Problem 48 - Chemistry and Chemical Reactivity

Question

Argon gas is 10 times denser than helium gas at the same temperature and pressure. Which gas is predicted to effuse faster? How much faster?

Step-by-Step Solution

Answer

Helium effuses 3.16 times faster than argon.

Step 1: Understanding Effusion

Effusion is defined as the process through which a gas escapes through a tiny hole into a vacuum. According to Graham's law, the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

Step 2: Graham's Law Formula

Graham's law is mathematically represented by the formula: \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]where \(r_1\) and \(r_2\) are the rates of effusion of gas 1 and gas 2 and \(M_1\) and \(M_2\) are their molar masses.

Step 3: Densities and Molecular Weights

It's given that argon is 10 times denser than helium at the same temperature and pressure, which means: \[ \frac{d_{\text{Argon}}}{d_{\text{Helium}}} = 10 \]Since density \(d\) is directly proportional to molar mass \(M\):\[ \frac{M_{\text{Argon}}}{M_{\text{Helium}}} = 10 \]

Step 4: Calculating Effusion Rate Ratio

Apply the density ratio to Graham's law:\[ \frac{r_{\text{Helium}}}{r_{\text{Argon}}} = \sqrt{\frac{M_{\text{Argon}}}{M_{\text{Helium}}}} \]\[ \frac{r_{\text{Helium}}}{r_{\text{Argon}}} = \sqrt{10} \approx 3.16 \]

Step 5: Conclusion

Helium gas effuses faster than argon gas. Based on the calculations, helium effuses approximately 3.16 times faster than argon.

Key Concepts

Effusion

Effusion is a fascinating process in the world of gases. It occurs when gas particles escape through a tiny opening into a vacuum. This movement is spontaneous and depends on the speed at which the particles are moving.

The particles of lighter gases move faster, aiding the process of effusion. This is where Graham's Law of Effusion comes into play. The law states that the rate of effusion is inversely proportional to the square root of the molar mass of the gas.

• The smaller the molar mass, the faster the rate of effusion.
• Lighter gases effuse quicker than heavier gases.

In the case of argon and helium, helium effuses faster because its particles are lighter and move more quickly than those of argon.

Molar Mass

Molar mass is a crucial concept when discussing gases and their behaviors. It refers to the mass of one mole of a substance, typically measured in grams per mole (g/mol).

Understanding molar mass helps us predict how different gases will behave, especially in processes like effusion. According to Graham's Law, the rate of effusion is determined by the molar mass of the gas.
When comparing argon and helium:

• Helium has a molar mass of about 4 g/mol.
• Argon has a molar mass of about 40 g/mol.

This large difference means helium effuses faster since its molar mass is lower, leading to a higher rate of effusion.

Density of Gases

Density is a term used to describe how much mass a substance has in a given volume. In gases, density can greatly influence how a gas behaves in various processes.

The density of a gas is directly related to its molar mass and the conditions of temperature and pressure. For two gases at the same temperature and pressure, a heavier gas (with a higher molar mass) will have more density.

• Argon being 10 times denser than helium implies it has a higher molar mass.
• Greater density means argon's particles move slower than helium's, affecting effusion rate.

In practical terms, because argon is denser, it effuses slower compared to helium, aligning with the predictions from Graham's Law.

Rate of Effusion

The rate of effusion is a measure of how quickly a gas can escape through a small hole. This rate is critical when comparing different gases and is influenced by the gas's molar mass.

Using Graham's Law, one can determine the relative rates of effusion for two gases:

• The formula to use is: \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]
• This shows that a lesser molar mass increases the rate of effusion.

For helium and argon, helium effuses approximately 3.16 times faster than argon. This is due to helium's significantly lower molar mass, allowing its particles to escape more rapidly through a tiny opening. This makes helium a lighter, faster-effusing gas compared to argon.