Capillary height

v_pino

1. The problem statement, all variables and given/known data

Given the following information, derive and expression to calculate the capillary depression of mercury in a glass tube given that:

Contact angle = 140 degrees
Surface tension 'Gamma' = 0.476 Nm^-1
Tube diameter = 1.0mm
Density of mercury 'rho' = 13.58x10^3 kgm^-3 (ie. >> 'rho_0')

2. Relevant equations

I've derived the following equation for capillary, which I think is correct:

$$h=\frac{2\gamma cos \theta}{g(\rho -\rho_0)R}$$

So 'h' is the height of the mercury above the base of the tube.

3. The attempt at a solution

But this gives me h= -1.095x10^-5 m. This doesn't seem like a correct value for 'h' even if I take it as possible, because it's even smaller than the tube diameter.

Also, the equation was derived by taking 'h' as the height from base to bottom of meniscus. I'm guessing the capillary depression is the distance between the bottom of the meniscus and the maximum height of the fluid (ie. radius of the meniscus). How do I calculate this using the equation I derived? I can only get 'h'.

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CAF123

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Since you measured h relative to the base of the meniscus and mercury is depressed within the capillary, the negative makes sense, although I think the result is more negative than the number you got.

haruspex

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So 'h' is the height of the mercury above the base of the tube.
It's not, in general, the height above the base of the tube. It's the height above where it would have been were there no surface tension. E.g. if you push a narrow tube down into the surface of a wide reservoir of liquid, it tells you how much higher the level will be inside the tube than outside.
it's even smaller than the tube diameter.
Why does that bother you? The narrower the tube the bigger the effect.

"Capillary height"

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