gorovogpg

2021-12-14

The single proton that forms the nucleus of the hydrogen atom has a radius of approximately $1.0×{10}^{-13}$ cm. The hydrogen atom itself has a radius of approximately $52.9±$. What fraction of the space within the atom is occupied by the nucleus?

kalfswors0m

Radius of nucleus: $1.0×{10}^{-13}cm$
Now, covert the given values in a single unit of measurement:
$cm\to m$
$1.0×{10}^{-13}cm×\frac{1m}{100cm}=1.0×{10}^{-15}m$
$\pi cometer\to m$
$52.9×\frac{1.0×{10}^{-12}m}{1±}=5.29×{10}^{-11}m$
Get the volume of the nucleus and the volume of the atom using:
${V}_{sphere}=\frac{4}{3}\pi {r}^{3}$
For atom:
${V}_{a\to m}=\frac{4}{3}\pi {\left(5.29×{10}^{-11}m\right)}^{3}$
${V}_{a\to m}=6.2009×{10}^{-31}{m}^{3}$
For nucleus
${V}_{\nu c\le us}=\frac{4}{3}\pi {\left(1.0×{10}^{-15}m\right)}^{3}$
${V}_{\nu c\le us}=4.1888×{10}^{-45}{m}^{3}$
Now, use this frmula to find the fraction of the space within the atom is occupied by the nucleus: $X=\frac{{V}_{\nu c\le us}}{{V}_{a\to m}}$
$X=\frac{4.1888×{10}^{-45}{m}^{3}}{6.2009×{10}^{-31}{m}^{3}}$
$X=6.7551×{10}^{-15}$

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