Problem 66 - Chemistry: The Central Science in SI Units, Global Edition

Question

A plasma-screen TV contains thousands of tiny cells filled 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

The number of Xe, Ne, and He atoms in the given plasma cell can be calculated using the Ideal Gas Law and partial pressures. First, calculate the volume of the cell: \(V = 2.7 \times 10^{-7} m^3\). Next, find the partial pressures for each gas: \(P_{Xe} = 2.6664 \mathrm{kPa}\), \(P_{Ne} = 32.0008 \mathrm{kPa}\), and \(P_{He} = 32.0008 \mathrm{kPa}\). Then, calculate the number of moles (n) for each gas using the Ideal Gas Law, assuming temperature remains constant throughout the calculation. Finally, use Avogadro's number to convert the number of moles to the number of atoms for each gas. The assumptions made include considering the gases as ideal gases and that the temperature of the gas mixture remains constant.

Step 1: Calculate the volume of the cell

Given dimensions are \(0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm}\). We need to calculate the volume (V) and convert it into the SI unit, which is cubic meters: V = Length × Width × Height V = (0.900 × 10^{-3} m) × (0.300 × 10^{-3} m) × (10 × 10^{-3} m) V = 2.7 × 10^{-7} m^3

Step 2: Calculate the partial pressures

We are given that the total pressure of the gas mixture is \(66.66 \mathrm{kPa}\). In order to calculate the partial pressures of the individual gases, we need to first determine their percentage share in the total pressure: - Xe: 4% of total pressure - Ne: 48% of total pressure (since it's 1:1 ratio with He, and the remaining pressure is split between Ne and He) - He: 48% of total pressure Now, we can calculate the partial pressures for each gas: P_Xe = 0.04 × 66.66 kPa = 2.6664 kPa P_Ne = 0.48 × 66.66 kPa = 32.0008 kPa P_He = 0.48 × 66.66 kPa = 32.0008 kPa

Step 3: Calculate the number of moles

We can now use the ideal gas law to calculate the number of moles (n) for each gas. Since we don't have the temperature (T) given, we'll assume that temperature remains constant throughout the calculation. PV = nRT, solving for n: n = PV / RT We need to use the gas constant R in the appropriate units, which is \(8.314 \mathrm{J/(mol\cdot K)}\). n_Xe = (2.6664 × 10^3 Pa)(2.7 × 10^{-7} m^3) / (8.314 J/(mol·K) × T) n_Ne = (32.0008 × 10^3 Pa)(2.7 × 10^{-7} m^3) / (8.314 J/(mol·K) × T) n_He = (32.0008 × 10^3 Pa)(2.7 × 10^{-7} m^3) / (8.314 J/(mol·K) × T) We can see that temperature (T) gets cancelled out, so we don't need its value.

Step 4: Calculate the number of atoms

Now, to convert the number of moles to the number of atoms, we can use Avogadro's number (\(N_A = 6.022\times 10^{23}\) atoms/mole): Number of Xe atoms = n_Xe × \(N_A\) Number of Ne atoms = n_Ne × \(N_A\) Number of He atoms = n_He × \(N_A\) With these equations, you can now calculate the number of Xe, Ne, and He atoms in the cell.

Key Concepts

Partial Pressure

When dealing with a mixture of gases, partial pressure is an essential concept. It represents the pressure that each gas in a mixture would exert if it occupied the entire volume on its own. In essence, even when gases are mixed in a container, each behaves as if it is alone within that space.

The total pressure of a gas mixture is the sum of the partial pressures of each individual gas within it. For example, in a plasma TV cell that contains a mixture of Xe, Ne, and He gases, the total pressure is 66.66 kPa. To find the partial pressure for each gas, we apply their percentage composition to the total pressure.

• Xe, being 4% of the mix, contributes a partial pressure of \(2.6664\) kPa.
• Ne and He, being in a 1:1 ratio, each exert a partial pressure of \(32.0008\) kPa - making up the remaining pressure after accounting for Xe.

This breakdown helps in calculating other important parameters, like the number of moles and atoms in the mixtures.

Moles of Gas

The concept of moles is critical when we want to quantify the amount of gas present in a particular volume at a specific pressure and temperature. A mole is essentially a very large number of entities, such as atoms or molecules, specifically Avogadro's number, which is \(6.022 \times 10^{23}\).

To calculate the number of moles in a gas, we use the Ideal Gas Law, expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. By rearranging this formula, the number of moles can be expressed as \(n = \frac{PV}{RT}\).

• Using the calculated partial pressure for Xe, Ne, and He and the known volume of the plasma cell, we can determine the number of moles for each gas.
• Even without a specific temperature, the calculation simplifies since it cancels out across the gases.

This allows us to understand how many moles of each type of gas are in the cell, which is pivotal for further calculations.

Volume Calculation

Volume calculation is the first step in many gas-related problems as it sets the stage for subsequent calculations involving gases. The volume of the plasma cell needs to be calculated from its given dimensions.

For our problem, the dimensions of the cell are \(0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm}\). To find the volume in cubic meters, we multiply these values while converting each dimension from millimeters to meters by dividing by 1000. The calculated volume is \(2.7 \times 10^{-7} \mathrm{~m}^3\).

• This volume is crucial as it directly influences the calculated moles of gas via the Ideal Gas Law.
• Accurate volume measurements ensure the reliability of gas quantity estimations in any application, from scientific experiments to industrial processes.

Understanding how to calculate and convert volume correctly is fundamental when working with the Ideal Gas Law and analyzing gas mixtures.