Diffusive flow

Classical Mechanics Level 3

Molecules in liquid can randomly diffuse from one point to another due to thermal energy. Concentrations, \(c(\vec{x})\), that are distributed randomly will tend to stay distributed randomly, however, regions of higher concentration are more likely to donate a molecule to a region of lower concentration than the reverse. Formally, regions of high concentration exhibit net flux into regions of low concentration in direct proportion to the concentration gradient:

\[\displaystyle J = -D\nabla c(\vec{x})\]

Because leaves maintains a high sucrose concentration relative to the other parts of a tree, sucrose can make it from the leaves to the sites of energy consumption by diffusion alone. However, this mode of transport isn't very fast, which is one of the reasons that vascular systems are an evolutionary boon for large plants.

If we look at the units of the diffusion constant, \(D\), we see that they must be equal the units of \(J x/c\), which are L\(^2\)/T. This suggests that diffusing molecules take time \(t_{diff}\sim l^2/D\) to move a distance \(l\). For regular flow, at speed \(u_{flow}\), it takes the time \(t_{flow} \sim l/u_{flow}\) to move a distance \(l\). The ratio of time taken by the vascular system, vs the time taken by diffusion is then \[\frac{t_{flow}}{t_{diff}} = \frac{D}{u_{flow}l}\]

Suppose that sieve cells are \(\sim l_{sieve}\) long and that the diffusion constant for sucrose is \(D\). With what effective speed will sucrose molecules diffuse across a sieve cell (in \(\mu\)m/s)?


  • \(l_{sieve} = 91\mbox{ } \mu\)m
  • \(D = 10^{-8}\) m\(^2\)/s

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