Question

Helium is the lightest noble gas and the second most abundant element (after hydrogen) in the universe. (a) The radius of a helium atom is \(3.1 \times 10^{-11} \mathrm{~m} ;\) the radius of its nucleus is \(2.5 \times 10^{-15} \mathrm{~m} .\) What fraction of the spherical atomic volume is occupied by the nucleus ( \(V\) of a sphere \(\left.=\frac{4}{3} \pi r^{3}\right) ?\) (b) The mass of a helium-4 atom is \(6.64648 \times 10^{-24} \mathrm{~g}\), and each of its two electrons has a mass of \(9.10939 \times 10^{-28} \mathrm{~g}\). What fraction of this atom's mass is contributed by its nucleus?

Short answer

The fraction of the atomic volume occupied by the nucleus is approximately 2.1 × 10^(-14) and the fraction of the mass contributed by the nucleus is approximately 0.999102.

Step-by-step solution

Volume of the helium atom

Calculate the volume of the helium atom using the given radius. The formula for the volume of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] Given the radius (\r\text{_atom}) = \(3.1 \times 10^{-11} \text{ m}\): \[ V_{\text{atom}} = \frac{4}{3} \pi (3.1 \times 10^{-11})^3 \]\

Volume of the helium nucleus

Calculate the volume of the helium nucleus using the given radius. Given the radius (\r\text{_nucleus}) = \(2.5 \times 10^{-15} \text{ m}\): \[ V_{\text{nucleus}} = \frac{4}{3} \pi (2.5 \times 10^{-15})^3 \]

Fraction of volume occupied by the nucleus

Determine the fraction of the atomic volume occupied by the nucleus by dividing the volume of the nucleus by the volume of the atom: \[ \text{Fraction} = \frac{V_{\text{nucleus}}}{V_{\text{atom}}} \]

Mass of electrons in a helium-4 atom

Calculate the total mass of the electrons in the helium-4 atom. Since there are 2 electrons: \[ \text{Mass}_{\text{electrons}} = 2 \times 9.10939 \times 10^{-28} \text{ g} \]

Mass of the nucleus in a helium-4 atom

Calculate the mass of the nucleus by subtracting the mass of the electrons from the total mass of the helium-4 atom: \[ \text{Mass}_{\text{nucleus}} = 6.64648 \times 10^{-24} \text{ g} - \text{Mass}_{\text{electrons}} \]

Fraction of mass contributed by the nucleus

Determine the fraction of the atom’s mass contributed by its nucleus by dividing the mass of the nucleus by the total atomic mass: \[ \text{Fraction}_{\text{mass}} = \frac{\text{Mass}_{\text{nucleus}}}{\text{Mass}_{\text{atom}}} \]

Key concepts

Atomic Volume Calculation

To understand the helium atom, we need to calculate its atomic volume first. The formula for the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \).
Using the radius of the helium atom provided (\r\text{_atom} = 3.1 \times 10^{-11} m), we substitute this value into the formula:
\ V_{\text{atom}} = \frac{4}{3} \pi (3.1 \times 10^{-11})^3 \.
This calculation gives us the volume occupied by the entire helium atom.

• The radius (r) is the distance from the center to the edge of the atom.
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