The answer is
6.8 * 10^-15
The explanation:
1- we have to convert all measurements to the same units:
Conversions:
when 1 m = 100 cm
and 1 m = 10^12 pm
So,
proton radius: 1.0*10^-13 cm * (1m / 100 cm) = 10^-15 m
proton volume: 4/3 * pi * r^3 = 4/3 * pi * (10^-15 m)^3 = 4.2 * 10^-45 cu. meters
and
H atom radius: 52.9 pm * (1m / 10^12 pm) = 5.29 * 10^-11 m
H atom volume: 4/3 * pi * r^3 = 4/3 * pi * (5.29 * 10^-11 m)^3 = 6.2 × 10^-31 cu. meters
So,
2- Fraction of space occupied by nucleus = proton volume / H atom volume
= (4.2 * 10^-45 cu. meters) / (6.2 × 10^-31 cu. meters)
= 6.8 * 10^-15
So, the "fraction" would be 6.8 * 10^-15 out of 1.
6.8 x 10⁻¹⁵
Given:
Question:
The fraction of the space within the atom is occupied by the nucleus.
The Process:
Step-1: Converting the radius of hydrogen atom from pm to cm
From [tex]\boxed{ \ 1 \ pm = 10^{-12} \ m \ } \ and \ \boxed{ \ 1 \ m = 10^2 \ cm \ }[/tex], we prepare [tex]\boxed{ \ 1 \ pm = 10^{-10} \ cm \ } [/tex]
So, the radius of hydrogen atom is [tex]\boxed{ \ 52.9 \ pm = 52.9 \times 10^{-10} \ cm \ }[/tex]
Step-2: The relationship between the volume sphere and the cubic radius.
We need to assume that both the nucleus and the atom are spheres. [tex]\boxed{ \ Volume \ of \ sphere = \frac{4}{3} \pi R^3 \ }[/tex]
[tex]\boxed{\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{R_N}{R_A} \bigg)^3 \ }}[/tex]
[tex]\frac{4}{3} \pi[/tex] has been crossed out.
Remember that the volume of spheres is proportional to the cubic radius.
Step-3: Determine the fraction
Let's substitute into fraction.
[tex]\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{1.0 \times 10^{-13} \ cm}{52.9 \times 10^{-10} \ cm} \bigg)^3 \ }[/tex]
[tex]\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{1.0 \times 10^{-13} \ cm}{52.9 \times 10^{-10} \ cm} \bigg)^3 \ }[/tex]
[tex]\boxed{ \ \frac{V_N}{V_A} = (1.89 \times 10^{-5})^3 \ }[/tex]
Therefore, fraction of the space in an atom occupied by the nucleus is equal to [tex]\boxed{\boxed{ \ 6.755 \times 10^{-15} \ rounded \ to \ 6.8 \times 10^{-15} \ }}[/tex]
Keywords: the single proton, that forms, the nucleus of the hydrogen atom, has a radius of approximately, 1.0×10−13 cm, 52.9 pm, fraction of the space, volume of sphere, cubic radius
