a) 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.

6262 sec

no

its 6264

To find the time it takes for 0.001 moles of CO2 to leak from the container at steady-state, we can use Fick's first law of diffusion:

J = -D * (dC/dx)

Where:
J is the flux of CO2 (moles/(cm^2 * s)),
D is the diffusivity of CO2 in the polymer (cm^2/s),
dC/dx is the concentration gradient of CO2 (moles/cm^3/cm).

To find the flux, we can use the equation:

J = (P * S * solubility) / thickness

Where:
P is the pressure of CO2 in the vessel (atm),
S is the effective area of the membrane (cm^2),
solubility is the solubility of CO2 in the polymer at the given pressure and temperature (moles/cm^3),
thickness is the thickness of the membrane (cm).

In steady-state, the flux of CO2 will be constant. Therefore, we can equate the two equations for J:

(P * S * solubility) / thickness = -D * (dC/dx)

Rearranging the equation, we can solve for dC/dx:

dC/dx = - (P * solubility) / (D * thickness)

Now, we can substitute the given values:

P = 10 atm
S = 100 cm^2
solubility = 6.97 * 10^-4 moles/cm^3
thickness = 100 microns = 0.01 cm
D = 2.28 * 10^-8 cm^2/s

Plugging in these values, we get:

dC/dx = - (10 atm * 6.97 * 10^-4 moles/cm^3) / ( 2.28 * 10^-8 cm^2/s * 0.01 cm)

Simplifying the equation, we get:

dC/dx = - 3.059 * 10^5 moles/cm^4s

Now, we need to find the time it takes for 0.001 moles of CO2 to leak through an area of 100 cm^2. We can divide the amount of CO2 by the flux:

Time = Amount of CO2 / Flux

Amount of CO2 = 0.001 moles
Flux = - dC/dx * S

Plugging in the values:

Amount of CO2 = 0.001 moles
Flux = 3.059 * 10^5 moles/cm^4s * 100 cm^2

Simplifying the equation, we get:

Time = (0.001 moles) / (3.059 * 10^5 moles/cm^4s * 100 cm^2)

Time = 3.27 * 10^-10 s

Therefore, it will take approximately 3.27 * 10^-10 seconds for 0.001 moles of CO2 to leak from the container at steady-state.