A pressure vessel contains a large volume of CO2 gas at 10 atm pressure. A membrane composed of a poly(ether ketone) polymer with thickness 100 microns and net effective area of 100 cm2 covers a small perforated port in the container. The solubility of CO2 at 10 atm is 6.97 x 10-4 moles/cm3 at 35 \(^\circ\)C. The diffusivity of CO2 in the polymer is known to be 2.29 x 10-8 cm2/s at this temperature. How long will it take for 0.001 moles of CO2 to leak from the container at steady-state? Assume that the amount of carbon dioxide in the surroundings is insignificant. Express your answer in hours
Use this:
http://en.wikipedia.org/wiki/Fick's_laws_of_diffusion
second?
please drwls, what is the answer of a and b
a) 1.74
b) 0.551
a) It's 6264
To find the time it takes for CO2 to leak from the container at steady-state, we can use Fick's Law of Diffusion. The equation is as follows:
Rate of diffusion = (Diffusion coefficient x Area x Concentration difference) / Thickness
First, let's calculate the concentration difference. We need to find the difference between the concentration of CO2 inside the container and the concentration outside, assuming that the amount of CO2 in the surroundings is insignificant.
In the pressure vessel, the initial concentration of CO2 is given by the solubility, which is 6.97 x 10^(-4) moles/cm^3. Since the net effective area of the membrane is 100 cm^2, we can calculate the total moles of CO2 present inside the container:
Total moles = Concentration x Volume
Total moles = (6.97 x 10^(-4) moles/cm^3) x (100 cm^2)
Total moles = 6.97 x 10^(-2) moles
The concentration difference will be the initial concentration (6.97 x 10^(-4) moles/cm^3) minus the final concentration (which will decrease as CO2 leaks out).
Now, we can calculate the final concentration:
Final concentration = Initial concentration - (Amount leaked / Volume)
Final concentration = 6.97 x 10^(-4) moles/cm^3 - (0.001 moles / (100 cm^2 x 100 microns))
Final concentration = 6.97 x 10^(-4) moles/cm^3 - (0.001 moles / (100 cm^2 x 1 x 10^(-2) cm))
Final concentration = 6.97 x 10^(-4) moles/cm^3 - 0.1 moles/cm^3
Final concentration = 0.0993 moles/cm^3
Now we can plug in the values into Fick's Law of Diffusion:
Rate of diffusion = (Diffusion coefficient x Area x Concentration difference) / Thickness
Rate of diffusion = (2.29 x 10^(-8) cm^2/s) x (100 cm^2) x (0.0993 moles/cm^3 - 6.97 x 10^(-4) moles/cm^3) / (100 microns)
Rate of diffusion = (2.29 x 10^(-8) cm^2/s) x (100 cm^2) x (0.09833 moles/cm^3) / (100 microns)
Rate of diffusion = 2.25417 x 10^(-7) moles/s
To find the time it takes for 0.001 moles of CO2 to leak at steady-state, we can use the following equation:
Time = Amount leaked / Rate of diffusion
Time = 0.001 moles / (2.25417 x 10^(-7) moles/s)
Time = 4435.78 s
Finally, we can convert seconds to hours:
Time in hours = Time in seconds / (60 seconds/minute x 60 minutes/hour)
Time in hours = 4435.78 s / (60 s/min x 60 min/hr)
Time in hours ≈ 1.232 hours
So, it will take approximately 1.232 hours for 0.001 moles of CO2 to leak from the container at steady-state.