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We all have seen the meniscus of a small graduated cylinder in the lab. The reasons for the occurrence of capillary action are clear, but why does it occur more profusely in smaller-rimmed containers?

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You have to look which forces act where. You have a force that tries pull the liquid up the capillary, which stems from the gain in energy due to adhesion. At the same time, however, gravity acts against this force. In equilibrium, both forces have the same value:

$$F_\text{g} = F_{\text{surf}}$$ $$ mg=\rho Vg= \rho R^2\pi h g=2\pi R\sigma_s \cos(\varphi)$$

Which leads to the formula presented by Babounet in another answer:

$$ h = \frac{2\sigma_s\cos(\varphi)}{R\rho g}$$

So, to boil it down, if you lower the radius $R$, you lower the force due to gravity $F_g$, but the force due to surface energy $F_\text{surf}$ does not fall as fast, which leads to a higher height $h$.

Sketch (source, $W=F_g$ here, naming isn't perfect, $F_{\text{surf}}$ is not drawn into)

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A meniscus is formed by the opposing forces of adhesion between the fluid and the walls of the container drawing the fluid up and gravity pulling the fluid down. The adhesive forces are proportional to the diameter of the tube while the gravitational effect due to the liquid's weight is proportional to the square of the diameter. This results in the greater height of the meniscus for narrower tubes.

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  • $\begingroup$ Welcome to chemistry.se! If you have questions about how to beautify your posts, have a look at the help center. Do you want to know more about this site, please take the tour. $\endgroup$ – Martin - マーチン Jan 5 '15 at 4:26
  • $\begingroup$ Remember that it isn't only gravity and adhesion, but also cohesion among the liquid molecules in the container. Now, something I would like an explanation for is why "The adhesive forces are proportional to the diameter of the tube while the gravitational effect due to the liquid's weight is proportional to the square of the diameter." Can you elucidate how these truths were derived? Thanks! $\endgroup$ – user11629 Jan 5 '15 at 16:29
  • $\begingroup$ You are correct @user11629, cohesive forces are also important. The difference in strength between adhesion and cohesion for the liquid can also change the shape of the meniscus, for example giving the inverted shape seen for Mercury. Unfortunately I can't find a clear explanation for the different relationships to diameter. $\endgroup$ – lazappi Jan 6 '15 at 0:25
  • $\begingroup$ If you look at John's answer, you can see that the weight has to do with the volume of the liquid. For a cylinder, it's the area of the cross section times the height. The area increases with r^2. The force pulling the liquid up (or down) is the surface tension times the length of the three-phase interface (contact line), so it's proportional to the circumference of the tube, which increases linearly with radius. N.B. This force acts in the direction of of the contact angle, which is where the cos() term comes in to extract the vertical component. $\endgroup$ – Michael DM Dryden Jan 6 '15 at 21:23
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According to the Jurin's Law

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

With :

$h$ the height of the fluid
$\gamma$ the surface tension
$\theta$ the contact angle with the tube
$r$ the tube radius
$\rho$ the liquid density
$g$ the gravitational constant

So, the $h$ is inversly proportional to $r$. In other words, the larger the tube, the lower the liquid climbs.

Note that this law is only valid if $r < \lambda_\text{c}$ with $\lambda_\text{c} = \sqrt{\frac{\gamma}{\rho g}} $

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  • $\begingroup$ This is a nice answer, but really only restates the observations stated in the question; that the meniscus is higher in smaller diameter tubes. Why is this so? $\endgroup$ – long Jan 4 '15 at 23:08
  • $\begingroup$ Agreed, long. Thanks for the reference to Jurin's Law, Babounet; I hadn't heard about it before. But I was asking specifically about the chemical explanation about why it is true that this phenomenon occurs. $\endgroup$ – user11629 Jan 4 '15 at 23:18
  • $\begingroup$ Capillary is more or less a chemical-physics topic. Not only about "chemistry" so it's hard to give a more "chemical" explanation :/ you can try to look after surface tension concept, it will help you I think. (You'll see why capillary act on the reverse way for Mercury for instance) $\endgroup$ – Babounet Jan 5 '15 at 21:30

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