Problem 66 - Chemistry: The Central Science

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.

Step-by-Step Solution

Answer

The number of Xe, Ne, and He atoms in the plasma cell can be calculated by following these steps: 1. Convert the total pressure to atmospheres: \(P_{atm} \approx 0.6579 \mathrm{~atm}\). 2. Calculate the partial pressures of the gases: \(P_{Xe} = 0.04 \times P_{total}\), \(P_{Ne} = P_{He} = 0.5 \times (1 - 0.04) \times P_{total}\). 3. Calculate the volume of the plasma cell: \(V \approx 0.00270 \mathrm{~cm}^3\). 4. Use the Ideal Gas Law to find the amount of each gas in moles, assuming room temperature (298 K) and gas constant R to be 0.0821 L atm/mol K: \(n_{Xe} = \frac{P_{Xe} \times V}{R \times T}\), \(n_{Ne} = \frac{P_{Ne} \times V}{R \times T}\), \(n_{He} = \frac{P_{He} \times V}{R \times T}\). 5. Convert moles to the number of atoms using Avogadro's number, \(N_A = 6.022 \times 10^{23} \mathrm{atoms/mol}\): \(N_{Xe} = n_{Xe} \times N_A\), \(N_{Ne} = n_{Ne} \times N_A\), \(N_{He} = n_{He} \times N_A\). Assumptions made in the calculation include assuming the temperature to be room temperature, the Ideal Gas Law is applicable, and Ne and He have equal partial pressures.

Step 1: To use the Ideal Gas Law, we need the pressure in atmospheres. Given 500 torr, let's convert to atmospheres using the conversion factor: 1 atm = 760 torr. \(P_{atm} = \frac{500torr}{760torr/atm} \approx 0.6579 \mathrm{~atm}\) #Step 2: Calculate partial pressures of the gases#

Using the percentage of Xe and the 1:1 Ne:He mixture, we can calculate the partial pressure for each gas: \(P_{Xe} = 0.04 \times P_{total}\) \(P_{Ne} = P_{He} = 0.5 \times (1 - 0.04) \times P_{total}\) #Step 3: Calculate the volume of the plasma cell#

Step 2: We are given the dimensions of the plasma cell: \(0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm}\). Firstly, convert the dimensions to cm and then find the volume: \(V = (0.0900 \mathrm{~cm}) \times (0.0300 \mathrm{~cm}) \times (1.00 \mathrm{~cm}) \approx 0.00270 \mathrm{~cm}^3\) #Step 4: Use the Ideal Gas Law to find the amount of each gas in moles#

The Ideal Gas Law is given as \(PV = nRT\), where P is pressure, V is volume, n is the amount in moles, R is the gas constant, and T is temperature. Here, we assume the temperature to be room temperature (298 K) and the gas constant R to be 0.0821 L atm/mol K. For each gas, rearrange the Ideal Gas Law to find the amount in moles: \(n_{Xe} = \frac{P_{Xe} \times V}{R \times T}\) \(n_{Ne} = \frac{P_{Ne} \times V}{R \times T}\) \(n_{He} = \frac{P_{He} \times V}{R \times T}\) #Step 5: Convert moles to the number of atoms#

Step 3: To find the number of atoms for each gas, multiply the moles by Avogadro's number, \(N_A = 6.022 \times 10^{23} \mathrm{atoms/mol}\). \(N_{Xe} = n_{Xe} \times N_A\) \(N_{Ne} = n_{Ne} \times N_A\) \(N_{He} = n_{He} \times N_A\) #Step 6: Identify assumptions made in the calculation#

1. We assumed the temperature to be room temperature (298 K), as it was not provided in the given information. 2. We assumed the Ideal Gas Law is applicable to the given problem. 3. We assumed both Ne and He have equal partial pressures. Now, go ahead and plug in the values obtained in step 1 and step 3 to get the desired number of Xe, Ne, and He atoms in the plasma cell.

Key Concepts

Partial Pressure

When dealing with gas mixtures, like in the plasma-screen TV example, understanding partial pressure is crucial. The total pressure of a gas mixture is the sum of the partial pressures of each individual gas in the mixture. Partial pressure refers to the pressure exerted by a single type of gas within the mixture.

For Xenon (Xe), given that it makes up 4% of the total gas mixture, its partial pressure is calculated as 4% of the total pressure. Ne and He share the remaining 96% equally since they are in a 1:1 ratio. This means their partial pressures are 50% each of the remaining 96% of the total pressure.

Calculating partial pressures is essential as it allows us to apply the Ideal Gas Law to each gas individually, thereby helping to determine the number of moles, and eventually the number of atoms for each gas type in the plasma cell.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, crucial for converting between moles and number of atoms. It is defined as the number of constituent particles (usually atoms or molecules) in one mole of any substance and is equal to approximately \(6.022 \times 10^{23}\) particles per mole.

In the context of the plasma-screen TV problem, once we have calculated the moles of each gas in the plasma cell using the Ideal Gas Law, Avogadro's number is used to convert these moles into the actual number of gas atoms. This conversion is simple yet powerful:

• Multiply the number of moles by Avogadro's number to yield the total number of atoms.
• This helps bridge the microscopic world of atoms and molecules with the macroscopic quantities we can measure.

Understanding and utilizing Avogadro's number therefore allows chemists to better grasp the scale and proportions of reactions involving gases.

Room Temperature Assumption

The assumption of room temperature, typically about 298 K (25°C), is often used in gas law calculations when no specific temperature is provided. This simplification makes it feasible to calculate the behavior of gases under typical conditions without needing precise environmental data.

In this plasma cell problem, assuming room temperature is key when applying the Ideal Gas Law, as it influences the calculated volume and pressure relationship of the gasses.

• Assuming room temperature permits the use of conventional constants (e.g., the gas constant \(R\) which at room temperature is \(0.0821\) L atm/mol K).
• It provides a standardized method for comparison across different problems or scenarios.

This assumption simplifies complex computations and retains accuracy for physics students and chemists, making it an invaluable tool for introductory courses and practical applications alike.