A cylinder of oxygen has a gauge pressure of 2000psi and a volume of 16L at 70 degrees F. If the oxygen is used as a constant flow rate of 2 L/minute, measured at 1 atm of pressure, how long will the tank last? Hint: Calculate the mass (grams) of oxygen in the tank and the mass flow rate (number of grams/min) flowing out the tank at 1 atm of pressure.
To calculate how long the tank will last, we need to find the mass of oxygen in the tank and the mass flow rate of oxygen flowing out of the tank.
First, let's convert the volume of the tank from liters to cubic meters, as the unit for pressure is given in pascals (Pa):
V = 16 L = 0.016 m^3
Next, we need to convert the gauge pressure from psi to pascals. 1 psi is approximately equal to 6894.76 pascals:
P = 2000 psi = 2000 * 6894.76 Pa ≈ 13,789,520 Pa
Now, we need to convert the temperature from Fahrenheit to Kelvin. The conversion formula is:
T(K) = (T(°F) - 32) * (5/9) + 273.15
T(°F) = 70°F
T(K) = (70 - 32) * (5/9) + 273.15 ≈ 294.26 K
To calculate the mass of oxygen in the tank, we can use the ideal gas law:
PV = nRT
Where:
P = pressure in pascals (13,789,520 Pa)
V = volume in cubic meters (0.016 m^3)
n = number of moles
R = ideal gas constant (8.314 J/(mol*K))
T = temperature in Kelvin (294.26 K)
Solving for n, the number of moles:
n = PV / (RT)
= (13,789,520 Pa * 0.016 m^3) / (8.314 J/(mol*K) * 294.26 K) ≈ 8.486 moles
Next, we need to calculate the mass of oxygen in grams. The molar mass of oxygen is approximately 32 g/mol:
mass = n * molar mass
≈ 8.486 moles * 32 g/mol
≈ 271.55 g
Now, let's calculate the mass flow rate of oxygen flowing out of the tank at a rate of 2 L/minute. First, convert 2 L/min to m^3/s:
flow_rate = 2 L/min = 0.002 m^3/s
Finally, we can calculate the mass flow rate of oxygen in grams/minute:
mass_flow_rate = flow_rate * density
= flow_rate * (mass / volume)
= 0.002 m^3/s * (271.55 g / 0.016 m^3)
≈ 33.903 g/min
To find out how long the tank will last, divide the mass of oxygen in the tank by the mass flow rate:
time = mass / mass_flow_rate
= 271.55 g / 33.903 g/min
≈ 8 minutes
Therefore, the tank will last approximately 8 minutes if the oxygen is used at a constant flow rate of 2 L/minute, measured at 1 atm of pressure.