Mechanics behind capillary depression.

[harjyot]

As the question states I have hard time understanding that how does a liquid reach equilibrium in case of capillary depression (mercury for example ). I know how it goes down but not how it 'stops'.

like in case of liquids which have more force of adhesion ,as the liquid goes up because of the forces, it's balanced by it's own weight.
what balances the force pulling mercury down? details please?

 

[A.T.]

as the liquid goes up because of the forces, it's balanced by it's own weight. what balances the force pulling mercury down?

The weight of the mercury outside of the capillary?

 

[harjyot]

can you please elaborate a bit on it?

 

[Chestermiller]

Notice the angle that the water makes with the wall of the capillary compared to the mercury. Each fluid has its own characteristic contact angle. In the case of a wetting fluid like water, the contact angle is positive, and, in the case of a non-wetting fluid like mercury, the contact angle is negative. (Actually, the contact angle depends on the surface involved, and the presence of contaminants in the liquid). If the contact angle is positive, the fluid will rise, and if the contact angle is negative, the fluid level will fall. Do the force balance and see how it works.

 

[harjyot]

actually I tried the force balance but where I got stuck was that which force balanced the declining mercury in the capillary?

 

[A.T.]

can you please elaborate a bit on it?

Pushing the mercury in the tube even further down requires pushing the mercury outside of the tube even further up. Creating a level difference in a fluid is always opposed by the pressure difference from the differential weight of the fluid column.

 

[Chestermiller]

actually I tried the force balance but where I got stuck was that which force balanced the declining mercury in the capillary?

If you do a force balance at the air interface at the top of the mercury column in the capillary, and take into account the fact that the surface tension force acts as if a membrane were present over the top of the mercury, you will find that the pressure at the interface is discontinuous between the mercury and the air, and the pressure in the mercury just below the interface is higher than atmospheric. All you need to do is imagine that the surface tension acts like a stretched balloon surface between the air and the mercury.

 

[harjyot]

If you do a force balance at the air interface at the top of the mercury column in the capillary, and take into account the fact that the surface tension force acts as if a membrane were present over the top of the mercury, you will find that the pressure at the interface is discontinuous between the mercury and the air, and the pressure in the mercury just below the interface is higher than atmospheric. All you need to do is imagine that the surface tension acts like a stretched balloon surface between the air and the mercury.

I read that in a capillary containing mercury, considering a point a outside that and a point be in the capillary along the same line, there's a horizontal pressure difference and this makes it go down. HOW exactly?

 

[Chestermiller]

I read that in a capillary containing mercury, considering a point a outside that and a point be in the capillary along the same line, there's a horizontal pressure difference and this makes it go down. HOW exactly?

I have no idea what you are referring to, so I can't comment on this. Look, just think of the surface tension as a very thin stretched membrane over the interface between the mercury within the capillary and the air above. Consider the schematic in the figure of post #2 for the case of mercury. In this geometry, if there is a thin stretched membrane over the interface, it will act to force the mercury downward. If you use your hands to pull a knit cap down over your head, you will feel a downward force on your head.

 

[harjyot]

I have no idea what you are referring to, so I can't comment on this. Look, just think of the surface tension as a very thin stretched membrane over the interface between the mercury within the capillary and the air above. Consider the schematic in the figure of post #2 for the case of mercury. In this geometry, if there is a thin stretched membrane over the interface, it will act to force the mercury downward. If you use your hands to pull a knit cap down over your head, you will feel a downward force on your head.

look it at this way, as you're saying the surface tension acts downward in that case, fine I agree. now why don't u do this, break the surface tension into it's horizontal and vertical components. you will see that there's a net down Ward force. so in equilibrium, the mercury has some excess pressure to balance this downward extra surface tension. now, when the mercury goes down in a capillary, it's because of this surfaces downward component. but don't we see that it reaches equilibrium after a certain decrease.so obviously something balances this surface tension. so the simple Question ,how is it balanced

 

[Chestermiller]

look it at this way, as you're saying the surface tension acts downward in that case, fine I agree. now why don't u do this, break the surface tension into it's horizontal and vertical components. you will see that there's a net down Ward force. so in equilibrium, the mercury has some excess pressure to balance this downward extra surface tension. now, when the mercury goes down in a capillary, it's because of this surfaces downward component. but don't we see that it reaches equilibrium after a certain decrease.so obviously something balances this surface tension. so the simple Question ,how is it balanced

It's balanced by the higher pressure imposed from the column of mercury outside the capillary. The column inside the capillary is in communication with the column outside the capillary at the base of the capillary. Check out A.T.'s second post.

 

[harjyot]

okay folks, after racking my brains I think I know what's happening but I just wanted to see if my assumptions are right I'm looking at it this way.

a beaker contains mercury and a capillary is introduced. now due to surface tension and the convex meniscus, there's excess pressure at a point b just below the level of meniscus in the capillary. at the same level outside the pressure is say P. taking this as point a

pressure at point a = P
pressure at point b = P + 2*sigma/R where R is the radius of the meniscus.

so in order to balance this pressure, the mercury is pushed down till it reaches a point where the outside pressure too is P+2*sigma*R.

 

[Chestermiller]

Yes. That sounds right.

 

[harjyot]

yes. but now I think I found another doubt, now in the way that I'm imagining it, there can be two paradoxes.

1)what happens to the pressure of the air column. does it change? or is it just negligible?

2)when the liquid in capillary goes up/down doesn't the height of the liquid outside changes? or is it again too small for considerations.

 

[Chestermiller]

yes. but now I think I found another doubt, now in the way that I'm imagining it, there can be two paradoxes.

1)what happens to the pressure of the air column. does it change? or is it just negligible?

2)when the liquid in capillary goes up/down doesn't the height of the liquid outside changes? or is it again too small for considerations.

I will answer #1 with a question: What is the density of air compared with the density of mercury?

Answer to #2: What really matters is the difference in height between the mercury outside the capillary and the mercury outside the capillary, and not the absolute heights.

 

[harjyot]

I will answer #1 with a question: What is the density of air compared with the density of mercury?

Answer to #2: What really matters is the difference in height between the mercury outside the capillary and the mercury outside the capillary, and not the absolute heights.

You are an angel! :D haha thank you.