Problem 63 - Chemistry The Molecular Science

Question

If equal amounts of the four inert gases \(\mathrm{Ar}, \mathrm{Ne}, \mathrm{Kr},\) and Xe are released at the same time at one end of a long, evacuated tube, which gas will reach the other end of the tube first? Explain your answer.

Step-by-Step Solution

Answer

Neon (Ne) will reach the other end of the tube first due to its lowest molar mass, resulting in the highest molecular speed.

Step 1: Understand Molecular Speed

The speed of a gas is determined by the root mean square (RMS) velocity, which depends inversely on the square root of the molecular mass. A lighter gas will have a higher molecular speed.

Step 2: Determine the Molar Masses

Find the approximate molar masses of the gases: Argon (Ar) = 40 g/mol, Neon (Ne) = 20 g/mol, Krypton (Kr) = 84 g/mol, and Xenon (Xe) = 131 g/mol. Neon, being the lightest gas, has the smallest molar mass.

Step 3: Apply Graham's Law

According to Graham's Law, the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass. Since Neon has the lowest molar mass, it will diffuse the fastest.

Step 4: Conclusion Based on Calculations

Since Neon's speed is the greatest due to its lowest molar mass, Neon will reach the other end of the tube before the other gases.

Key Concepts

Root Mean Square Velocity

The root mean square velocity (RMS velocity) is a key concept in understanding how gas molecules move. It is a measure of the average speed of gas particles. The RMS velocity links the microscopic world of gas particles to macroscopic properties like temperature.
The formula for RMS velocity is given by:\[ v_{ rms } = \sqrt{\frac{3RT}{M}} \] Where:

• \( v_{ rms } \) is the root mean square velocity.
• \( R \) is the ideal gas constant (8.314 J/mol·K).
• \( T \) is the temperature in Kelvin.
• \( M \) is the molar mass of the gas in kilograms per mole.

This formula shows that RMS velocity is dependent on both the temperature of the gas and its molar mass. Specifically, the RMS velocity is inversely proportional to the square root of the molar mass. This means that lighter molecules, which have a smaller molar mass, typically move faster compared to heavier ones at the same temperature. This relationship is crucial for predicting behaviors in various gas-related scenarios, such as diffusion and effusion.

Graham's Law

Graham's Law of Effusion provides insights into why different gases move at different rates. According to Graham's Law, the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, this is expressed as:\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]Where:

• \( r_1 \) and \( r_2 \) are the effusion rates of gas 1 and gas 2, respectively.
• \( M_1 \) and \( M_2 \) are the molar masses of gas 1 and gas 2, respectively.

This principle illustrates why lighter gases, such as Neon in the exercise, diffuse more rapidly than heavier gases. Since Neon has a relatively low molar mass when compared to Argon, Krypton, and Xenon, it effuses and diffuses more quickly through a medium.By understanding Graham's Law, one can predict which gas will move the fastest, especially in processes requiring separation or rapid mixing of gases. Knowing how each gas behaves allows for accurate predictions and the design of processes in chemistry and industry.

Molar Mass

Molar mass is a fundamental concept in the study of gases, determining key properties like speed and rate of diffusion or effusion. It is defined as the mass of one mole of a substance, and its unit is typically grams per mole (g/mol).
In the context of molecular speed and gases, the molar mass has a significant impact. According to both the RMS velocity formula and Graham's Law, the molar mass directly influences a gas's speed and effusion behavior. In our example, the molar masses of inert gases are crucial:

• Argon (Ar): 40 g/mol
• Neon (Ne): 20 g/mol
• Krypton (Kr): 84 g/mol
• Xenon (Xe): 131 g/mol

Since Neon has the smallest molar mass, it moves fastest among the gases in the evacuated tube scenario. Understanding how to determine molar masses and their effect on molecular speed is crucial for many applications in chemistry, from industrial processes to anticipating gas behavior in different environments.