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 A.) Why is this ratio so large?

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Given that one mole of an ideal gas at standard temperature and pressure (STP) occupies 22.4 liters (molar volume), we need to determine the atomic volume of a mole of hydrogen and then calculate the ratio.

The atomic volume of a mole of hydrogen can be found by considering the size of a hydrogen molecule, which is approximately 1 angstrom (A). Since a hydrogen molecule is composed of two hydrogen atoms, each with an approximate radius of 0.5 A, the volume occupied by a mole of hydrogen atoms can be calculated using the formula for the volume of a sphere:

Vatom=43πr3Vatom=34πr3

Substituting the radius (r=0.5 Ar=0.5A) into the formula yields:

Vatom=43π(0.5)3 A3Vatom=34π(0.5)3A3

Vatom=43π(0.125) A3Vatom=34π(0.125)A3

Vatom=16π A3Vatom=61πA3

Now, let's calculate the ratio of molar volume to the atomic volume of a mole of hydrogen:

Ratio=22.4 LVatomRatio=Vatom22.4L

Ratio=22.4×103 cm316π A3Ratio=61πA322.4×103cm3

Ratio=22.4×10316π cm3A3Ratio=61π22.4×103A3cm3

Ratio=22.4×6πRatio=π22.4×6

Ratio≈134.43.14Ratio≈3.14134.4

Ratio≈42.75Ratio≈42.75

So, the ratio of molar volume to the atomic volume of a mole of hydrogen is approximately 42.75.

Now, why is this ratio so large? This is primarily because the molar volume of a gas represents the volume occupied by a large number of gas molecules, while the atomic volume refers to the volume occupied by individual atoms. In the case of hydrogen gas, the molar volume is significantly larger because the gas molecules are not only composed of two hydrogen atoms but also exhibit considerable intermolecular space between them. This intermolecular space contributes to the larger molar volume compared to the atomic volume of hydrogen atoms.