Problem 66 - Chemistry The Central Science In Si Units

Question

A plasma-screen TV contains thousands of tiny cells tilled with a mixture of Xe, Ne, and He gases that emits light of specific wavelengths when a voltage is applied. A particular plasma cell, \(0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm}\), contains \(4 \%\) Xe in a 1:1 Ne:He mixture at a total pressure of \(66.66 \mathrm{kPa}\). Calculate the number of Xe, Ne, and He atoms in the cell and state the assumptions you need to make in your calculation.

Step-by-Step Solution

Answer

Xe atoms: \(1.74 \times 10^{16}\), Ne atoms: \(2.09 \times 10^{17}\), He atoms: \(2.09 \times 10^{17}\). Assumed constant temperature and ideal gas behavior.

Step 1: Determine Gas Volumes

First, calculate the volume of the plasma cell using the given dimensions: \(0.900 \text{ mm} \times 0.300 \text{ mm} \times 10.0 \text{ mm}\). Convert these dimensions to meters: \((0.900 \times 10^{-3}\text{ m}) \times (0.300 \times 10^{-3}\text{ m}) \times (10.0 \times 10^{-3}\text{ m})\). The volume \(V\) of the cell is given by \(V = 0.900 \times 10^{-3} \times 0.300 \times 10^{-3} \times 10.0 \times 10^{-3}\), which yields \(V = 2.70 \times 10^{-9}\text{ m}^3\).

Step 2: Calculate Moles of Gas

Using the ideal gas equation \(PV = nRT\), solve for \(n\), the number of moles: \[ n = \frac{PV}{RT} \]Where:- \(P = 66.66 \text{kPa} = 66.66 \times 10^{3} \text{Pa}\)- \(R = 8.314 \text{ J/mol} \cdot \text{K}\)- \(T = 298 \text{ K}\,\) (Assuming room temperature.)- \(V=2.70 \times 10^{-9}\,\text{m}^3\)\( n = \frac{66.66 \times 10^3 \times 2.70 \times 10^{-9}}{8.314 \times 298} = 7.22 \times 10^{-7} \text{ mol}.\)

Step 3: Determine Moles of Each Gas

Since the cell contains \(4\%\) Xe and a 1:1 mixture of Ne and He:- Moles of Xe: \(0.04 \times 7.22 \times 10^{-7} = 2.89 \times 10^{-8}\text{ mol}\)- Moles of Ne and He individually: \(0.96 \times 7.22 \times 10^{-7} \times 0.5 = 3.47 \times 10^{-7} \text{ mol each}\)

Step 4: Calculate Number of Atoms

Convert moles to number of atoms using Avogadro's number \(N_A = 6.022 \times 10^{23} \text{ atoms/mol}\):- Xe atoms: \(2.89 \times 10^{-8} \times 6.022 \times 10^{23} = 1.74 \times 10^{16} \text{ atoms.}\)- Ne atoms: \(3.47 \times 10^{-7} \times 6.022 \times 10^{23} = 2.09 \times 10^{17} \text{ atoms.}\)- He atoms: \(3.47 \times 10^{-7} \times 6.022 \times 10^{23} = 2.09 \times 10^{17} \text{ atoms.}\)

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, symbolized by \(N_A\), which represents the number of constituent particles (usually atoms or molecules) in one mole of a substance. Avogadro's number is approximately equal to \(6.022 \times 10^{23}\), and it allows chemists to work with macroscopic amounts of material without dealing with extremely large numbers of atoms or molecules.

• This number is crucial because it provides a bridge between the atomic scale and the lab scale, making calculations of quantities in chemistry feasible.
• In practical terms, Avogadro's number lets us convert moles of a substance to the actual number of atoms or molecules present.

This conversion is crucial when you need to determine the exact quantity of elements, like xenon (Xe), neon (Ne), and helium (He) atoms in a gas mixture, as shown in this exercise.

Moles Calculation

In chemistry, the mole is a unit that measures the amount of substance. It allows chemists to count particles (atoms, molecules, etc.) by weighing them. To find the number of moles in a given volume of gas, we often use the ideal gas law: \[PV = nRT\]Where:

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the temperature in Kelvin.

Using this formula, you can rearrange it to calculate the number of moles when you know the pressure, volume, and temperature: \[n = \frac{PV}{RT}\]This equation is used in the step-by-step solution to find the total moles of the gas mixture inside the plasma cell.After obtaining the total moles, we can determine the moles for each individual gas based on their concentration ratios. This methodology is essential when you need to know the specific number of atoms or molecules in chemical reactions or gas mixtures.

Gas Volume Conversion

Converting gas volume between different units is an essential skill in solving chemistry problems involving gases. This exercise required converting measurements of a plasma cell from millimeters to meters to find the volume in cubic meters. Here's why these conversions matter:

• Many scientific calculations require standard units like meters and cubic meters, ensuring consistency and compatibility with values like the gas constant \(R\), which uses SI units.
• Accurate volume conversions are vital when using equations like the ideal gas law to derive other quantities, such as moles of gas.

For volume conversion, note the following:- 1 millimeter (mm) is \(1 \times 10^{-3}\) meters (m).By converting each dimension of the cell from millimeters to meters and then calculating the total volume in standard units, we ensure accurate results in subsequent moles calculations.