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 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 number of Xe, Ne, and He atoms in the plasma cell can be calculated using the Ideal Gas Law and the given gas composition. First, convert the total pressure to atm (500 Torr = \( \frac{500}{760} \) atm) and find the partial pressures of the gases: Xe is 4% of total pressure and Ne and He are each 48% of total pressure. Next, calculate the volume of the plasma cell in liters using the given dimensions. Use the Ideal Gas Law to determine the moles of each gas assuming a constant temperature of 298 K, and then calculate the number of atoms for each gas using Avogadro's number. The assumptions made in this calculation include the applicability of the Ideal Gas Law, a constant temperature of 298 K, and the accuracy of the given gas composition.

Step-by-step solution

Convert Pressure to atm

The total pressure given in Torr; we need to convert it into the standard unit of pressure, atm. 1 atm = 760 Torr Total Pressure = 500 Torr = (500/760) atm

Find the Partial Pressures of the Gases

Given that the gas mixture is composed of 4% of Xe and 1:1 Ne:He, we need to find the partial pressures of each of the gases. Partial Pressure of Xe = 4% of Total Pressure = 0.04 * (500/760) atm Partial Pressure of Ne = Partial Pressure of He = 48% of Total Pressure = 0.48 * (500/760) atm

Calculate the Volume of the Plasma Cell

The dimensions of the cell are given as: \(0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm}\). We need to convert these dimensions into liters and then find the total volume of the plasma cell. Volume of Plasma Cell = Length x Width x Depth Volume = (0.900 mm x 0.300 mm x 10.0 mm) x \((\frac{1 \mathrm{~cm}}{10 \mathrm{~mm}})^3\) x \((\frac{1 \mathrm{~L}}{1000 \mathrm{~cm^3}})\)

Use the Ideal Gas Law to Determine the Moles of Each Gas

We can now use the Ideal Gas Law (PV = nRT, where n is the number of moles) to find the moles of each gas in the cell. The gas constant R is 0.0821 L atm/mol K. Assume the temperature in the cell is constant at 298 K (room temperature). Moles of Xe = \(\frac{\text{Partial Pressure of Xe}\times\text{Volume}}{R\times T}\) Moles of Ne = \(\frac{\text{Partial Pressure of Ne}\times\text{Volume}}{R\times T}\) Moles of He = \(\frac{\text{Partial Pressure of He}\times\text{Volume}}{R\times T}\)

Calculate the Number of Atoms of Each Gas

Now that we have the moles of each gas, we can calculate the number of atoms using Avogadro's number (6.022 x 10^23 atoms/mol). Number of Xe atoms = Moles of Xe x Avogadro's number Number of Ne atoms = Moles of Ne x Avogadro's number Number of He atoms = Moles of He x Avogadro's number

State the Assumptions

The assumptions made in the calculation are as follows: 1. The Ideal Gas Law is applicable to the gas mixture in the plasma cell. 2. The temperature of the cell is constant at 298 K (room temperature). 3. The given percentage composition of the gases in the cell is accurate. After performing these calculations, we will have the number of Xenon, Neon, and Helium atoms present in the plasma cell.

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial for working with gases in chemistry. This law relates the pressure (P), volume (V), and temperature (T) of a gas to the amount of substance present in moles (n) via the equation: \[ PV = nRT \] where R is the universal gas constant with a value of 0.0821 L atm/mol K. It is an approximation that applies best under low pressure and high temperature conditions.

In practical terms, the Ideal Gas Law allows us to predict how a gas will behave under different conditions. When solving problems involving gas mixtures, like in our plasma cell example, it tells us the relationship between the quantities of gas present and the space they occupy, considering they behave ideally. The calculations assume that gas particles act independently without interactions and that the volume of gas particles is negligible.

Partial Pressure

Partial pressure is a concept that describes the pressure exerted by a single type of gas in a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Mathematically, this can be expressed as: \[ P_{total} = P_{1} + P_{2} + P_{3} + ... \] where each \(P_i\) represents the partial pressure of a gas in the mixture.

In our plasma cell example, we calculated the partial pressures of Xenon (Xe), Neon (Ne), and Helium (He) based on their percentage contributions to the total pressure. This is important because knowing the partial pressures allows us to use the Ideal Gas Law to determine the amount (moles) of each gas separately.

Avogadro's Number

Avogadro's number, often denoted as \(N_A\), is a constant that represents the number of particles in one mole of substance and is equal to approximately \(6.022 \times 10^{23}\). This number is named after the scientist Amedeo Avogadro, and it provides the link between macroscopic quantities measured in the laboratory and the microscopic scale of atoms and molecules.

In gas law problems, after finding the amount of gas in moles, we use Avogadro's number to convert this amount to actual numbers of atoms or molecules. For example, if we have one mole of any gas, we know it contains about \(6.022 \times 10^{23}\) particles, whether they be atoms in an element or molecules in a compound. In the plasma-screen TV cells, we used this number to find out exactly how many atoms of Xe, Ne, and He are present.

Gas Mixture

A gas mixture comprises more than one type of gas combined together without reacting chemically, such as the Xe, Ne, and He mixture in the plasma cell. Gas mixtures are common in both natural and industrial environments, and their behavior can often be described by the same laws that apply to pure gases.

Key properties, such as pressure, volume, and temperature, still play significant roles. But when dealing with mixtures, we need to address the proportion of each gas present. Understanding partial pressures as explained allows us to dissect the gas mixture into components and analyze each separately, assuming the gases do not affect each other's behavior.

Stoichiometry

Stoichiometry is a section of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It's based on the conservation of mass and the concept of the mole. With stoichiometry, we can calculate quantities such as masses, volumes, and number of particles involved in chemical reactions.

In the context of gases, stoichiometry is applied together with gas laws to find out the relationships between different gases in a reaction. For example, if a gas is produced in a reaction, we can predict its volume at a certain temperature and pressure. This is directly relevant to our exercise, as we used stoichiometric calculations to determine the number of atoms of each gas present in the plasma cell, after first finding the moles using the Ideal Gas Law.