Question

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

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

Step-by-step solution

Write down the given values

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

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: \(r = 69 \mathrm{pm} \cdot \frac{1 \mathrm{cm}}{1 \times 10^{12} \mathrm{pm}} = 69 \times 10^{-12} \mathrm{cm}\)

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. \(V = \frac{4}{3} \pi (69 \times 10^{-12} \mathrm{cm})^3\)

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 \(density = \frac{mass}{volume}\). \(density = \frac{3.35 \times 10^{-23} \mathrm{g}}{\frac{4}{3} \pi (69 \times 10^{-12} \mathrm{cm})^3}\)

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.

Key concepts

neon atom

A neon atom is a very small particle that makes up the element neon, which is a noble gas in the periodic table. Neon atoms are so tiny that we typically measure their features on a scale of picometers (pm), where 1 pm is equal to one trillionth of a meter. This small size helps neon to remain colorless and odorless, and it's often used in brightly colored signs due to its ability to glow in vibrant reds and oranges when charged with electricity. Perhaps surprisingly, neon is quite common in the universe, even though it's rare on Earth.
In chemistry, understanding the structure of atoms like neon helps us learn how they interact with each other and form various compounds. This is important for many applications, including in electronics and neon lights where neon's inertness is a beneficial property.

volume of a sphere

The volume of a sphere is a measurement of how much three-dimensional space the sphere occupies. To calculate this, you need to know the sphere's radius, which is the distance from the center of the sphere to any point on its surface. The formula to compute the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \), where \( V \) is the volume and \( r \) is the radius.Let's break this down:

• \( \pi \) is a constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.
• The radius \( r \) needs to be in the correct unit for the calculation's output to be meaningful, in this case, centimeters.

Calculating the volume of a sphere helps in various scientific calculations, including finding out how tightly packed atoms are within a material, which is critical in density calculations like the one in the exercise.

unit conversion

Unit conversion is the process of changing a measurement from one unit to another. It's essential to perform calculations accurately, especially in scientific contexts where measurements are taken in various units. In the example of calculating the density of the neon atom, we initially had the radius in picometers (pm) and needed to convert it to centimeters (cm) to continue our calculations.Here's how unit conversion works for this problem:

• We know that 1 cm equals \( 1 \times 10^{12} \) pm. This factor is important because it allows us to convert between these units.
• Using this factor, the radius of 69 pm is changed to \( 69 \times 10^{-12} \) cm.

By converting the radius to centimeters, we can ensure that our calculation of the sphere's volume will be in cubic centimeters, keeping all subsequent measurements consistent, especially important when subsequently calculating density.

atomic radius

The atomic radius is a measure of the size of an atom. It describes the distance from the center of the nucleus to the outer boundary of the surrounding cloud of electrons. Atomic radius is typically measured in picometers (pm), and it varies across the elements in the periodic table. Neon, for instance, has an atomic radius of 69 pm, a relatively small size, which is typical for noble gases. Elements with a larger atomic radius usually have more electron shells around the nucleus, while those with a smaller radius, like neon, have fewer. As you move left to right across a period in the periodic table, elements often have a smaller atomic radius, due to increased nuclear charge, which pulls the electron cloud closer to the nucleus. Understanding atomic radius is crucial in many fields of chemistry as it affects how atoms will interact, bond, and structure themselves in molecules. For problems involving physical dimensions, like our exercise on density calculation, knowing the atomic radius helps in determining the volume of atoms that are treated as spheres.