A spherical ballon of 21 cm diameter is to be filled with `H_(2)` at NTP from a cylinder containing the gas at 20 atm at `27^(@)C` .If the cylinder can hold 2.80 L of water , calculate the number of ballons that can be filled up .

Options

A. 2
B. 4
C. 8
D. 10

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the volume of one balloon
Given the diameter of the balloon is 21 cm, we first find the radius:
\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{21 \text{ cm}}{2} = 10.5 \text{ cm} \]
Next, we calculate the volume \( V \) of the balloon using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the value of \( r \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times (10.5)^3 \]
Calculating \( (10.5)^3 \):
\[ (10.5)^3 = 1157.625 \text{ cm}^3 \]
Now substituting this back into the volume formula:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 1157.625 \approx 4851 \text{ cm}^3 \]

### Step 2: Convert the volume of one balloon to liters
Since 1 liter = 1000 cm³, we convert the volume:
\[ V = \frac{4851 \text{ cm}^3}{1000} = 4.851 \text{ L} \]

### Step 3: Calculate the total volume of hydrogen gas in the cylinder
The cylinder can hold 2.80 L of water. The total volume of hydrogen gas that can be used to fill the balloons is:
\[ \text{Total Volume} = 2.80 \text{ L} + n \times 4.851 \text{ L} \]
Where \( n \) is the number of balloons.

### Step 4: Use the ideal gas law to find the volume of hydrogen gas at NTP
At Normal Temperature and Pressure (NTP), the conditions are:
- Pressure \( P_1 = 1 \text{ atm} \)
- Temperature \( T_1 = 273 \text{ K} \)

The gas in the cylinder is at:
- Pressure \( P_2 = 20 \text{ atm} \)
- Temperature \( T_2 = 300 \text{ K} \)

Using the ideal gas law:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
Substituting the known values:
\[ \frac{1 \times (4.851n + 2.80)}{273} = \frac{20 \times 2.80}{300} \]

### Step 5: Solve for \( n \)
Calculating the right side:
\[ \frac{20 \times 2.80}{300} = \frac{56}{300} = 0.1867 \]
Now substituting back into the equation:
\[ \frac{4.851n + 2.80}{273} = 0.1867 \]
Cross-multiplying gives:
\[ 4.851n + 2.80 = 0.1867 \times 273 \]
Calculating the right side:
\[ 0.1867 \times 273 \approx 51.00 \]
So we have:
\[ 4.851n + 2.80 = 51.00 \]
Subtracting 2.80 from both sides:
\[ 4.851n = 51.00 - 2.80 = 48.20 \]
Now dividing by 4.851:
\[ n = \frac{48.20}{4.851} \approx 9.92 \]
Rounding gives:
\[ n \approx 10 \]

### Conclusion
The number of balloons that can be filled is approximately **10**.

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