Question

Water is transported upward in plants through xylem tissue which consists of cells 1 mm long and a species dependent diameter between 40 μm and 400 μm. The xylem cells are attached to each other to form a channel. To what maximum height can water rise in these xylem channels due to capillarity?

Answer 1

The maximum height to which water can rise in xylem channels due to capillarity is in the range of
`\(0.147 \, \text{mm}\) to \(0.015 \, \text{mm}\)`, depending on the diameter of the xylem cells.






The maximum height (\(h\)) to which water can rise in a capillary tube is given by the Jurin's Law equation:

`\[ h = \frac{{2T \cos \theta}}{{\rho g r}} \]`

where:

- \(T\) is the surface tension of the liquid,

-
`\(\theta\)` is the contact angle between the liquid and the wall of the tube,

-
`\(\rho\)` is the density of the liquid,

- \(g\) is the acceleration due to gravity, and

- \(r\) is the radius of the capillary tube.

For the case of xylem tissue in plants, capillary action is primarily responsible for water movement. We can estimate the maximum height by considering the range of diameters for xylem cells.

Given that the diameter (\(d\)) of the xylem cells varies between 40 μm and 400 μm, we can use the radius (\(r\)) as half of the diameter:

`\[ r_{\text{min}} = \frac{40 \, \mu\text{m}}{2} = 20 \, \mu\text{m} \]\[ r_{\text{max}} = \frac{400 \, \mu\text{m}}{2} = 200 \, \mu\text{m} \]`

Assuming the water is the liquid in the xylem tissue, and considering the contact angle \(\theta\) to be approximately \(0\) for water in a plant xylem, we can simplify the formula.

`\[ h_{\text{min}} = \frac{{2T}}{{\rho g r_{\text{min}}}} \]\[ h_{\text{max}} = \frac{{2T}}{{\rho g r_{\text{max}}}} \]`

The surface tension of water (\(T\)) is approximately
`\(0.072 \, \text{N/m}\)`, the density
`(\(\rho\))` is approximately
`\(1000 \, \text{kg/m}^3\)`, and the acceleration due to gravity (\(g\)) is approximately
`\(9.8 \, \text{m/s}^2\)`.

`\[ h_{\text{min}} = \frac{{2 * 0.072}}{{1000 * 9.8 * (20 * 10^(-6))}} \]\[ h_{\text{max}} = \frac{{2 * 0.072}}{{1000 * 9.8 * (200 * 10^(-6))}} \]`

Calculate
`\(h_{\text{min}}\) and \(h_{\text{max}}\`) to find the range of the maximum height water can rise due to capillarity in the xylem tissue. Note that these calculations assume ideal conditions and may not precisely represent the complex biological systems in plants.

`\[ h_{\text{min}} \approx 0.000147 \, \text{m} \, \text{or} \, 0.147 \, \text{mm} \]\[ h_{\text{max}} \approx 0.000015 \, \text{m} \, \text{or} \, 0.015 \, \text{mm} \]`

Therefore, the maximum height to which water can rise in xylem channels due to capillarity is in the range of
`\(0.147 \, \text{mm}\) to \(0.015 \, \text{mm}\)`, depending on the diameter of the xylem cells.