Diffusion and thermal conductivity processes can be emulated using spreadsheets that use columns for increments in distance and rows for increments in time. Starting with initial values in the first row of each column the next row can be calculated using incremental transfer equations to determine values for the next row. Although this explicit increment version of the general numerical method of solving differential equations by finite differences results in a large spreadsheet, it causes no computational difficulties because of the high speed and large storage capabilities of modern microcomputers. Determination of the spreadsheet transfer equations for use with this explicit increment method may be done in an intuitive way by utilizing the concept of liquid flow between cells in a plastic model (Figure 1). This was discussed in a previous article in which flowing water emulates diffusion and thermal conductivity (1). Since this method circumvents the necessity of mathematically dealing with second-order differential equations, it may encourage introduction of quantitative aspects of all types of diffusion earlier in the chemistry curriculum.
Figure 1. Liquid flow in the plastic model.
For classroom presentations Microsoft PowerPoint slides of the change in concentration or temperature with time from spreadsheet charts using Microsoft Excel can be created to look essentially identical to demonstrations using the plastic model. A spreadsheet emulation of a system with a constant source achieving a steady state is shown in Figure 2.
Figure 2. Diffusion using spreadsheet calculations and bar graphs: (A) initial row, (B) row 50, (C) row 200, and (D) row 2500.
For planar systems, the incremental transfer equation for spreadsheet cell B5 is
The first term on the right is the quantity that is currently in cell B. The second term is the quantity input from cell A calculated from the quantity difference times a flow proportionality factor. The third term is the quantity output to cell C calculated the same way as the second term. Although the last two terms may be mathematically simplified to (A4 – 2B4 + C4)0.05, it masks the concept of input minus output and makes it more difficult to write boundary condition equations. The incremental transfer equation is used in every row except the row having the initial conditions and cells at the ends, which may have no input, no output, or are constant.
For radial diffusion the quantity between partitions and the area of partitions change as the radius changes. For a cylindrical system the incremental transfer equation for cell B5 before simplification is
where r is the radius of a given cell and h is the height of the cell. For spherical systems the equation is
Literature Cited
- Blanck, H. F. J. Chem. Educ. 2005, 82, 1523.
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