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The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å. 1 Å = 10⁻¹⁰ m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m³ of a mole of hydrogen atoms?


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Last updated date: 13th Jun 2024
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Answer
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Hint: Let's convert the volume of a hydrogen atom from angstroms to meters using the given conversion factor, and then multiply that volume by the number of atoms in a mole. Then use the formula for the volume of a sphere $V = \dfrac{4}{3}\pi r^3$ to calculate the volume of a hydrogen atom.


Solution:

Equation

$V = \dfrac{4}{3}\pi r^3$

where:

  • V is the volume of the hydrogen atom

  • r is the radius of the hydrogen atom


To calculate the total atomic volume of a mole of hydrogen atoms, we first need to determine the volume of a single hydrogen atom and then multiply it by Avogadro's number.


The volume of a single hydrogen atom can be calculated using the formula for the volume of a sphere:

$V = \dfrac{4}{3}\pi r^3$

Substituting the given value of

r (0.5 Å), we get:

$V = \dfrac{4}{3}\pi (0.5 \times 10^{-10} \text{ m})^3$

$V = 1.25 \times 10^{-23} \text{ m}^3$


To calculate the total atomic volume of a mole of hydrogen atoms, we multiply the volume of a single atom by Avogadro's number (6.022 × 10²³ atoms/mole):

$\text{Total Volume} = V \times N$

$\text{Total Volume} = 1.25 \times 10^{-23} \text{ m}^3/\text{atom} \times 6.022 \times 10^{23} \text{ atoms}/\text{mole}$

$\text{Total Volume} = 3.157 \times 10^{-7} \text{ m}^3/\text{mole}$


Answer

The total atomic volume of a mole of hydrogen atoms is approximately 3.157 × 10⁻⁷ m³. Rounding off the value of 3.157 × 10⁻⁷ m³ to two decimal places gives us 3.16 × 10⁻⁷ m³.


Note

This is just a simplified solution to the problem. In reality, the calculation of atomic volume is more complex, as it takes into account the packing arrangement of atoms in a solid.

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