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As a science teacher, I always explain kids about how water rises in a capillary tube: Capillary action occurs when the adhesion to the walls is stronger than the cohesive forces between the liquid molecules. We know that surface tension is the cohesive forces between molecules. First I thought that surface tension will decrease with a temperature increase and hence water will rise more in a capillary tube (More adhesion force will become dominant than cohesive force). But the formula says different story.

$$ h = \frac{2\gamma \cos \theta}{\rho g r} $$

Neglecting the change of density with temperature, the height of water column should decrease with the surface tension with a temperature increase. But this is a contradictory to my earlier saying that the cohesive force will become less effective so adhesion force will attract more liquid upward. I wonder what is wrong with my logic.

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Capillary effects involve two things: the surface tension of the liquid itself (gamma in your equation) and the contact angle of the liquid on a solid surface (theta in your equation). High surface tension means the liquid is capable of clinging strongly to itself. Low contact angle means good "wetting" of the surface by the liquid and trigonometrically maximizes the effectiveness of that pull.

This means that high surface tension and low contact angle mean a high capillary draw up the tube. These effects are magnified in small tubes and diminished by large tubes, as expressed by the r in the denominator of your equation.

Now, as the temperature of water increases, the surface tension of the water diminishes. This makes it easier for attractive forces between water molecules and other surfaces in the vicinity to pull water into capillary crevices and pores in those surfaces and wet them out. This is why hot water does a better job rinsing dirt out of your clothes than cold water does: it reduces the contact angle between the water and the solid surfaces, enhancing the wettability of those surfaces.

But note that if we reduce the surface tension of the water, we reduce the available pull force it can exert to drag itself up a vertical surface against gravity. So although the hot water wets that surface better, it cannot climb as far up that surface, and the capillary draw goes down in response.

We make up for this by agitating the surfaces in contact with the water while washing them, so as to assist the water to get in there and do its job. This is why we put agitators in washing machines.

We can do much the same at room temperature by adding detergents to the water which "break" its surface tension without heat. Now it wets the surfaces like crazy (cosine theta goes to 1) while at the same time gamma gets (typically) cut in half.

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