Question

A plasma-screen TV contains thousands of tiny cells filled with a mixture of \(\mathrm{Xe}, \mathrm{Ne}\) , 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: Ne: He mixture at a total pressure of 500 torr. Calculate the number of Xe, Ne, and He atoms in the cell and state the assumptions you need to make in your calculation.

Short answer

The volume of the plasma cell is \(2.7 \times 10^{-6} m^3\), and the total pressure in atm is 0.6579 atm. Partitioning the pressure, we find that the partial pressures for Xe, Ne, and He are 0.0263 atm, 0.3023 atm, and 0.3023 atm, respectively. Using the Ideal Gas Law, we calculate the number of moles for Xe, Ne, and He. Finally, by multiplying the moles by Avogadro's number, we find the number of Xe, Ne, and He atoms in the cell. The assumptions made during the calculation are that the Ideal Gas Law is applicable, the room temperature is 298 K, and the mixture contains gases in a 1:1:1 ratio.

Step-by-step solution

Calculate the volume of the cell

Given the dimensions of the plasma cell, we can simply multiply them to get the volume of the cell. \(Volume = 0.900 mm \times 0.300 mm \times 10.0 mm\) Note that we have to convert the dimensions to meters to use in the Ideal Gas Law. \(Volume = (0.900 \times 10^{-3} m) \times (0.300 \times 10^{-3} m) \times (10.0 \times 10^{-3} m)\) Calculate the volume: \(Volume = 2.7 \times 10^{-6} m^3\)

Convert total pressure to atm

Given the total pressure is 500 torr, we need to convert it to atm. \(500 \, torr \times \frac{1 \, atm}{760 \, torr} = 0.6579 \, atm\)

Partition the pressure of each gas

The gases are mixed in a 1:1:1 ratio, so they will have equal parts of the total pressure. Since Xe is 4% of the total pressure, we can determine the percentage of Ne and He in the mixture. For Ne and He: \(50\% - 4\% = 46\%\) of the total pressure each. Calculate partial pressures: \(P_{Xe} = 0.6579 \, atm \times 0.04\\ P_{Ne} = 0.6579 \, atm \times 0.46\\ P_{He} = 0.6579 \, atm \times 0.46\)

Calculate the number of moles for each gas using Ideal Gas Law

Using the Ideal Gas Law (PV = nRT), we can calculate the number of moles for each gas. Rearrange the equation to solve for n: \(n = \frac{PV}{RT}\), where R is the ideal gas constant (0.0821 L atm K^-1 mol^-1) and T is the temperature in Kelvin (assumed to be the room temperature, 298 K). We also need to convert the volume from m³ to liters (1 m³ = 1000 L): \(Volume = 2.7 \times 10^{-6} m^3 \times \frac{1000 \, L}{1 \, m^3} = 2.7 \times 10^{-3} L\) Calculate moles: \(n_{Xe} = \frac{P_{Xe} \times Volume}{R \times T} \\ n_{Ne} = \frac{P_{Ne} \times Volume}{R \times T} \\ n_{He} = \frac{P_{He} \times Volume}{R \times T}\)

Calculate the number of atoms for each gas

The number of atoms can be found by multiplying the moles of each gas by Avogadro's number (\(6.022 \times 10^{23}\)). Calculate atoms: \(Xe \, atoms = n_{Xe} \times 6.022 \times 10^{23}\\ Ne \, atoms = n_{Ne} \times 6.022 \times 10^{23}\\ He \, atoms = n_{He} \times 6.022 \times 10^{23}\) Now, we have the number of atoms for Xe, Ne, and He in the plasma cell. Assumptions made during the calculation: 1. The Ideal Gas Law is applicable. 2. The room temperature is 298 K. 3. The mixture contains gases in a 1:1:1 ratio.

Key concepts

Partial Pressure

Partial pressure is the pressure that a single type of gas in a mixture of gases would exert if it occupied the entire volume by itself. It is an important concept when dealing with mixtures of gases, like in our plasma cell problem.
The total pressure of a gas mixture can be expressed as the sum of the individual partial pressures of the gases in that mixture. The formula for this can be written as:\[ P_{total} = P_{Xe} + P_{Ne} + P_{He} \]This tells us how much each gas in the mixture contributes to the overall pressure inside the plasma cell. For our problem, though the gases Xe, Ne, and He are mixed, only Xe has a specific percentage attributed to it, 4% of the total pressure. The remaining 96% is equally divided between Ne and He.
Knowing the partial pressure can help calculate how much of each gas is contributing to the behavior of the whole gas mixture.

Volume Conversion

When dealing with gas calculations, it's crucial to ensure that all measurements are in consistent units. The dimensions of the plasma cell were given in millimeters, but the ideal gas law requires volume in liters. Hence, a conversion is necessary. The basic conversion relationship to remember is:- 1 meter = 1000 millimeters- 1 m³ = 1000 L
The plasma cell's dimensions need to be converted from millimeters to meters first. This can be done by multiplying each dimension by \(10^{-3}\). Once the dimensions are in meters, the volume can be calculated. From cubic meters, it is then converted into liters by multiplying by 1000. This conversion ensures accuracy in further calculations involving the ideal gas law.

Mole Calculation

The mole is an essential unit in chemistry that measures the quantity of a substance. In the context of gases, the Ideal Gas Law is often used to compute the number of moles present. The Ideal Gas Law is given by:\[ PV = nRT \]where:

• \( P \) is the pressure in atm,
• \( V \) is the volume in liters,
• \( n \) is the moles,
• \( R \) is the ideal gas constant \( (0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1}) \),
• and \( T \) is the temperature in Kelvin.

To find the number of moles of each gas in the plasma cell, we rearrange the formula to solve for \( n \): \[ n = \frac{PV}{RT} \]The partial pressure and converted volume are plugged into this equation, along with the standard constant \( R \) and assumed temperature \( T = 298 \, \text{K} \). This calculation gives the moles of each gas, an important step in determining the actual quantity in atom counts.

Gas Mixture Ratio

In this exercise, understanding the gas mixture ratio is key to solving the problem. A gas mixture ratio tells us the proportions of each component in the mixture. In a typical 1:1:1 ratio, each gas would contribute equally unless specified otherwise.
In the current plasma cell example, the gases Xe, Ne, and He are assumed to have equal molar amounts in the mixture, as indicated by the 1:1:1 ratio, but we know Xe represents only 4% of the total mixture. Thus, the rest (96%) splits equally between Ne and He. This makes each Ne and He 46% contributors to the total partial pressure.
Understanding the mixture's ratio allows one to divide the partial pressures appropriately among the gases. This step is crucial for further calculations, such as determining the number of moles and eventually the number of atoms for each gas in the sample.