Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

Four rubber tubes are respectively filled with \(\mathrm{H}_{2}, \mathrm{He}\), \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2} .\) Which tube will be reinflated first? (a) \(\mathrm{H}_{2}\) filled tube (b) \(\mathrm{N}_{2}\) filled tube (c) He filled tube (d) \(\mathrm{O}_{2}\) filled tube

Step-by-Step Solution

Answer

(a) \( \mathrm{H}_2 \) filled tube will be reinflated first.

Step 1: Understanding the Problem

We need to identify which gas will diffuse out of its respective rubber tube the fastest. This will determine which tube will need to be reinflated first.

Step 2: Applying Graham's Law

Graham's law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass: \( \text{Rate} \propto \frac{1}{\sqrt{M}} \), where \( M \) is the molar mass of the gas.

Step 3: Calculating Molar Masses

Let's calculate the molar masses of the gases: - For \( \mathrm{H}_2 \), \( M = 2 \ g/mol \) - For \( \mathrm{He} \), \( M = 4 \ g/mol \) - For \( \mathrm{N}_2 \), \( M = 28 \ g/mol \) - For \( \mathrm{O}_2 \), \( M = 32 \ g/mol \)

Step 4: Comparing Rates of Effusion

Using the calculated molar masses, we see that the lighter the gas, according to Graham's Law, the faster it will diffuse. \( \mathrm{H}_2 \) has the lowest molar mass, and thus will diffuse the fastest.

Step 5: Conclusion

Since \( \mathrm{H}_2 \) will diffuse out of the tube the fastest due to its low molar mass, the \( \mathrm{H}_2 \)-filled tube will need to be reinflated first.

Key Concepts

Gas Diffusion

Gas diffusion is the process by which gas molecules spread out in response to a concentration gradient. Imagine opening a bottle of perfume in one corner of a room; eventually, you will smell it on the other side. This is because gas molecules are in constant motion and will move from areas of higher concentration to areas of lower concentration.
In the context of the exercise, gas diffusion helps us understand how gases escape from the rubber tubes. Each gas moves from inside the tube (higher concentration) to the outside air (lower concentration). The rate at which this occurs is influenced by the gas's properties, specifically its molar mass.

Molar Mass Calculation

Molar mass is a measure of the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all the atoms in a molecule. Knowing the molar mass of a gas is essential when applying Graham's Law of Effusion.
Here's a quick breakdown for the gases in the exercise:

• Hydrogen (  _2 ) has a molar mass of 2 g/mol.
• Helium (  e ) has a molar mass of 4 g/mol.
• Nitrogen (  _2 ) has a molar mass of 28 g/mol.
• Oxygen (  _2 ) has a molar mass of 32 g/mol.

The calculation of these molar masses involves looking up the atomic mass of each element in the periodic table and multiplying it by the number of atoms in the molecule. This information is vital for understanding how quickly a gas will effuse.

Effusion Rate

Effusion is the process where gas escapes through a small hole or pores in a container. Graham's Law of Effusion explains that the rate of effusion is inversely proportional to the square root of the molar mass of the gas. In mathematical terms: \[ \text{Rate} \propto \frac{1}{\sqrt{M}} \]This means, the lighter a molecule (lower molar mass), the faster it escapes through a small opening.
Applying this concept to our exercise, we will see that hydrogen (  _2 ), with its low molar mass, effuses the fastest compared to the other gases. As a result, the hydrogen-filled tube will lose its gas contents more quickly than the others, needing to be reinflated first.

Chemical Properties of Gases

Gases exhibit unique chemical properties that influence their behaviors like diffusion and effusion. The kinetic molecular theory helps explain these behaviors, noting that gas particles are in constant and random motion, colliding with each other and the walls of their container.
Understanding these properties is crucial when studying gases:

• Gas particles move faster when they have less mass.
• Collisions between gas particles are perfectly elastic, meaning they don't lose energy.
• The capacity of a gas to fill the shape and volume of its container allows it to diffuse rapidly and evenly when unconfined.
• Temperature influences the speed of gas particles and, consequently, their diffusion rates.

These properties combine to affect how quickly gases like hydrogen, helium, nitrogen, and oxygen will diffuse out of their respective tubes, ultimately determined by their individual molar masses and characteristics.