Problem 45 - Physics

Question

In a mixture of argon (atomic mass \(=39.9 \mathrm{u}\) ) and neon (atomic mass \(=20.2 \mathrm{u}\) ), the speed of sound is \(363 \mathrm{~m} / \mathrm{s}\) at \(3.00 \times 10^{2} \mathrm{~K}\). Assume that both monatomic gases behave as ideal gases. Find the percentage of the atoms that are argon and the percentage that are neon.

Step-by-Step Solution

Answer

Argon: 31.4%, Neon: 68.6%

Step 1: Understand the Context

A mixture of argon and neon gases transmits sound at a certain speed. Each gas behaves ideally, and we need to find out what fraction, in percent, of the total mixture each gas constitutes; specifically, how much is argon and how much is neon.

Step 2: Use the Speed of Sound Formula for Gases

The speed of sound in an ideal gas is given by the formula \( v = \sqrt{\frac{\gamma RT}{M}} \), where \( v \) is the speed of sound, \( \gamma \) is the adiabatic index (1.67 for monatomic gases), \( R \) is the gas constant (8.314 J/mol·K), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the mixture.

Step 3: Express Molar Mass of Mixture

Since we are dealing with a mixture, the molar mass \( M \) is the weighted average based on the fractions of each gas: \( M = x \times 39.9 + (1-x) \times 20.2 \), where \( x \) is the mole fraction of argon.

Step 4: Solve for Molar Mass Using Given Speed

Given \( v = 363 \text{ m/s} \) and \( T = 300 \text{ K} \), we set up the equation: \( 363 = \sqrt{\frac{1.67 \times 8.314 \times 300}{M}} \). Simplifying this gives us: \( M = \frac{1.67 \times 8.314 \times 300}{363^2} \approx 26.4 \text{ g/mol} \).

Step 5: Calculate the Mole Fraction of Argon

Substitute \( M = 26.4 \) into the weighted molar mass equation: \( 26.4 = x \times 39.9 + (1-x) \times 20.2 \). Solve the equation to find \( x \).

Step 6: Solve for x

Rearrange and solve for \( x \): \[ 26.4 = 39.9x + 20.2 - 20.2x \]\[ 26.4 = 39.9x - 20.2x + 20.2 \]\[ 26.4 - 20.2 = 19.7x \]\[ 6.2 = 19.7x \]\[ x = \frac{6.2}{19.7} \approx 0.314 \]

Step 7: Calculate Percentage Composition

The percentage of argon atoms is \( x \) as a percentage: \( 0.314 \times 100 \approx 31.4\% \).The percentage of neon atoms is \( 100\% - 31.4\% = 68.6\% \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of ideal gases. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant (8.314 J/mol·K)
• \( T \) is the temperature in Kelvin

This law helps us understand how gases behave under various conditions of pressure, volume, and temperature. In our context, both argon and neon gases are considered ideal, meaning they perfectly obey this law.
This enables us to use related equations, like the speed of sound formula for gases, which depends on the temperature and molar mass of the gas mixture.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, usually given in grams per mole (g/mol). To find the molar mass of a gas mixture, like our argon and neon mixture, we use the weighted average of the molar masses of each component gas.
The formula is:

• \( M = x \times M_{\text{Ar}} + (1-x) \times M_{\text{Ne}} \)

where \( x \) is the mole fraction of argon, \( M_{\text{Ar}} \) is the molar mass of argon (39.9 g/mol), and \( M_{\text{Ne}} \) is the molar mass of neon (20.2 g/mol).
This formula helps us determine how the presence of different gases affects the overall properties of the mixture, like the speed of sound.

Mole Fraction

The mole fraction is a way of expressing the composition of a gas mixture. It's defined as the ratio of the number of moles of one component to the total number of moles of all components. In our scenario:

• Let's say \( x \) is the mole fraction of argon.
• This implies neon has a mole fraction of \( 1-x \).

Mole fractions are dimensionless numbers that help us understand the proportion of each gas in the mixture.
For our calculation, we solved for the mole fraction of argon to find it was approximately 0.314, meaning about 31.4% of the mixture is argon by mole.

Gas Mixtures

Gas mixtures are combinations of different gases that exist in the same space. The properties of a gas mixture, such as pressure, volume, temperature, and speed of sound, depend on the properties of its individual components and their relative amounts.

• The speed of sound in a gas mixture is influenced by the molar mass.
• Individual gases contribute to the overall behavior based on their mole fractions.

In the exercise, a mixture of argon and neon behaves as an ideal gas, allowing us to use equations that consider the average molecular weight (or molar mass) and the uniform temperature.
Understanding the composition of a gas mixture helps in predicting how the mixture will behave under given conditions.