What will be the passenger carrying capacity of a balloon of diameter 20 m and mass 100 kg filled with helium at 1.0 atm at `27^(@)C`. Density of air is `1.2 "kg"//m^(3)` and average weight of a passenger is 65 kg.

Options

A. 55
B. 60
C. 65
D. 70

Explanation

To solve the problem of determining the passenger carrying capacity of a balloon, we will follow these steps:

### Step 1: Calculate the Volume of the Balloon
The volume \( V \) of a sphere (balloon) can be calculated using the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Given the diameter of the balloon is 20 m, the radius \( r \) will be:
\[ r = \frac{20}{2} = 10 \, \text{m} \]
Now substituting the value of \( r \) into the volume formula:
\[ V = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi (1000) = \frac{4000}{3} \pi \approx 4188.79 \, \text{m}^3 \]

### Step 2: Calculate the Mass of Air Displaced
The mass of the air displaced can be calculated using the formula:
\[ \text{Mass of air displaced} = \text{Volume} \times \text{Density of air} \]
Given the density of air is \( 1.2 \, \text{kg/m}^3 \):
\[ \text{Mass of air displaced} = 4188.79 \, \text{m}^3 \times 1.2 \, \text{kg/m}^3 \approx 5026.55 \, \text{kg} \]

### Step 3: Calculate the Mass of Helium in the Balloon
Using the Ideal Gas Law, we can calculate the number of moles of helium in the balloon:
\[ PV = nRT \]
Where:
- \( P = 1.0 \, \text{atm} = 101325 \, \text{Pa} \)
- \( V = 4188.79 \, \text{m}^3 \)
- \( R = 0.0821 \, \text{L atm/(K mol)} = 8.314 \, \text{J/(K mol)} \) (convert to appropriate units)
- \( T = 27^\circ C = 300 \, \text{K} \)

First, convert volume to liters:
\[ V = 4188.79 \, \text{m}^3 = 4188790 \, \text{L} \]
Now, substituting into the ideal gas equation:
\[ n = \frac{PV}{RT} = \frac{(1 \, \text{atm})(4188790 \, \text{L})}{(0.0821 \, \text{L atm/(K mol)})(300 \, \text{K})} \approx 170344 \, \text{mol} \]

The molar mass of helium is approximately \( 4 \, \text{g/mol} \):
\[ \text{Mass of helium} = n \times \text{molar mass} = 170344 \, \text{mol} \times 0.004 \, \text{kg/mol} \approx 681.38 \, \text{kg} \]

### Step 4: Calculate the Total Mass of the Balloon
The total mass of the balloon filled with helium is:
\[ \text{Total mass} = \text{mass of helium} + \text{mass of balloon} = 681.38 \, \text{kg} + 100 \, \text{kg} = 781.38 \, \text{kg} \]

### Step 5: Calculate the Payload Capacity
The payload capacity is the difference between the mass of air displaced and the total mass of the balloon:
\[ \text{Payload} = \text{mass of air displaced} - \text{total mass of balloon} = 5026.55 \, \text{kg} - 781.38 \, \text{kg} \approx 4245.17 \, \text{kg} \]

### Step 6: Calculate the Number of Passengers
Given the average weight of a passenger is 65 kg, the number of passengers that can be carried is:
\[ \text{Number of passengers} = \frac{\text{Payload}}{\text{average weight of passenger}} = \frac{4245.17 \, \text{kg}}{65 \, \text{kg}} \approx 65.00 \]

### Final Answer
The passenger carrying capacity of the balloon is approximately **65 passengers**.

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