Cumulative Uptake by Diffusion for the Semi-Infinite Domain

Illustration 4.5 Cumulative Uptake by Diffusion for the Semi-Infinite Domain 

Here we calculate the amount of material per unit area that has diffused into a semi-infinite medium to a certain point in time t. We proceed in two steps by first calculating the rate cf dijfusion at the base plane and then integrating that rate over time. 

Calculation of the rate requires taking the derivative of the error function in Equation 4.9a. Let us consider the special (and usual) case of zero initial concentration C = 0. The need is then to evaluate 

Combining Equations 4.11a and 4.11c we obtain, for this intermediate step. 

This is the expression that now must be integrated with respect to time. We obtain immediately 

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