Question

Helium gas is stored at \(293 \mathrm{~K}\) in a 3-m-outer-diameter spherical container made of 3 -cm-thick Pyrex. The molar concentration of helium in the Pyrex is \(0.00069 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner surface and negligible at the outer surface. Determine the mass flow rate of helium by diffusion through the Pyrex container.

Short answer

Answer: The mass flow rate of helium through the Pyrex container is approximately -2.09 x 10^(-12) kg/s.

Step-by-step solution

Determine the surface area of the container

To find the surface area of the spherical container, use the following formula: \[A =4 \pi r^{2}\] where \(A\) is the surface area, and \(r\) is the radius of the sphere. Since the sphere has a diameter of 3 meters, its radius is 1.5 meters. \[A =4 \pi (1.5)^{2} = 9\pi \mathrm{m}^{2}\]

Calculate the molar flux

To calculate the molar flux, we need to use Fick's first law of diffusion, given by: \[J_{A} = -D_{AB} \frac{dC_{A}}{dx}\] where \(J_{A}\) is the molar flux, \(D_{AB}\) is the diffusion coefficient, \(dC_{A}\) is the change in molar concentration, and \(dx\) is the thickness of the container. In this problem, we are considering the diffusion of helium gas through Pyrex. We can assume that the molar concentration of helium is linear between the inner and outer surfaces. We are given that the molar concentration is \(0.00069 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner surface and negligible at the outer surface, and the thickness of Pyrex is 3 cm. First, convert the thickness into meters: \[d=3 \times 10^{-2}\,\mathrm{m}\] Now, find the molar concentration gradient: \[\frac{dC_{A}}{dx} = \frac{0.00069-0}{0.03} =0.023\,\mathrm{kmol} / \mathrm{m}^{4}\] We are now able to determine the diffusion coefficient, \(D_{AB}\), for helium gas through Pyrex. From external sources like literature or handbooks, let us assume a diffusion coefficient value of \(D_{AB} = 8\times10^{-10} \mathrm{m^2/s}\) for helium in Pyrex at 293 K. Now, we can calculate the molar flux: \[J_{A} = -D_{AB} \frac{dC_{A}}{dx} = -(8\times10^{-10})(0.023) =-1.84\times 10^{-11} \mathrm{kmol/m^{2}s}\] Please note that the negative sign indicates that the direction of diffusion is opposite to the direction of the concentration gradient. Here, it indicates that the helium gas is diffusing from inside to outside of the container.

Calculate the mass flow rate

Now, we multiply the molar flux by the surface area to obtain the total molar flow rate of helium, \(\dot{n}_A\): \[\dot{n}_A = J_{A}\times A=-1.84\times 10^{-11} \times 9\pi \mathrm{m}^{2}=-5.23\times 10^{-10}\,\mathrm{kmol/s}\] Finally, multiply the molar flow rate by the molar mass of helium, \(M_A = 4 \mathrm{g/mol}\) (or \(4 \times 10^{-3} \mathrm{kg/kmol}\)) to find the mass flow rate, \(\dot{m}_A\). \[\dot{m}_A =\dot{n}_A\times M_A =-5.23\times 10^{-10}\times 4 \times 10^{-3} \,\mathrm{kg/s}=-2.09\times 10^{-12}\,\mathrm{kg/s}\] The mass flow rate of helium by diffusion through the Pyrex container is approximately \(-2.09\times 10^{-12} \mathrm{kg/s}\). The negative sign indicates that the helium gas is diffusing from inside to outside of the container.