A spherical balloon of `21` cm diameter is to be filled up with hydrogen at 1 atm, 273 K from a cylinder containing the gas at 20 atm and `27^(@)C`. If the cylinder can hold 2.82 litre of water, calculate the number of balloons that can be filled up completely.

Answer

To solve the problem of how many balloons can be filled with hydrogen gas from a cylinder, we will follow these steps:

### Step 1: Calculate the Volume of the Balloon
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 sphere. Given the diameter of the balloon is 21 cm, the radius \( r \) is:
\[ r = \frac{21 \text{ cm}}{2} = 10.5 \text{ cm} \]
Now, substituting the value of \( r \) into the volume formula:
\[ V = \frac{4}{3} \pi (10.5)^3 \]
Calculating this gives:
\[ V \approx \frac{4}{3} \times 3.14 \times 1157.625 \approx 4846.59 \text{ cm}^3 \]

### Step 2: Convert Volume to Liters
To convert the volume from cubic centimeters to liters, we use the conversion:
\[ 1 \text{ L} = 1000 \text{ cm}^3 \]
Thus,
\[ V \approx \frac{4846.59 \text{ cm}^3}{1000} \approx 4.846 \text{ L} \]

### Step 3: Calculate Moles of Hydrogen Required for One Balloon
Using the Ideal Gas Law \( PV = nRT \), we can find the number of moles \( n \) of hydrogen gas needed to fill one balloon at 1 atm and 273 K:
- \( P = 1 \text{ atm} \)
- \( V = 4.846 \text{ L} \)
- \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \)
- \( T = 273 \text{ K} \)

Rearranging the ideal gas equation to solve for \( n \):
\[ n = \frac{PV}{RT} \]
Substituting the values:
\[ n = \frac{(1 \text{ atm})(4.846 \text{ L})}{(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1})(273 \text{ K})} \]
Calculating this gives:
\[ n \approx 0.216 \text{ moles} \]

### Step 4: Calculate Moles of Hydrogen Available in the Cylinder
Now, we need to calculate how many moles of hydrogen are available in the cylinder:
- The cylinder's pressure is 20 atm, volume is 2.82 L, and temperature is 27 °C (which is 300 K).

Using the Ideal Gas Law again:
\[ n = \frac{PV}{RT} \]
Substituting the values:
\[ n = \frac{(20 \text{ atm})(2.82 \text{ L})}{(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1})(300 \text{ K})} \]
Calculating this gives:
\[ n \approx 2.289 \text{ moles} \]

### Step 5: Calculate the Number of Balloons That Can Be Filled
Finally, to find the number of balloons that can be filled, we divide the total moles of hydrogen available by the moles required for one balloon:
\[ \text{Number of Balloons} = \frac{\text{Total moles in cylinder}}{\text{Moles required per balloon}} = \frac{2.289}{0.216} \]
Calculating this gives:
\[ \text{Number of Balloons} \approx 10.589 \]
Since we can only fill whole balloons, we round down to get:
\[ \text{Number of Balloons} = 10 \]

### Final Answer
The number of balloons that can be filled completely is **10**.