⁻¹⁰ m Radius (r) = 0.5×100.5 \times 10⁻¹⁰ m Volume of one molecule = (4/3)πr³ = (4/3) × 3.14 × (0.5×100.5 \times 10⁻¹⁰)³ = 5.24×105.24 \times 10⁻³¹ m³

Step 3: Calculate Atomic Volume of one mole of hydrogen Atomic volume of 1 mole = Volume of one molecule × Avogadro number = 5.24×105.24 \times 10⁻³¹ × 6.022×106.022 \times 10²³ = 3.15×103.15 \times 10⁻⁷ m³

Step 4: Calculate the Ratio Ratio = Molar volume / Atomic volume = (22.4×1022.4 \times 10⁻³) / (3.15×103.15 \times 10⁻⁷) ≈\approx 7.1×107.1 \times 10⁴

This ratio is approximately 10⁴.

Why is this ratio so large? This ratio is large because in the gaseous state, the molecules are far apart from each other. Most of the volume occupied by a gas is empty space between molecules. The actual volume occupied by the molecules themselves is very small compared to the total volume the gas occupies. This shows that gases have large intermolecular spaces and are highly compressible.

Exercise: EXERCISES | Q: 1.15 | (Page 11)
For More Understanding

Explanation

This question from NCERT Physics Class 11 (Chapter 2: Units and Measurements) tests the student's understanding of molar volume, atomic volume, and the nature of gaseous state. The calculation involves converting units properly (L to m³, Å to m), using the formula for volume of a sphere, and understanding Avogadro's number. The large ratio (~10⁴) demonstrates that gases consist mostly of empty space, which is a fundamental concept in understanding the behaviour of gases and their compressibility.

Solution Steps

  1. Step 1: Convert molar volume from L to m³ (22.4 L = 22.4×1022.4 \times 10⁻³ m³)

  2. Step 2: Calculate volume of one hydrogen molecule using sphere formula with radius = 0.5 Å

  3. Step 3: Multiply by Avogadro's number (6.022×106.022 \times 10²³) to get atomic volume of 1 mole

  4. Step 4: Divide molar volume by atomic volume to get ratio ≈\approx 7.1×107.1 \times 10⁴

  5. Step 5: Explain that large ratio is due to empty space between gas molecules

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