chemistry 121 homework about density and diameter

Question

An atom has a diameter of 2.50 Å and the nucleus of that atom has a diameter of 9.50×10−5 Å . Determine the fraction of the volume of the atom that is taken up by the nucleus. Assume the atom and the nucleus are a sphere.

fraction of atomic volume:?

Calculate the density of a proton, given that the mass of a proton is 1.0073 amu1.0073 amu and the diameter of a proton is 1.69×10−15 m1.69×10−15 m.

density:?

Answer 1

The volume of a sphere is 4/3πr³ so the ratio or "fraction of the volume" is [4/3π(r_nucleus)³]/[4/3π(r_atom)³

Notice that the 4/3 and the π cancel out so the fraction is (r_nucleus)³]/(r_atom)³

r_nucleus = d_nucleus/2 = 9.50x10⁻⁵/2 = 4.75x10^-5

r_atom = d_atom/2 = 2.5/2 = 1.25

Fraction = (4.75x10^-5)³/1.25³ = 5.49x10^-14

To calculate density (in units of g/cm³), first convert 1.0073 amu to grams: 1 amu is equivalent to 1.66x10^-24 grams so the mass in grams is (1.0073)(1.66x10^-24) = 1.672x10^-24 grams

To calculate the volume of the proton, divide the diameter by 2 to get radius = 1.69x10⁻¹⁵/2 = 8.45x10^-16 m. Now convert meters to cm by multiplying by 100 so r = 8.45x10^-14 cm. That means V = 4/3πr³ = 4/3π(8.45x10^-14)³ = 2.527x10^-39 cm³

So density = (1.672x10^-24 grams)/(2.527x10^-39 cm³) = 6.62 x 10¹⁴ g/cm³

If your answer needs to be in kg/m³, you will need to make the appropriate conversion.