Question

The single proton that forms the nucleus of the hydrogen atom has a radius of approximately 1.0×10−13cm. The hydrogen atom itself has a radius of approximately 52.9 pm
What fraction of the space within the atom is occupied by the nucleus?

Answer 1

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.

Answer 2

6.8 x 10⁻¹⁵

Further explanation

Given:

• The radius of nucleus = 1.0 x 10⁻¹³ cm
• The radius of hydrogen atom = 52.9 pm

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 $\boxed{ \ 1 \ pm = 10^{-12} \ m \ } \ and \ \boxed{ \ 1 \ m = 10^2 \ cm \ }$, we prepare $\boxed{ \ 1 \ pm = 10^{-10} \ cm \ }$

So, the radius of hydrogen atom is $\boxed{ \ 52.9 \ pm = 52.9 \times 10^{-10} \ cm \ }$

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. $\boxed{ \ Volume \ of \ sphere = \frac{4}{3} \pi R^3 \ }$

• $\boxed{ \ V_N = volume \ of \ nucleus \ }$
• $\boxed{ \ V_A = volume \ of \ atom \ }$
• $\boxed{ \ R_N = radius \ of \ nucleus \ }$
• $\boxed{ \ R_A = radius \ of \ atom \ }$

$\boxed{\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{R_N}{R_A} \bigg)^3 \ }}$

$\frac{4}{3} \pi$ has been crossed out.

Remember that the volume of spheres is proportional to the cubic radius.

Step-3: Determine the fraction

• $\boxed{ \ R_N = 1.0 \times 10^{-13} \ cm \ }$
• $\boxed{ \ R_A = 52.9 \times 10^{-10} \ cm \ }$

Let's substitute into fraction.

$\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{1.0 \times 10^{-13} \ cm}{52.9 \times 10^{-10} \ cm} \bigg)^3 \ }$

$\boxed{ \ \frac{V_N}{V_A} = \bigg( \frac{1.0 \times 10^{-13} \ cm}{52.9 \times 10^{-10} \ cm} \bigg)^3 \ }$

$\boxed{ \ \frac{V_N}{V_A} = (1.89 \times 10^{-5})^3 \ }$

Therefore, fraction of the space in an atom occupied by the nucleus is equal to $\boxed{\boxed{ \ 6.755 \times 10^{-15} \ rounded \ to \ 6.8 \times 10^{-15} \ }}$

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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