Question

The lunar surface reaches \(370 \mathrm{~K}\) at midday. The atmosphere consists of neon, argon, and helium at a total pressure of only \(2 \times 10^{-14} \mathrm{~atm} .\) Calculate the \(\mathrm{rm}\) speed of each component in the lunar atmosphere. [Use \(R=8.314 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})\) and express \(\mathscr{A}\) in \(\mathrm{kg} / \mathrm{mol} .]\)

Short answer

Neon: 674.45 m/s, Argon: 480.75 m/s, Helium: 1519.26 m/s.

Step-by-step solution

Identify the Required Formula

To calculate the root mean square (rms) speed of a gas, use the formula: \[ v_{rms} = \sqrt{ \frac{3RT}{M} } \] where \( R \) is the gas constant (8.314 J/(mol·K)), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kg/mol.

Convert Molar Mass to kg/mol

The molar masses of the gases are: - Neon (Ne): 20.18 g/mol - Argon (Ar): 39.95 g/mol - Helium (He): 4.00 g/mol Convert these to kg/mol by dividing each by 1000: - Neon: \( M = 0.02018 \ \text{kg/mol} \) - Argon: \( M = 0.03995 \text{kg/mol} \) - Helium: \( M = 0.004 \text{kg/mol} \)

Calculate the rms Speed for Neon

For Neon (Ne): \[ v_{rms} = \sqrt{ \frac{3 \times 8.314 \times 370}{0.02018} } \] Compute the value inside the square root first: \[ \frac{3 \times 8.314 \times 370}{0.02018} = 455047.142 \] Then take the square root: \[ v_{rms} = \sqrt{455047.142} = 674.45 \ \text{m/s} \]

Calculate the rms Speed for Argon

For Argon (Ar): \[ v_{rms} = \sqrt{ \frac{3 \times 8.314 \times 370}{0.03995} } \] Compute the value inside the square root first: \[ \frac{3 \times 8.314 \times 370}{0.03995} = 231097.2 \] Then take the square root: \[ v_{rms} = \sqrt{231097.2} = 480.75 \ \text{m/s} \]

Calculate the rms Speed for Helium

For Helium (He): \[ v_{rms} = \sqrt{ \frac{3 \times 8.314 \times 370}{0.004} } \] Compute the value inside the square root first: \[ \frac{3 \times 8.314 \times 370}{0.004} = 2308152.5 \] Then take the square root: \[ v_{rms} = \sqrt{2308152.5} = 1519.26 \ \text{m/s} \]

Key concepts

Molar Mass Conversion

Understanding molar mass conversion is crucial when dealing with gas calculations. Molar mass is typically given in grams per mole (g/mol). However, for certain formulas, including the root mean square speed (rms speed) of gases, molar mass must be in kilograms per mole (kg/mol). This is because the gas constant, R, is often used in units that require kg/mol for correct dimensionality.
To convert molar mass from g/mol to kg/mol, simply divide by 1000. For instance:

• Neon (Ne) has a molar mass of 20.18 g/mol, which converts to 0.02018 kg/mol.
• Argon (Ar), with a molar mass of 39.95 g/mol, converts to 0.03995 kg/mol.
• Helium (He) at 4.00 g/mol converts to 0.004 kg/mol.

This straightforward conversion ensures accuracy in subsequent calculations.

Gas Constant

The gas constant (R) is an essential component in gas law equations. Its value is 8.314 J/(mol·K), which stands for joules per mole per kelvin. This constant relates the energy scale to the temperature scale when dealing with gases.
Joules (J) signify the unit of energy, while mole (mol) is a common unit for the amount of substance, and Kelvin (K) represents the temperature scale. Thus, the gas constant bridges these units, enabling calculations of properties such as pressure, volume, and, notably, the rms speed of gas molecules. Remembering this constant and how to use it correctly is key to solving gas-related problems efficiently.

Temperature in Kelvin

Temperature is a critical factor when calculating the rms speed of gases. The temperature must be in Kelvin (K) because the Kelvin scale is absolute and aligns with the gas constant's units (J/(mol·K)). The Kelvin scale starts at absolute zero, the point at which all molecular motion stops.
For instance, the lunar surface temperature at midday is given as 370 K. It's important to ensure temperatures are in Kelvin when substituting them into formulas, especially for rms speed calculations:
\(v_{rms} = \sqrt{ \frac{3RT}{M} } \)
By using Kelvin, you ensure the accuracy and consistency of your calculations, aligning with the units of R and keeping the mathematical relationships intact.

rms speed calculation

The root mean square (rms) speed of gas molecules is a measure of their average speed at a given temperature. Calculate the rms speed using the formula:
\[ v_{rms} = \sqrt{ \frac{3RT}{M} } \]
Here, R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and M is the molar mass in kg/mol.
Follow these steps to calculate the rms speed for different gases:

• Neon's rms speed: \[ v_{rms, Ne} = \sqrt{ \frac{3 \cdot 8.314 \cdot 370}{0.02018} } = 674.45 \text{m/s} \]
• Argon's rms speed: \[\text{v_{rms, Ar}} = \sqrt{ \frac{3 \cdot 8.314 \cdot 370}{0.03995} } = 480.75 \text{m/s} \]
• Helium's rms speed: \[\text{v_{rms, He}} = \sqrt{ \frac{3 \cdot 8.314 \cdot 370}{0.004} } = 1519.26 \text{m/s} \]

Substitute the correct values, carefully perform each calculation, and you'll find the average speed of gas molecules. This method is essential for predicting gas behavior under different conditions.