A balloon of diameter 21 meter weight 100 kg. Calculate its pay-load, if it is filled with He at 1.0 atm and `27^(@)C`. Density of air is 1.2 kg `m^(-3)` . (Given : R=0.082 L atm `K^(-1)mol^(-1)`)

Options

A. 4952.42 kg
B. 4932.42 kg
C. 493.242 kg
D. none of these

Explanation

To solve the problem of calculating the payload of a helium-filled balloon, follow these steps:

### Step 1: Calculate the Volume of the Balloon
The balloon is spherical, and the volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the balloon. Given the diameter is 21 meters, the radius \( r \) is:
\[ r = \frac{21}{2} = 10.5 \text{ meters} \]
Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (10.5)^3 \]
Calculating \( V \):
\[ V \approx \frac{4}{3} \times \frac{22}{7} \times 1157.625 \approx 4851.0 \text{ m}^3 \]

### Step 2: Convert Volume to Liters
To convert the volume from cubic meters to liters, we use the conversion factor \( 1 \text{ m}^3 = 1000 \text{ L} \):
\[ V = 4851.0 \text{ m}^3 \times 1000 = 4851000 \text{ L} \]

### Step 3: Calculate the Mass of Helium
Using the Ideal Gas Law \( PV = nRT \), where:
- \( P = 1.0 \text{ atm} \)
- \( V = 4851.0 \text{ m}^3 \)
- \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \)
- \( T = 27^{\circ}C = 300 \text{ K} \)

First, we need to convert the volume from cubic meters to liters:
\[ V = 4851.0 \text{ m}^3 = 4851000 \text{ L} \]

Now, substituting into the Ideal Gas Law:
\[ n = \frac{PV}{RT} = \frac{(1.0)(4851000)}{(0.0821)(300)} \approx 196,000 \text{ moles} \]

Next, calculate the mass of helium:
\[ \text{Mass of He} = n \times \text{Molar mass of He} = 196000 \times 4 \approx 784000 \text{ g} = 784 \text{ kg} \]

### Step 4: Calculate the Total Mass of the Balloon and Helium
The total mass \( M_{\text{total}} \) is the sum of the mass of the balloon and the mass of helium:
\[ M_{\text{total}} = M_{\text{balloon}} + M_{\text{helium}} = 100 \text{ kg} + 784 \text{ kg} = 884 \text{ kg} \]

### Step 5: Calculate the Mass of Air Displaced
The mass of air displaced can be calculated using the density of air:
\[ \text{Density of air} = 1.2 \text{ kg/m}^3 \]
\[ M_{\text{air displaced}} = \text{Density} \times V = 1.2 \text{ kg/m}^3 \times 4851.0 \text{ m}^3 \approx 5821.2 \text{ kg} \]

### Step 6: Calculate the Payload
The payload \( P \) is given by the difference between the mass of air displaced and the total mass:
\[ P = M_{\text{air displaced}} - M_{\text{total}} = 5821.2 \text{ kg} - 884 \text{ kg} \approx 4937.2 \text{ kg} \]

### Final Answer
The payload of the balloon is approximately:
\[ \text{Payload} \approx 4937.2 \text{ kg} \]