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

To find the number of balloons, calculate the volume each displaces using the ideal gas, then the lift force using displaced air mass minus the balloon's own gas mass, and divide the total weight by this lift per balloon.

Step 1: Calculate the Total Weight to be Lifted

First, determine the total weight that needs to be lifted by the balloons. This includes the weight of the man and the weight of the balloons themselves. Weight of man = \(72.15\, \mathrm{kg}\)Weight of one balloon = \(20\, \mathrm{kg}\)Convert balloon weight to the same unit as the total weight through addition, but since it's per balloon, and we're looking for the number, treat as one cycle.Thus, total weight if one cycle comprises man and one such loaded balloon is \(72.15 + 20 = 92.15\, \mathrm{kg}\). This weight needs to be multiplied by the number of balloons for a floating effect if one cannot lift it all without calculating buoyant specifics.

Step 2: Calculate Volume of One Balloon

We use the ideal gas law to find the volume of one balloon. The ideal gas law is given by \[PV = nRT\]Where:- \(P\) is pressure in atm- \(V\) is volume in liters- \(n\) is the number of moles of gas- \(R\) is the ideal gas constant, \(0.0821 \text{ L atm/mol K}\)- \(T\) is temperature in KelvinConvert temperature from Celsius to Kelvin: \[T = 27 + 273.15 = 300.15\, \mathrm{K}\]Given for one balloon:- \(n = 50\, \text{moles}\)- \(P = 0.05\, \text{atm}\)Plug into the equation to solve for \(V\):\[V = \frac{nRT}{P} = \frac{50 \times 0.0821 \times 300.15}{0.05}\] Calculate the numerical result to find volume \(V\).

Step 3: Determine the Lifting Force of One Balloon

The capacity of the balloon to lift is determined by the difference in the density between the air and the hydrogen gas, producing a buoyant force.Calculate the mass of air displaced by volume \(V\):Mass of air displaced = density of air \( \times \) volume of one balloon\[\text{Mass of air displaced} = 1.25\, \text{g/L} \times V = 1.25 \times V \text{ g (convert to kg)}\]This mass of air equates to the lifting force applied by one balloon because each gram of displaced air contributes to lift an equivalent mass.

Step 4: Calculate the Actual Lift from Hydrogen's Mass

The actual lift is reduced by the weight of the hydrogen gas inside each balloon. Calculate the mass of hydrogen in one balloon:The molar mass of hydrogen \((\text{H}_2)\) is approximately \(2\, \text{g/mol}\).Thus, \(50\, \text{moles} \times 2\, \text{g/mol} = 100\, \text{g}\) or \(0.1\, \text{kg}\) as the total weight of the gas in one balloon.Actual lift = Mass of air displaced (in kg) - Mass of hydrogen in a balloon = Total weight lifted by the balloon.

Step 5: Determine Number of Balloons Needed

Now, determine the number of balloons needed to lift the total weight calculated in Step 1 using the actual lift capacity of one balloon from Step 4.Let actual lift be \(X\, \mathrm{kg/balloon}\) from prior step:\[\text{Number of balloons} = \frac{\text{Total mass}}{X}\] This result will tell you how many balloons are necessary for the man to fly in the sky.

Key Concepts

Buoyant Force

Imagine trying to lift something heavy out of water, like a large rock submerged in a pool. It feels lighter. This is due to the buoyant force. Buoyant force is an upward force exerted by a fluid, like water or air, which opposes the weight of an object immersed in it. In our scenario of the man with balloons, this force is created by the difference in density between the lighter gas inside the balloon and the denser surrounding air.

This force allows objects to float or rise, like the balloons filled with hydrogen. This force is governed by Archimedes' Principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. This principle is why boats float or why the balloons can lift the man. Understanding this principle helps in determining how many balloons are needed for the lift-off. Without adequately displacing the air, the balloons wouldn't be able to rise.

• The buoyant force must overcome the gravitational force to lift the man.
• This requires calculating both the volume of air displaced and its weight compared to the contents of the balloon.

Mole Concept

The mole concept is a fundamental chemistry framework to assess quantities in chemical reactions and solutions. A mole is a unit used to measure the amount of a substance. It is based on Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles, atoms, or molecules. In this exercise, the mole concept helps us quantify the amount of hydrogen gas in each balloon.

Understanding moles allows us to convert between mass, number of particles, and volume. In our problem, each balloon contains 50 moles of hydrogen gas. Knowing this, we can find the mass using the molar mass of hydrogen gas, which is about 2 g/mol. Thus, the total weight of the hydrogen gas is critical in moving forward with determining the lifting capability of the balloons.

• 50 moles of hydrogen help calculate how much hydrogen is in each balloon.
• This aids in understanding the weight contribution of hydrogen to the overall buoyancy effects.

Gas Laws

Gas laws are essential in predicting how gases behave under various conditions of temperature, pressure, and volume. Here, we use the Ideal Gas Law, which connects these properties with the equation: \[PV = nRT\]
This equation allows us to calculate the volume of gas inside a single balloon by rearranging the formula to solve for \(V\) (volume), given the pressure \(P\), number of moles \(n\), Ideal Gas Constant \(R\), and temperature \(T\).

For the problem at hand, the application of the Ideal Gas Law helps determine the volume of hydrogen in the balloons, which is crucial for understanding their lifting power.

• The law allows us to find how much space the gas occupies.
• It connects temperature and pressure conditions with the amount of gas.

Density of Air

Density refers to the mass of a substance per unit volume. For gases, including air, density tells us how much mass is present in a certain volume. In this exercise, knowing the density of air is vital because it allows us to calculate the buoyant force exerted by the air displaced by the hydrogen gas within the balloons.

The given density of air is 1.25 g/L under the conditions specified in the problem. By determining the volume of the balloon using the ideal gas law, we can find out how much mass of air is displaced, which in turn is used to calculate the upward buoyant force exerted by the balloon. This is crucial for evaluating how many balloons are necessary to lift the total weight of the man and the balloons.

• This knowledge allows us to calculate the lifting force exerted by the displaced air.
• Understanding air density is fundamental to applying the Archimedes’ principle in gases.