Calculate Capillary Depression of Mercury in Glass Tube

[v_pino]

Homework Statement

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')

Homework 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.

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'.

 

[SteamKing]

Check this article: http://en.wikipedia.org/wiki/Capillary_action

You will see that for mercury, the meniscus is depressed from the surface of the surrounding mercury.

 

[CAF123]

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]

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.

 

[Nickie]

I would first check the units of all the values given and make sure they are consistent. In this case, it seems like the units for surface tension and density are in SI units, while the units for tube diameter are in millimeters. This could be causing an error in the calculation.

I would also double check the derivation of the equation for capillary depression to make sure all the variables and assumptions are correct.

Assuming that the units are corrected, the equation you have derived should give the correct value for the capillary depression. However, as you have mentioned, this value might be smaller than the tube diameter. In this case, it is possible that the mercury will not form a complete meniscus and the equation may not be applicable. In this case, it might be necessary to use a different equation or to make some assumptions about the shape of the meniscus.

To calculate the distance between the bottom of the meniscus and the maximum height of the fluid, you can use the equation:

h = R - Rcos(theta)

Where R is the radius of the tube and theta is the contact angle. This equation assumes a circular meniscus. However, if the meniscus is not circular, more complex equations may be needed to calculate the distance.

Overall, it is important to double check all the values and equations used, and to make sure they are applicable to the specific situation. If there are any doubts, it is always best to consult with a colleague or a mentor for clarification.