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Chapter 1: Problem 114

The radius of a neon atom is \(69 \mathrm{pm},\) and its mass is \(3.35 \times\) \(10^{-23} \mathrm{g} .\) What is the density of the atom in grams per cubic centimeter \(\left(\mathrm{g} / \mathrm{cm}^{3}\right) ?\) Assume the nucleus is a sphere with volume \(=\frac{4}{3} \pi r^{3}\)

Short Answer

Expert verified
The density of the neon atom is approximately \(0.324 \mathrm{g/cm^3}\).

Step by step solution

01

Write down the given values

We are given the radius (r) of the neon atom as 69 picometers (pm) and its mass (m) as \(3.35 \times 10^{-23} \mathrm{g}\).
02

Convert the radius to centimeters

To get the volume and density in the appropriate units (cubic centimeters), we need to convert the given radius from picometers to centimeters. Since 1 centimeter is equal to \(1 \times 10^{12}\) picometers, we can convert the radius (r) as follows: \(r = 69 \mathrm{pm} \cdot \frac{1 \mathrm{cm}}{1 \times 10^{12} \mathrm{pm}} = 69 \times 10^{-12} \mathrm{cm}\)
03

Calculate the volume of the neon atom

Assuming the nucleus is spherical, we can compute its volume (V) using the formula \(V = \frac{4}{3} \pi r^3\), where r is the radius in centimeters. \(V = \frac{4}{3} \pi (69 \times 10^{-12} \mathrm{cm})^3\)
04

Calculate the density of the neon atom

We now have the mass and volume of the neon atom. To find the density, simply divide the mass by the volume using the formula \(density = \frac{mass}{volume}\). \(density = \frac{3.35 \times 10^{-23} \mathrm{g}}{\frac{4}{3} \pi (69 \times 10^{-12} \mathrm{cm})^3}\)
05

Simplify and solve for density

Now, perform the calculations and simplify to obtain the density of the neon atom in grams per cubic centimeter: \(density = \frac{3.35 \times 10^{-23} \mathrm{g}}{\frac{4}{3} \pi (69 \times 10^{-12} \mathrm{cm})^3} = 0.324 \mathrm{g/cm^3}\) Hence, the density of the neon atom is approximately 0.324 g/cm³.

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