Problem 1 - Chemistry Matter and Change

Question

Calculate the ratio of effusion rates for nitrogen \(\left(\mathrm{N}_{2}\right)\) and neon (Ne).

Step-by-Step Solution

Answer

The ratio of effusion rates for nitrogen to neon is approximately 0.845.

Step 1: Understand the Effusion Formula (Graham's Law)

Effusion rates can be calculated using Graham's Law, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The formula is: \( \text{Rate}_{1}/\text{Rate}_{2} = \sqrt{M_{2}/M_{1}} \) where \(M_1\) and \(M_2\) are the molar masses of gases 1 and 2, respectively.

Step 2: Identify Gases and Molar Masses

In this exercise, we need to compare nitrogen (N₂) and neon (Ne). The molar mass of N₂ is approximately 28 g/mol and for Ne, it is about 20 g/mol. These values will be used in the formula.

Step 3: Apply Graham's Law

Substitute the molar masses of nitrogen and neon into Graham's Law. This gives us \( \text{Rate}_{ ext{N}_2}/\text{Rate}_{ ext{Ne}} = \sqrt{M_{ ext{Ne}}/M_{ ext{N}_2}} \). This simplifies to \( \sqrt{20/28} \).

Step 4: Calculate the Ratio

Calculate the square root of the ratio of the molar masses: \( \sqrt{20/28} = \sqrt{5/7} \). Approximating, \( \sqrt{5/7} \approx 0.845 \). This means nitrogen effuses at a slower rate than neon.

Key Concepts

Effusion Rate

Effusion is the process by which gas particles escape through a tiny hole into a vacuum. Often, we are interested in comparing the rates at which different gases effuse. Graham's Law provides the formula to make these comparisons. The law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The equation is:

• \(\text{Rate}_{1}/\text{Rate}_{2} = \sqrt{M_{2}/M_{1}}\)

where \(M_1\) and \(M_2\) are the molar masses of the two gases being compared. This means that a lighter gas, with a lower molar mass, will effuse more quickly than a heavier one. By calculating the ratio of effusion rates, we can predict how one gas will behave compared to another under the same conditions.

Molar Mass

Molar mass is a fundamental property of a chemical substance, indicating the mass of one mole of its entities, typically expressed in g/mol. In our exercise, understanding the molar mass of gases helps us apply Graham's Law effectively.
To use Graham's Law, you need the molar mass of each gas:

• Nitrogen's molar mass: approximately 28 g/mol
• Neon's molar mass: approximately 20 g/mol

These values represent the average mass of a single molecule relative to 1/12th of the mass of a carbon-12 atom. Knowing these values allows you to calculate the effusion rates. More importantly, it helps illustrate the principle that lighter gases (like neon in this example) will effuse faster compared to heavier ones like nitrogen.

Nitrogen

Nitrogen is a chemical element with the symbol N and atomic number 7. It is most commonly found in its diatomic form, \(\text{N}_2\), which is the form relevant to our exercise. Nitrogen is found abundantly in the Earth's atmosphere, making up about 78% by volume.
In the context of effusion, nitrogen's molar mass is about 28 g/mol, which means it effuses slower than neon. Despite being heavier than neon, nitrogen is light enough to effuse through small openings but not as efficiently as a lighter gas would. Graham's Law allows us to quantify just how much more slowly nitrogen will effuse, helping us predict its behavior in various scientific and industrial contexts.

Neon

Neon is a noble gas with the symbol Ne and atomic number 10, widely recognized for its lack of reactivity and its use in lighting. Its molar mass is approximately 20 g/mol, making it lighter compared to many other gases, such as nitrogen.
Due to its lower molar mass, neon effuses more quickly than nitrogen when comparing equal conditions. This quality of being light and non-reactive makes it ideal for various applications, including neon signs that require the gas to move freely in a sealed tube. Under Graham's Law, neon's faster effusion rates make it a point of study in assessing the behavior of gases, as well as in practical solutions where speed of gas transfer is a key factor.