When examining diffusion out of (or into) a semi-infinite membrane, we applied the assumptions that there...

 

Transcribed Image Text:

When examining diffusion out of (or into) a semi-infinite membrane, we applied the assumptions that there is no reaction in the membrane, we are not at steady-state, C is constant in space and time, there and there is only diffusion in the x-direction, in order to reduce the general form of Fick's 2nd law to the following differential equation: 0 = c-C C-C Our goal is to solve for c(x,t), but we cannot assume that c(x,t) is separable, in other words c(x, t)#f(x)g(t), because the boundary condition at x=0 is not homogeneous, c(0, t) = C. Therefore, we performed a substitution to turn this second order partial differential equation into a dimensionless ordinary differential equation. The substitution for c, x, and t is defined below. n = 4Dt t = D. - ac x Show that when 0 and n are substituted into the differential equation above, the resulting differential equation is: de dn Show every step and provide a brief explanation when implementing a rule or identity. de 2- dn When examining diffusion out of (or into) a semi-infinite membrane, we applied the assumptions that there is no reaction in the membrane, we are not at steady-state, C is constant in space and time, there and there is only diffusion in the x-direction, in order to reduce the general form of Fick's 2nd law to the following differential equation: 0 = c-C C-C Our goal is to solve for c(x,t), but we cannot assume that c(x,t) is separable, in other words c(x, t)#f(x)g(t), because the boundary condition at x=0 is not homogeneous, c(0, t) = C. Therefore, we performed a substitution to turn this second order partial differential equation into a dimensionless ordinary differential equation. The substitution for c, x, and t is defined below. n = 4Dt t = D. - ac x Show that when 0 and n are substituted into the differential equation above, the resulting differential equation is: de dn Show every step and provide a brief explanation when implementing a rule or identity. de 2- dn