Understanding Buoyancy & Its Causes

[TheLil'Turkey]

consider an open cylindrical container of water. Increase its temperature. Has the density at the bottom changed? Has the pressure?

Good question! Assuming it starts out at room temperature and you heat it from there, I'd say that the density at the bottom decreases (and therefore the average time that a water molecule travels between collisions increases), and the average speed of a water molecule increases. Furthermore, the density and speed change in just such a way so that the pressure stays the same.

 

[russ_watters]

Density decreases: correct.

For the rest, you are saying that the average time between collisions increases and their average energy increases and the two effects cancel out. Since the molecules do not break contact with each other, there are no collisions, per se, but otherwise, it is correct.

So you have just shown that pressure is not a function of compressibility in this case.

 

[chingel]

How would the increased density and decreased distance between molecules result in increased pressure?

A molecule is constantly being bulled down by gravitational force. For a molecule to stay roughly in it's position, the average force it hits the bottom molecules or gets hit by the bottom molecules has to be equal to the gravitational force (minus the average force it gets hit by the molecules above of course), it doesn't matter what the distance is.

If you put pressure on a molecule, it's mass doesn't change, it can apply more average force by hitting more often and/or with more speed. Probably both happen as the pressure increases in a liquid.

 

[TheLil'Turkey]

Density decreases: correct.

For the rest, you are saying that the average time between collisions increases and their average energy increases and the two effects cancel out. Since the molecules do not break contact with each other, there are no collisions, per se, but otherwise, it is correct.

If you would prefer to say something like "average time that a molecule travels before it changes direction" rather than "average time that a molecule travels between collisions" that's perfectly fine. In fact I'd appreciate it if someone could tell me what the standard terminology is.

So you have just shown that pressure is not a function of compressibility in this case.

This follows from what I've been saying all along. I kept temperature constant in my earlier explanations because I thought that way it would be easier to understand.

If you put pressure on a molecule, it's mass doesn't change, it can apply more average force by hitting more often and/or with more speed. Probably both happen as the pressure increases in a liquid.

My hypothesis is based on my assumption that increased pressure is due (at least almost entirely) to the increased frequency of the collisions if the temperature is constant. That frequency is directly proportional to the the average distance a molecule travels before it changes direction, and as that distance decreases, density increases. I assume that the molecules only gain speed if you increase the temperature.

 

[Doc Al]

My hypothesis is based on my assumption that increased pressure is due (at least almost entirely) to the increased frequency of the collisions if the temperature is constant. That frequency is directly proportional to the the average distance a molecule travels before it changes direction, and as that distance decreases, density increases.

If your hypothesis were true, then density would be proportional to pressure. But as has been stated several times, the density of fluids such as water doesn't change much even when pressure changes greatly.

Yes, when fluids are under increased pressure the molecules are forced together a bit. But it doesn't take much to resist the increased pressure, due to strong short-range repulsive forces between the molecules. That's why the bulk modulus of water is so high.

 

[sophiecentaur]

So in your own, personal atomic model, the molecules in a solid or liquid are perfectly still regardless of the temperature?

What you say, implies that hydrostatic pressure should be a (first order) function of temperature?

 

[sophiecentaur]

If your hypothesis were true, then density would be proportional to pressure. But as has been stated several times, the density of fluids such as water doesn't change much even when pressure changes greatly.

Yes, when fluids are under increased pressure the molecules are forced together a bit. But it doesn't take much to resist the increased pressure, due to strong short-range repulsive forces between the molecules. That's why the bulk modulus of water is so high.

I agree. Modelling a liquid as a gas will not get anywhere useful.

 

[russ_watters]

If you would prefer to say something like "average time that a molecule travels before it changes direction" rather than "average time that a molecule travels between collisions" that's perfectly fine. In fact I'd appreciate it if someone could tell me what the standard terminology is.

The molecules are vibrating, but are always in contact with each other. That's the key that you're missing.

This follows from what I've been saying all along.

No, it doesn't:

My hypothesis is based on my assumption that increased pressure is due (at least almost entirely) to the increased frequency of the collisions if the temperature is constant.

I know - and it is still wrong. Again:

1. There are no collissions since water molecules are in constant contact with each other and compression has no effect on the number of molecules in contact with each other.
2. The oscillation of the molecules due to their temperature does not affect how much force they apply to each other.

That frequency is directly proportional to the the average distance a molecule travels before it changes direction, and as that distance decreases, density increases. I assume that the molecules only gain speed if you increase the temperature.

In determining the pressure in an open container, it just plain doesn't matter how fast they are vibrating, as the example with temperature showed. Think of the molecules as springs with masses stuck in the middle of them. Place one of these spring-masses on a table and it applies a certain force. Set it oscillating and it applies a variable force, but the average force doesn't change. Set it oscillating faster and the average force still doesn't change. The oscillation just doesn't have anything to do with the average force.

Then do it again, but this time push down on the top of the spring. The force applied to the table increases. Set the spring-mass in motion: the average force remains at the new value, but the oscillation frequency is higher than it was without you pushing down on the spring. So what caused what? The push-down on the top of the spring both causes the frequency to change and causes the force on the table to change. The frequency does not cause the force to change.

[snip] I kept temperature constant in my earlier explanations because I thought that way it would be easier to understand.

Indeed - but by putting temperature back in, you proved that compression/expansion is an effect of temperature change, not a cause of pressure change. Compression/expansion is an effect of pressure change, but the inverse is not true: pressure change is not an effect of compression/expansion in this case.

Just to make sure we stay grounded to the OP, here's the first sentence:

I think buoyancy is caused by the increase in density with depth.

In other words: density increase causes pressure increase, causes buoyancy increase.

Trouble is, even for a hypothetical incompressible fluid, pressure is still a function of depth. How can A be the cause of B if eliminating A doesn't eliminate B?

 

[TheLil'Turkey]

If your hypothesis were true, then density would be proportional to pressure. But as has been stated several times, the density of fluids such as water doesn't change much even when pressure changes greatly.

Did you even read the OP or post 13? The bolded couldn't be more incorrect. If my hypothesis were true, then the change in density would be proportional to pressure.

And looking at the graph at http://www.engineeringtoolbox.com/fluid-density-temperature-pressure-d_309.html, the change in the density of water is about -0.05 kg per cubic meter per bar at constant temperature. I think that with this graph, everyone can now agree that I'm right.

 

[TheLil'Turkey]

I agree. Modelling a liquid as a gas will not get anywhere useful.

While true, that has nothing to do with this thread. A gas is modeled with the assumption that the volume of the molecules is 0, where as in my model, the radius of a molecule is assumed to be huge relative to the average distance it will travel before changing direction.

 

[Doc Al]

Did you even read the OP or post 13? The bolded couldn't be more incorrect. If my hypothesis were true, then the change in density would be proportional to pressure.

Did you read what you actually wrote that I responded to? (You're bouncing all over the place.)

 

[TheLil'Turkey]

Did you read what you actually wrote that I responded to? (You're bouncing all over the place.)

I assume you're talking about this

as that distance decreases, density increases

This does not imply that the density is directly proportional to the pressure. Because the volume of a molecule is not zero, it implies that the change in density is directly proportional to the pressure. At this point I can only recommend that you read or reread all my posts in this thread.

 

[sophiecentaur]

But what has any of this bickering got to do with a simple explanation of hydrostatic pressure? Water is not incompressible BUT, when you fall in the water and you float, its compressibility has no measurable effect on the depth to which you sink anddoesn't need to be brought into the argument. There is a very simple explanation which involves a linear increase in pressure with depth.

Detailed Gas models do not assume point molecules. They take into account, the intermolecular forces just as models of liquids do. It just so happens that the modulus of a gas is very much lower than that of a liquid. But, except in World-scale (/extreme) situations, the variations of density of water is hardly relevant.
The OP is clearly not correct because there is a better and more accurate explanation. Arguing that the density variation is relevant just seems to me to show a reluctance to acknowledge - being based on the ideas used in the gas model. Sometimes arm-waving has to yield to accepted and reliable (appropriate) models.

 

[TheLil'Turkey]

How does the vibration frequency of molecules in a liquid or solid vary with pressure? The answer to that determines if I'm right.

 

[sophiecentaur]

For a linear force law will the frequency of vibration be independent of the pressure. (Simple harmonic oscillator theory). For moderate pressures, the force law will be linear.

But how is this at all relevant or how does it justify the contents of the OP, which don't make sense? You are using a model that just doesn't apply to 'condensed matter'. You seem to be hung up on a version of 'the Kinetic Theory of Gases' and it doesn't fit the situation. If you are looking for a Classical Model then a network of masses, separated by springs, is a far better model than point (/small radius) masses flying around at high speed for most of the time (as in a gas). The volume in a gas is pretty much directly proportional to the Temperature under constant pressure - not so for a liquid. The RMS movement of molecules in a liquid is about the same as for a gas but the mean free path is only about the diameter of a molecule (a 'springy ball'). The frequency of oscillation about the mean position will be the same over a large range of pressures (/individual forces). What counts is the mean force, which relates just to the weight of liquid above. This mean force is applied to the sides of the immersed object.

Let'f face it. That's what we can actually observe every day!

 

[TheLil'Turkey]

If you are looking for a Classical Model then a network of masses, separated by springs, is a far better model than point (/small radius) masses flying around at high speed for most of the time (as in a gas).

I've been advocating what you describe here as "a Classical Model" with one difference: that the velocity of the water molecules is constant between "collisions." Thinking about it for a moment, I'd guess that the spring model is closer to reality than mine, but they both make the same prediction which is point of this thread: that pressure and density are inextricably linked in 1 substance.

From my OP:

I crudely visualize my model as water molecules being huge (and in constant motion) with tiny spaces between them. This would mean that a tiny percentage increase in density would lead to an enormous percentage decrease in the average distance from the "edge" of one molecule to the "edge" of another.

 

[TheLil'Turkey]

I thought about the classical model of condensed matter sophiecentaur described since it seems more realistic than mine. One way in which the 2 models differ is that the classical one predicts that the frequency of "collisions" doesn't change with pressure, but that the force of the "collisions" increases with pressure (if I'm understanding it right). This is the opposite of what mine predicts.

 

[sophiecentaur]

The liquid state is a weird, mid point between the vapour and solid phases. For most substances (?), where the pressures are not (planetary core) extreme the temperature range in which they exist as liquid is pretty limited. Pure metals have very narrow temperature ranges in the transition stage from one state to another.

Molecular separation is much the same in a liquid as in the solid so I should think that a solid-like model would be more likely to apply so a liquid would be more likely to behave like a 'very flexible' solid than a 'highly compressed' gas.

The term "force of collisions" doesn't go down well with me, I'm afraid. You need to talk of Momentum Change (Impulse) or Kinetic Energy because when does this force of yours apply?

I wish you'd comment on my query about the OP which seems so wrong that the rest of this thread can't be resolved without sorting out the issue.

 

[TheLil'Turkey]

I wish you'd comment on my query about the OP which seems so wrong that the rest of this thread can't be resolved without sorting out the issue.

I assume you're talking about this:

But how is this (referring to the relationship between the pressure and frequency of the molecules of a liquid or solid) at all relevant or how does it justify the contents of the OP, which don't make sense?

My model predicts that frequency increases with pressure. The classical model predicts that they're not related. I suspect that the latter is correct, but could someone please provide some data?

The term "force of collisions" doesn't go down well with me, I'm afraid. You need to talk of Momentum Change (Impulse) or Kinetic Energy

Ya, I should've used the term impulse.

 

[sophiecentaur]

No I was referring to the very first statement:
"I think that buoyancy is caused by the increase of density with depth (the deeper you go, the more molecules there are per unit volume). Therefore an object in a fluid will be hit by more of the fluid molecules from below than from above (even if the difference is only a tiny fraction of 1%). Is this correct?"

This cannot be right because the density increases at a minuscule rate with depth (incompressible, almost) yet the pressure is directly proportional to depth.

Also
"My model predicts that frequency increases with pressure. The classical model predicts that they're not related. I suspect that the latter is correct, but could someone please provide some data?"

I have already explained why the classical model predicts what it does.
The 'spaces between' the molecules that are present in a gas (where your kinetic ideas do apply) are not present in a liquid. In a liquid you just have (classically) coupled oscillators.If you have an almost linear force / displacement law then the frequency of oscillations will not change if there is an overall increase in that force - the oscillators just operate around a different mean point. (In the same way that a mass on a spring will oscillate at the same frequency when hanging down, standing up or in zero gravity).

 

[TheLil'Turkey]

No I was referring to the very first statement:
"I think that buoyancy is caused by the increase of density with depth (the deeper you go, the more molecules there are per unit volume). Therefore an object in a fluid will be hit by more of the fluid molecules from below than from above (even if the difference is only a tiny fraction of 1%). Is this correct?"

This cannot be right because the density increases at a minuscule rate with depth (incompressible, almost) yet the pressure is directly proportional to depth.

As I explained before, the model I proposed predicts the bolded. My second statement in the OP explains this. But another thing it predicts (which I now understand is wrong) is that the vibrational frequency of a condensed material in real life increases with pressure (for normal pressures).

I have already explained why the classical model predicts what it does.

I was talking about a real condensed substance, not a model. But I found something just as good: a graph of intermolecular attraction vs. distance between the centers of molecules. What was wrong with the model I proposed was that I imagined the distance axis of this type of graph being severely compressed.
edit: One thing that should have tipped us all off that my initial model is wrong is that if it were true, condensation would be impossible - all matter would be gas. I didn't realize it made that prediction until right now.

Now I finally feel happy with my understanding of pressure and density and I also see how Hooke's Law fits into the classical model of condensed matter. Even though this took so long, it was definitely worthwhile for me :)