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Capillary action

The tendency of a fluid to be raised or suppressed in a narrow tube, or capillary tube, is called capillary action.

If a capillary tube is placed vertically into a liquid, as shown in [link] , capillary action will raise or suppress the liquid inside the tube depending on the combination of substances. The actual effect depends on the relative strength of the cohesive and adhesive forces and, thus, the contact angle θ size 12{θ} {} given in the table. If θ size 12{θ} {} is less than 90º , then the fluid will be raised; if θ size 12{θ} {} is greater than 90º , it will be suppressed. Mercury, for example, has a very large surface tension and a large contact angle with glass. When placed in a tube, the surface of a column of mercury curves downward, somewhat like a drop. The curved surface of a fluid in a tube is called a meniscus . The tendency of surface tension is always to reduce the surface area. Surface tension thus flattens the curved liquid surface in a capillary tube. This results in a downward force in mercury and an upward force in water, as seen in [link] .

Mercury kept in a container into which a narrow tube is inserted lowers its level inside the tube relative to the level in the rest of the container. In a similar situation, water rises in the tube so that the water level in the tube is above the water level in the rest of the container. This phenomenon is due to the large contact angle of mercury with glass and the smaller contact angle of water with glass.
(a) Mercury is suppressed in a glass tube because its contact angle is greater than 90º . Surface tension exerts a downward force as it flattens the mercury, suppressing it in the tube. The dashed line shows the shape the mercury surface would have without the flattening effect of surface tension. (b) Water is raised in a glass tube because its contact angle is nearly . Surface tension therefore exerts an upward force when it flattens the surface to reduce its area.

Contact angles of some substances
Interface Contact angle Θ
Mercury–glass 140 º size 12{"140"°} {}
Water–glass 0 º size 12{0°} {}
Water–paraffin 107 º size 12{"107"°} {}
Water–silver 90 º size 12{"90"°} {}
Organic liquids (most)–glass 0 º size 12{0°} {}
Ethyl alcohol–glass 0 º size 12{0°} {}
Kerosene–glass 26 º size 12{"26"°} {}

Capillary action can move liquids horizontally over very large distances, but the height to which it can raise or suppress a liquid in a tube is limited by its weight. It can be shown that this height h size 12{h} {} is given by

h = cos θ ρ gr . size 12{h= { {2γ" cos"θ} over {ρ ital "gr"} } } {}

If we look at the different factors in this expression, we might see how it makes good sense. The height is directly proportional to the surface tension γ size 12{γ} {} , which is its direct cause. Furthermore, the height is inversely proportional to tube radius—the smaller the radius r , the higher the fluid can be raised, since a smaller tube holds less mass. The height is also inversely proportional to fluid density ρ , since a larger density means a greater mass in the same volume. (See [link] .)

The left image shows liquid in a container with four tubes of progressively smaller diameter inserted into the liquid. The liquid rises higher in the smaller-diameter tubes. The right image shows two containers, one holding a dense liquid and the other holding a less-dense liquid. Identical tubes are inserted into each liquid. The less-dense liquid rises higher in its tube than the more-dense liquid does in its tube.
(a) Capillary action depends on the radius of a tube. The smaller the tube, the greater the height reached. The height is negligible for large-radius tubes. (b) A denser fluid in the same tube rises to a smaller height, all other factors being the same.

Calculating radius of a capillary tube: capillary action: tree sap

Can capillary action be solely responsible for sap rising in trees? To answer this question, calculate the radius of a capillary tube that would raise sap 100 m to the top of a giant redwood, assuming that sap’s density is 1050 kg /m 3 size 12{"1050"`"kg/m" rSup { size 8{3} } } {} , its contact angle is zero, and its surface tension is the same as that of water at 20.0º C .

Strategy

The height to which a liquid will rise as a result of capillary action is given by h = cos θ ρ gr size 12{h= { {2γ" cos"θ} over {ρ ital "gr"} } } {} , and every quantity is known except for r size 12{r} {} .

Solution

Solving for r size 12{r} {} and substituting known values produces

r = cos θ ρ gh = 2 0.0728 N/m cos 1050 kg/m 3 9 . 80 m/s 2 100 m = 1.41 × 10 7 m. alignl { stack { size 12{r= { {2γ" cos"θ} over {ρ ital "gh"} } = { {2 left (0 "." "0728"`"N/m" right ) left (1 right )} over { left ("1050"`"kg/m" rSup { size 8{3} } right ) left (9 "." "80"`"m/s" rSup { size 8{2} } right ) left ("100"`m right )} } } {} #" "=1 "." "42" times "10" rSup { size 8{ - 7} } `m "." {} } } {}

Discussion

This result is unreasonable. Sap in trees moves through the xylem , which forms tubes with radii as small as 2 . 5 × 10 5 m size 12{2 "." 5 times "10" rSup { size 8{ - 5} } `m} {} . This value is about 180 times as large as the radius found necessary here to raise sap 100 m size 12{"100"`m} {} . This means that capillary action alone cannot be solely responsible for sap getting to the tops of trees.

Questions & Answers

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Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
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