To determine how many spherical balloons can be filled with hydrogen gas from the cylinder, we need to start by calculating the volume of one balloon and then compare that to the volume of gas available in the cylinder under the given conditions. Let's break this down step by step.
Step 1: Calculate the Volume of One Balloon
The formula for the volume of a sphere is given by:
V = (4/3)πr³
First, we need to find the radius of the balloon. The diameter is 21 cm, so the radius (r) is:
r = diameter / 2 = 21 cm / 2 = 10.5 cm
Now, substituting the radius into the volume formula:
V = (4/3)π(10.5 cm)³
Calculating this gives:
- 10.5 cm³ = 1157.52 cm³ (approximately)
Since 1 liter is equivalent to 1000 cm³, the volume of one balloon in liters is:
V_balloon = 1157.52 cm³ / 1000 = 1.15752 liters
Step 2: Determine the Volume of Hydrogen in the Cylinder
Next, we need to find out how much hydrogen gas is available in the cylinder. The cylinder can hold 2.82 liters of water at 1 atm and 273 K. Since the gas behaves ideally, we can use the ideal gas law to find the volume of hydrogen at the conditions in the cylinder.
The ideal gas law is:
PV = nRT
Where:
- P = pressure (in atm)
- V = volume (in liters)
- n = number of moles
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature (in Kelvin)
First, we need to convert the temperature from Celsius to Kelvin:
T = 27°C + 273 = 300 K
Now, we can calculate the volume of hydrogen gas at the cylinder's conditions:
P = 20 atm
Using the volume of the cylinder (2.82 liters) at 1 atm and 273 K, we can find the equivalent volume at 20 atm and 300 K using the combined gas law:
(P1V1/T1) = (P2V2/T2)
Substituting the known values:
(1 atm * 2.82 L / 273 K) = (20 atm * V2 / 300 K)
Solving for V2 gives:
V2 = (2.82 L * 20 atm * 300 K) / (1 atm * 273 K)
Calculating this results in:
- V2 ≈ 62.03 liters (approximately)
Step 3: Calculate the Number of Balloons
Now that we have the volume of hydrogen gas available in the cylinder (62.03 liters), we can find out how many balloons can be filled:
Number of Balloons = Total Volume of Gas / Volume of One Balloon
Number of Balloons = 62.03 liters / 1.15752 liters
Calculating this gives:
- Number of Balloons ≈ 53.67
Since we can't fill a fraction of a balloon, we round down to the nearest whole number. Therefore, the maximum number of balloons that can be filled is:
53 balloons
In summary, from the cylinder containing hydrogen gas, you can fill up to 53 spherical balloons of 21 cm diameter at the specified conditions. This calculation illustrates the application of the ideal gas law and the relationship between pressure, volume, and temperature in gas behavior.
