An empty gas cylinder of 40L is to be filled with air (ideal gas) to 5000psia and 25C. The gas cylinder is being filled by a pump that increases its flow by 20 moles per minutes after turned on. How long will it take the pump to fill the tank?
My work:
1st- I converted units to common terms and found that the total number of mols possible was 556 moles using PV=nRT.
2nd- I know that I will need a natural log function of e^(some number)t. I am just not sure how to set that up. This was my first attempt, but I did not feel it correct.
dln(N)=0.20
------
dt
e^(.20t)
I am starting to confuse myself now:(
Edit:
e^(.20t)=556
To solve this problem, you need to use the equation for the rate of change of the number of moles in the cylinder with respect to time:
dN/dt = constant rate of increase
You mentioned that the rate of increase is 20 moles per minute, so you can write:
dN/dt = 20
Next, you need to integrate this equation to find the total number of moles in the cylinder as a function of time:
∫dN = ∫20 dt
N = 20t + C
Here, C is the constant of integration which represents the initial number of moles in the cylinder. Since the cylinder is initially empty, C = 0.
Now, we know that the total number of moles in the cylinder when it is filled is 556 mol. So, we can set up the equation:
N = 20t
556 = 20t
Solving for t, we find:
t = 556/20
t ≈ 27.8 minutes
Therefore, it will take approximately 27.8 minutes for the pump to fill the gas cylinder.
To find how long it will take the pump to fill the tank, you need to use the ideal gas law equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
First, convert the pressure from psia to atm. 5000 psia is approximately 344.738 atm.
Next, convert the volume from liters to cubic meters. 40 liters is equal to 0.04 cubic meters.
Now you can plug in the values into the ideal gas law equation:
PV = nRT
344.738 atm * 0.04 cubic meters = n * 0.0821 L atm / (mol K) * (25C + 273.15K)
Simplifying the equation:
13.78952 = n * 20.2075
Now you can solve for n, which represents the total number of moles needed to fill the tank:
n = 13.78952 / 20.2075
n ≈ 0.682 moles
Since the pump increases the flow by 20 moles per minute, you can calculate how long it will take to fill the tank:
t = n / flow rate
t = 0.682 moles / 20 moles per minute
t ≈ 0.0341 minutes
Therefore, it will take approximately 0.0341 minutes (or about 2.046 seconds) for the pump to fill the tank.