Problem 30 - Chemistry: The Central Science

Question

To derive the ideal-gas equation, we assume that the volume of the gas atoms/molecules can be neglected. Given the atomic radius of neon, \(0.69 \AA\), and knowing that a sphere has a volume of \(4 \pi \mathrm{r}^{3} / 3\), calculate the fraction of space that Ne atoms occupy in a sample of neon at STP.

Step-by-Step Solution

Answer

The fraction of space occupied by Ne atoms in a sample of neon at STP can be calculated using the following steps: 1. Calculate the volume of a single Ne atom: \(V_{atom} = \dfrac{4}{3} \pi (0.69 \times 10^{-10})^3\) 2. Calculate the volume of 1 mole of Ne gas at STP: \(V_{stp} = 22.4 \times 10^{-3} m^3\) 3. Find the number of atoms in 1 mole of Ne: \(N_{atoms} = 6.022 \times 10^{23}\) 4. Compute the total volume of Ne atoms in 1 mole: \(V_{total} = V_{atom} \times N_{atoms}\) 5. Calculate the fraction of space that Ne atoms occupy in the sample at STP: \(Fraction_{occupied} = \dfrac{V_{total}}{V_{stp}}\) By plugging in the values calculated in previous steps and simplifying the expression, we can find the fraction of space occupied by Ne atoms.

Step 1: Calculate the volume of a single Neon atom

We know that the atomic radius of neon is \(0.69 \times 10^{-10}\) meters and a sphere has a volume of \(\dfrac{4}{3} \pi r^{3}\). So, we can compute the volume of a single Neon atom as follows: \[V_{atom} = \dfrac{4}{3} \pi (0.69 \times 10^{-10})^3\]

Step 2: Calculate the volume of 1 mole of Neon gas at STP

At Standard Temperature and Pressure (STP), 1 mole of any ideal gas occupies \(22.4\) liters of volume. So, we can calculate the volume of 1 mole of Neon at STP: \[V_{stp} = 22.4 L = 22.4 \times 10^{-3} m^3\]

Step 3: Find the number of atoms in 1 mole of Neon

In one mole of a substance, there are Avogadro's number (\(N_A\)) of particles (atoms, molecules, or ions). Avogadro's number is approximately \(6.022 \times 10^{23}\) particles per mole. Thus, in 1 mole of Neon, there are: \[N_{atoms} = N_A = 6.022 \times 10^{23}\]

Step 4: Compute the total volume of Neon atoms in 1 mole

To find the total volume occupied by the Neon atoms in 1 mole, we will multiply the volume of a single atom (calculated in step 1) by the number of atoms in 1 mole (calculated in step 3): \[V_{total} = V_{atom} \times N_{atoms}\]

Step 5: Calculate the fraction of space that Ne atoms occupy in the sample at STP

Now, we can calculate the fraction of space that Ne atoms occupy in the sample at STP by dividing the total volume occupied by Neon atoms (calculated in step 4) by the volume of 1 mole of Neon gas (calculated in step 2): \[Fraction_{occupied} = \dfrac{V_{total}}{V_{stp}}\] Calculate the fraction by plugging in the values calculated in previous steps and simplifying the expression.

Key Concepts

Atomic Radius

The atomic radius is the distance from the nucleus of an atom to the outer boundary of its electron cloud. This value helps us understand the size of individual atoms. Neon has an atomic radius of \(0.69 \text{ Å}\), which is equivalent to \(0.69 \times 10^{-10}\) meters.
Knowing the atomic radius is essential when calculating the volume of an atom, especially since many physical and chemical properties are influenced by atomic size. For neon, with its noble gas status, the atomic radius determines how it occupies space at the molecular level.

Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry to denote the number of particles in a mole of a substance. It is approximately \(6.022 \times 10^{23}\).
This number helps us convert between atoms/molecules and moles, making it easier to calculate quantities in chemical reactions and processes. When dealing with gases like Neon, knowing Avogadro's number allows us to find out how many neon atoms are in a sample.

• Useful for stoichiometry in chemistry.
• Allows transition between atomic scale and macroscopic measurements.

Volume of a Sphere

The formula to calculate the volume of a sphere is \( \dfrac{4}{3} \pi r^3 \). This formula helps determine how much space a spherical object, like an atom, occupies.
Given the atomic radius of neon, calculating a single atom's volume is crucial in understanding how much of a given volume of gas is actually filled by matter.

• Important in derivations involving gases and molecules.
• Helps visualize how much physical space atoms take.

Using the formula, the volume of a single neon atom is determined, which then contributes to understanding collective atomic arrangements in gases.

Standard Temperature and Pressure

Standard Temperature and Pressure (STP) is a reference point in chemistry used to define conditions of \(0^\circ\text{C}\) (273.15 K) and 1 atmosphere of pressure. At STP, one mole of an ideal gas occupies 22.4 liters of volume.
This standard is crucial when performing calculations involving gas volumes, because it provides a benchmark that relates to the ideal gas law.

• Assists in comparing gas behaviors under standard conditions.
• Facilitates calculations in varied scientific disciplines.

Understanding STP allows us to accurately measure and predict how gases like neon behave under these conditions.