Question

NOTE: pm is ”picometer” or 1 x 10-12 m.

Calculate the answer in units of g/cm3.

Calculate the average density of a single

atom by assuming that it is a uniform sphere

of radius 135.9 pm and its mass is 5.71 ×

10^−22 g. The volume of a sphere is 4

3

π r3

.

Answer in units of g/cm3

.

Answer 1

To calculate the density of a single atom modeled as a sphere, convert the radius to centimeters, calculate the volume using the formula for a sphere, and then divide the mass by the volume to obtain the density, which is 27.2 g/cm³.

Step-by-step explanation:

To calculate the average density (in g/cm3) of a single atom, we assume it is a uniform sphere with a radius of 135.9 pm (picometers). First, we must convert the radius from picometers to centimeters:

135.9 pm = 135.9 × 10^-12 m = 135.9 × 10^-10 cm.

Now, we use the formula for the volume of a sphere, V = ⅔πr3, where r is the radius:

V = ⅔π(135.9 × 10^-10 cm)3 = ⅔π(2.50 × 10^-8 cm)3

V = ⅔π(15.625 × 10^-24 cm3) = 2.094 × 10^-23 cm3.

Finally, to find the density (ρ), we divide the mass by the volume:

ρ = ⅗.71 × 10^-22 g / 2.094 × 10^-23 cm3 = 27.2 g/cm3.

The average density of a single atom, assuming it is a sphere and given the mass and radius provided, is therefore 27.2 g/cm3.

Answer 2

d = 54.4 g/cm³

Step-by-step explanation:

Given data:

Radius of sphere = 135.9 pm

Mass = 5.71 ×10⁻²² g

Volume = 4/3 ×πr³

Density = ?

Solution:

Volume = 4/3 ×πr³

Volume = 4/3×3.14× (135.9×10⁻¹²m)³

Volume = 4.2 × (135.9×10⁻¹²m)³

Volume = 1.05 ×10⁻²⁹m³

Volume = 1.05 ×10⁻²⁹×10⁶ cm³

Volume = 1.05 ×10⁻²³cm³

Density:

d = m/v

d = 5.71 ×10⁻²² g/1.05 ×10⁻²³cm³

d = 54.4 g/cm³