Problem 121 - Chemistry Matter and Change

Question

The radius of an argon atom is 94 \(\mathrm{pm}\) . Assuming the atom is spherical, what is the volume of an argon atom in cubic nanometers? \(V=4 / 3 \pi r^{3}\) (Chapter 3\()\)

Step-by-Step Solution

Answer

The volume of an argon atom is approximately \(3.48 \times 10^{-3}\,\text{nm}^{3}\).

Step 1: Convert the unit of the radius

We are given the radius of the argon atom in picometers (pm) and need to convert it to nanometers (nm) so the units in our answer will align. There is a relation between picometers and nanometers: \(1\,\text{nm} = 1000\,\text{pm}\). So to convert, divide the given picometers by 1000: \[\frac{94\,\text{pm}}{1000\,\text{pm/nm}} = 0.094\,\text{nm}\]

Step 2: Solve for the volume using the formula

Now that we have the radius in nanometers, we can use the given formula to find the volume: \[V = \frac{4}{3}\pi r^{3}\] Plug in the radius we found in the previous step: \[V = \frac{4}{3}\pi (0.094\,\text{nm})^{3}\]

Step 3: Evaluate the expression to find the volume

Evaluate the formula with the given radius: \[V = \frac{4}{3}\pi (0.094\,\text{nm})^{3} \approx 3.48 \times 10^{-3}\,\text{nm}^{3}\] So the volume of an argon atom is approximately \(3.48 \times 10^{-3}\,\text{nm}^{3}\).

Key Concepts

Unit Conversion

When working with measurements, unit conversion is an essential skill. Converting units allows us to work with different scales and units seamlessly. In our context, we started by having the radius of an argon atom given in picometers (pm). However, our desired unit for the final calculation was in nanometers (nm). Here's how we do this conversion:

• A picometer (pm) is a smaller unit than a nanometer (nm). Specifically, 1 nm equals 1000 pm.
• To convert from picometers to nanometers, we simply divide the length measurement in picometers by 1000. This method gives us the equivalent measurement in nanometers.
• In our problem, the radius was given as 94 pm, equating to 0.094 nm after conversion.

Mastering this conversion process allows for consistent units, making calculations like volume straightforward and comparable.

Radius of an Argon Atom

The radius of an atom is a critical parameter when calculating its volume. In our exercise, we focus on an argon atom, which has a radius of 94 picometers. Understanding the size of an atom requires an appreciation for these tiny units:

• An argon atom is relatively small, so scientists use special units like picometers to measure atomic radii.
• A picometer (pm) is one trillionth of a meter, showing how minute these dimensions are.
• In simpler terms, if human hair is about 80,000 to 100,000 nanometers thick, an atom like argon is drastically smaller.

This understanding of the radius informs how we can use the spherical volume formula to find the three-dimensional space occupied by the atom.

Volume Calculation

Calculating the volume of a sphere, like an argon atom, requires a precise formula. Once we have the radius in the correct unit, the calculation becomes straightforward. This is how we perform the calculation:

• The formula for the volume of a sphere is \( V = \frac{4}{3}\pi r^{3} \), where \( r \) represents the radius.
• In our solved exercise, the radius after conversion was 0.094 nm.
• By substituting this value into the formula, the calculation for volume in cubic nanometers is possible.

Understanding this step is crucial because it links a mathematical formula to a tangible concept—the volume that an atom can occupy in space.

Spherical Volume Formula

The spherical volume formula is at the heart of many calculations involving round shapes. This formula uses the principles of geometry to find the space within a three-dimensional sphere:

• The spherical volume formula is \( V = \frac{4}{3}\pi r^{3} \); this relates the volume \( V \) to the radius \( r \).
• \( \pi \) is a constant approximately equal to 3.14159, reflecting the ratio of a circle's circumference to its diameter.
• The cubic power of the radius emphasizes how volume increases dramatically with larger radii, given the three-dimensional nature of spheres.
• This formula is used universally across various fields, from chemistry in atom size calculations, like in this exercise, to astrophysics for planets and stars.

Grasping this formula allows us to understand how different dimensions interact to determine the space an object like an atom encompasses.