Problem 135 - A Complete Resource Book in Chemistry for JEE Main

Question

A man weigh \(72.15 \mathrm{~kg}\) and want to fly in the sky with the aid of balloons itself weighing \(20 \mathrm{~kg}\) and each containing 50 moles of \(\mathrm{H}_{2}\) gas at \(0.05 \mathrm{~atm}\) and \(27^{\circ} \mathrm{C}\). If the density of air at the given conditions is \(1.25 \mathrm{~g} / \mathrm{L}\), how many such types of balloons he is needed to fly in the sky.

Step-by-Step Solution

Answer

The man needs 4 balloons to fly.

Step 1: Calculate the total weight to lift

The total weight to lift includes the man's weight and the weight of the balloons. Thus, the total weight is \(72.15 + 20 = 92.15 \text{ kg} \) or \(92150 \text{ g} \).

Step 2: Determine the volume of one balloon

Using the ideal gas law \( PV = nRT \), where \( P = 0.05 \text{ atm} \), \( n = 50 \text{ moles} \), \( R = 0.0821 \text{ L atm/mol K} \), and \( T = 300 \text{ K} \), we find the volume \( V \) of one balloon:\[ V = \frac{nRT}{P} = \frac{50 \times 0.0821 \times 300}{0.05} = 24630 \text{ L} \].

Step 3: Calculate the buoyant force of one balloon

The buoyant force (lift) is equal to the weight of the air displaced by the balloon. The weight of the air displaced by one balloon is:\[ 24630 \text{ L} \times 1.25 \text{ g/L} = 30787.5 \text{ g} \].

Step 4: Determine the net lift force of one balloon

Calculate the net lift force by subtracting the balloon's weight from the weight of displaced air:\[ 30787.5 \text{ g} - (50 \text{ moles} \times 2 \text{ g/mol}) = 30787.5 - 100 = 30687.5 \text{ g} \].

Step 5: Calculate the number of balloons needed

Divide the total weight to lift by the net lift force of one balloon:\[ \frac{92150}{30687.5} \approx 3.00 \]. Therefore, he needs 4 balloons, as the calculation must be rounded up to ensure lift.

Key Concepts

Buoyant Force

The buoyant force is a concept from physics that explains why objects float or sink in a fluid, such as air or water. In this situation, balloons are filled with hydrogen gas to lift a man into the sky.

When a balloon is filled with a gas lighter than air, like hydrogen, it displaces a volume of air. According to Archimedes' principle, the buoyant force on the balloon is equal to the weight of the air displaced. To find this, multiply the volume of air displaced by its density:

• For one balloon, the volume displaced is 24630 L.
• The density of air is 1.25 g/L.

So, the weight of air displaced by one balloon is 30787.5 g. This is the buoyant force exerted by the air on the balloon.

The key idea is that if the buoyant force is greater than the weight of the object (here, the balloons plus the man), it will rise. If less, it will fall. Therefore, by ensuring enough balloons are used, each contributing to the total buoyant force, it is possible to lift the man.

Mole Concept

The mole is a standard unit in chemistry used to express amounts of a chemical substance. It lets us count particles by weighing them and is central to calculations involving chemical reactions and gases.

Each balloon in this problem contains 50 moles of hydrogen gas. To find out the weight of hydrogen gas in a balloon:

• The molar mass of hydrogen (H) is approximately 2 g/mol.
• Therefore, 50 moles of hydrogen weigh 100 g (50 moles x 2 g/mol).

This weight of hydrogen must be considered when calculating the net lift of the balloons.

The ideal gas law (PV = nRT) is used to find the volume each mole occupies under specific conditions. For this exercise, it helps determine the volume of the balloon, which is essential for calculating the buoyant force. Mole concept not only helps in gas calculations but is vital for understanding reactions and compositions in chemistry.

Density of Air

Density is a measure of how much mass is contained in a given volume, and for gases like air, it varies with temperature and pressure. In the given situation, the density of air is 1.25 g/L, which helps determine how much lift a balloon can provide.

Why is air density important? When calculating buoyant force and lifting capacity:

• High density means more mass in the same volume, increasing the buoyant force.
• Knowing the air density allows us to calculate how much mass each cubic meter of air can support.

For our calculation, this means:

• Knowing the density allows us to convert the volume of displaced air into mass (i.e., buoyant force).

The conditions given (e.g., air density at specific pressure and temperature) dictate the amount of air displaced by the balloons, crucial for calculating the lift needed to make the man airborne. Understanding density of air not only aids in this specific problem but also helps in various scientific and practical applications involving gases.