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 in the polymer at 10 atm is 6.97*10-4 moles/cm3 at 35⁰ C. The diffusivity of CO2 in the polymer is known to be 2.28*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 seconds.

To determine how long it will take for 0.001 moles of CO2 to leak from the container at steady-state, we can use Fick's law of diffusion. Fick's law states that the rate of diffusion of a substance through a material is proportional to the concentration gradient and the area for diffusion, while inversely proportional to the thickness or distance of diffusion. The equation for Fick's law of diffusion is:

J = -D * (dC/dx) * A

Where:
J is the diffusion rate (moles/s)
D is the diffusivity (cm2/s)
dC/dx is the concentration gradient (moles/cm3/cm)
A is the area for diffusion (cm2)

First, let's calculate the concentration gradient, dC/dx. Since the amount of carbon dioxide in the surroundings is insignificant, the concentration in the surroundings is assumed to be 0 moles/cm3. Therefore, the concentration gradient is equal to the concentration of CO2 in the polymer, which is given as 6.97*10-4 moles/cm3.

Next, we can rearrange Fick's law to solve for the diffusion rate, J:

J = (D * (dC/dx) * A)

Substituting the given values:
J = (2.28*10-8 cm2/s) * (6.97*10-4 moles/cm3) * (100 cm2)
J = 1.59276*10-12 moles/s

Now, we can determine the time it takes for 0.001 moles of CO2 to diffuse by dividing the amount of CO2 by the diffusion rate:

Time = Amount / Diffusion rate
Time = 0.001 moles / (1.59276*10-12 moles/s)
Time = 6.28*10+8 seconds

Therefore, it will take approximately 6.28*10+8 seconds for 0.001 moles of CO2 to leak from the container at steady-state.