# Molar flux through membrane

Consider a liquid of pure $$\ce{A}$$ covered by a composite membrane through which the vapor of $$\ce{A}$$ is diffusing. The membrane consists of two layers of thickness $$l_1$$ and $$l_2$$. The diffusion coefficient of $$\ce{A}$$ in the two layers are $$D_{\ce{A},1}$$ and $$D_{\ce{A},2}$$ respectively. On the lower side of the membrane, the concentration of $$\ce{A}$$ ($$c^s_\ce{A})$$ is controlled by the vapor pressure of $$\ce{A}$$. On the upper side the concentration of $$\ce{A}$$ is zero. The solubulity of $$\ce{A}$$ in the two membrane layers is the same.

(a) Derive the expression for the flux $$J_\ce{A}$$ through the composite membrane The flux through a membrane is given by the equation:

$$J_\ce{A} = \frac{D_{AB}}{L}\left(c^{(m)}_{\ce{A},F} - c^{(m)}_{\ce{A},P}\right)$$

where, $$c^{(m)}_{\ce{A},F}$$ and $$c^{(m)}_{\ce{A},P}$$ are the concentrations at the edges inside the membrane.

So I would assume that $$c^{(m)}_{\ce{A},F}$$ = $$c^s_\ce{A}$$ and $$c^{(m)}_{\ce{A},P}$$ = $$c_\ce{A}$$

and because there are two membranes I would have to combine the diffusion coefficients as well as the lengths, which I do by addition to get the equation:

$$J_\ce{A} = \frac{D_{\ce{A},1} +D_{\ce{A},2}}{l_1 +l_2} (c^s_\ce{A} - c_\ce{A})$$

I am unsure if this is the correct way of deriving the expression for the flux when there are two membranes. Am I thinking correct?

Edit

I named the third concentration in the membrane $$c_m$$. Then I made two equations for the molar flux.

$$J_\ce{A} = \frac{D_{A,1}}{l_1}(c^s_A - c_m)$$

$$J_\ce{A} = \frac{D_{A,2}}{l_2}(c_m - c_A)$$

When rearranging and solving for $$J_A$$, I get:

$$J_\ce{A} = \frac{(c^s_A - c_A)}{\frac{l_1}{D_{A,1}}+\frac{l_2}{D_{A,2}}}$$

But that is the same answer as I got before

• What you did is an exact equivalent of fractions addition in a "simple" way: $${1\over2}+{1\over3}={1+1\over2+3}={2\over5}.$$ Pity it never works like that. – Ivan Neretin Apr 30 '19 at 9:21
• @IvanNeretin Yeah, I thought that it couldn't be the correct way. How should it be derived? – lotte07 Apr 30 '19 at 9:26
• Imagine a very thin layer of vapor between the membranes (which is not really there, but hey, that's what imagination is for). Then you know the equation for the flux to this layer, and also from this layer, and the two are equal. Can you continue? – Ivan Neretin Apr 30 '19 at 9:40
• @IvanNeretin But when making two equations, one for the flux to the layer and one from the layer, how do I use the concentrations? Will I have $(c^s_\ce{A} - c_\ce{A})$ for both fluxes? – lotte07 Apr 30 '19 at 9:56
• There will be a third concentration, the one that exists between the membranes, initially unknown. – Ivan Neretin Apr 30 '19 at 10:01