A rigid tank having a volume of 0.100 m3 contains helium gas at 155 atm. How many balloons can be inflated by opening the valve at the top of the tank? Each filled balloon is a sphere 0.300 m in diameter at an absolute pressure of 1.15 atm. (Remember that the tank cannot be completely emptied of gas.)
To find out how many balloons can be inflated by opening the valve at the top of the tank, we need to determine the amount of helium gas in the tank and then divide it by the amount of gas required to inflate one balloon.
Step 1: Convert the tank's volume to liters.
The volume of the tank is given as 0.100 m^3. Since there are 1000 liters in a cubic meter, we can convert the volume as follows:
0.100 m^3 x 1000 L/m^3 = 100 L
Step 2: Convert the pressure of helium in the tank to atmospheres.
The pressure in the tank is given as 155 atm, which is already in atm.
Step 3: Use the ideal gas law to calculate the amount of helium gas in the tank.
The ideal gas law equation is given by:
PV = nRT
Where:
P = pressure in atm
V = volume in liters
n = amount of gas in moles
R = J/(mol·K) (universal gas constant)
T = temperature in Kelvin
Since the temperature is not given and is not needed for this calculation, we can rearrange the equation as follows:
n = PV / RT
We can use this equation to find the amount of helium gas in moles:
n = (155 atm) * (100 L) / (0.0821 atm·L/mol·K)
n = 1881.7 mol
Step 4: Determine the amount of gas required to inflate one balloon.
The volume of each balloon is given as 0.300 m. Since the volume of a sphere is given by the formula V = (4/3)πr^3, we can calculate the radius as follows:
Radius (r) = 0.300 m / 2 = 0.150 m
Using the volume formula, we can find the volume of one balloon:
V = (4/3)π(0.150 m)^3
V = 0.014 m^3
Step 5: Convert the volume of one balloon to liters.
Since there are 1000 liters in a cubic meter, we can convert the volume as follows:
0.014 m^3 x 1000 L/m^3 = 14 L
Step 6: Calculate the number of balloons that can be inflated.
To find the number of balloons that can be inflated, divide the amount of helium gas in the tank by the amount of gas required to inflate one balloon:
Number of balloons = (1881.7 mol) / (14 L/mole)
Number of balloons = 134.4 balloons
Therefore, approximately 134 balloons can be inflated by opening the valve at the top of the tank. Keep in mind that the tank cannot be completely emptied of gas.