Nutrients dissolved in water are carried to the upper parts of plants by tiny tubes partly because of the capillary effect. Determine how high the water solution will rise in a tree in a 0.0026-mm-diameter tube as a result of the capillary effect.  

Note: Treat the solution as water at 𝟐𝟎𝟎𝑪 with a contact angle 𝟏𝟓𝟎.  

Solution:

To determine the height (h) to which water will rise in a capillary tube, you can use the Young-Laplace equation:

h=  `{2Tcos(θ)}/{ρgr}`

where:

T is the surface tension of the liquid (water in this case),

θ is the contact angle between the liquid and the tube,

ρ is the density of the liquid,

g is the acceleration due to gravity, and

r is the radius of the capillary tube.

Given values:

Surface tension of water (T): 0.0728 N/m (at 20°C)

Contact angle (θ): 150 degrees

Density of water (ρ): 1000 kg/m³

Acceleration due to gravity (g): 9.8 m/s²

Radius of the capillary tube (r): 0.0000013 m (converted from 0.0026 mm)

Now plug in these values into the equation:

ℎ = `{2 × 0.0728 × cos(150^∘)}/{1000 × 9.8 × 0.0000013}`

Calculate this expression to find the height (h) to which water will rise in the capillary tube. Note that the cosine of 150 degrees is equal to `-\sqrt {3}/ 2`.

ℎ ≈ `{2 × 0.0728 × cos(-\sqrt3 / 2)}/{1000 × 9.8 × 0.0000013}`

ℎ ≈ `-0.1253/0.00001274`

ℎ ≈ -9827 m

The negative sign indicates that the water will actually descend in this capillary tube due to the chosen contact angle of 150 degrees. This result might seem unrealistic, and it's possible that the chosen contact angle is not suitable for the conditions. Please double-check the provided values and ensure that the contact angle is appropriate for the situation.