Pressure & Force: Gases, Liquids, Solids

[aspodkfpo]

Homework Statement

n/a

Relevant Equations

P=FA

Sorry if I'm asking too many questions, but I'm currently reviewing a textbook and I'm realising quite a few things that I'm conceptually unsure of, so I'm going to be spewing out questions.

If a bottle is shaken, dissolved gas comes out of solutions and builds up pressure inside a bottle.
As I understand it, Pressure = Force x area, individual molecules of a liquid and a gas are the same mass, but gases are more erratic in movement which results in more force. However, given that liquids and solids are tighter together, I hypothesise that the extra amount of particles in the given space helps to balance out the lesser movement thereby equalling the force. Now, if I had a bottle completely filled with water, would the pressure be similar to a bottle filled with gas or not?

Also, confused about how the buildup of dissolved gas results in the liquid rushing out as well.

Textbook also says that as the water rushes out, the bottle would go backwards. Makes sense according to conservation of momentum, but I can't quite understand the reasoning.

 

[etotheipi]

If a bottle is shaken, dissolved gas comes out of solutions and builds up pressure inside a bottle.
As I understand it, Pressure = Force x area, individual molecules of a liquid and a gas are the same mass, but gases are more erratic in movement which results in more force.

The gas just above the liquid in a sealed bottle of a fizzy drink is, in equilibrium, at the same pressure as the liquid at the surface. Both phases exert equal forces on each other!

Also, it is Force = Pressure x Area

 

[aspodkfpo]

The gas just above the liquid in a sealed bottle of a fizzy drink is, in equilibrium, at the same pressure as the liquid at the surface. Both phases exert equal forces on each other!

Also, it is Force = Pressure x Area

Oops, wrote the equation wrong.

What's the point of shaking the bottle to make it explode then, if liquid and gas produce the same effect?

 

[etotheipi]

For this effect, it might be helpful to look at Henry's law, i.e. the concentration of a dissolved gas is proportional to the partial pressure of that gas above the surface of the solvent. The constant of proportionality is $H^{cp}$ $$c = H^{cp}p$$When you release the lid, the gas already at the top of the bottle, which is at higher than atmospheric pressure, flows out of the bottle down a pressure gradient. As the partial pressure just above the solvent decreases, more gas (carbon dioxide) comes out of solution which causes the fizzing.

Then we need to ask why shaking the bottle makes it more 'fizzy'. To be honest, I'm not totally sure, but here's my guess!

To increase the radius of a bubble of gas by $\delta r$, you require energy $\delta E = 16 \gamma \pi r \delta r$, where $\gamma = \frac{\partial E}{\partial A}$ is the surface tension. The increase in volume of the bubble is $\delta V = 4\pi r^2 \delta r$, so the energy per unit increase in volume of the bubble is$$\frac{dE}{dV} = \frac{4\pi \gamma}{r}$$i.e. inversely proportional to the radius. This seems to imply that it is energetically favourable for gas to coalesce with existing, larger bubbles, than to produce brand new bubbles. Shaking the bottle would seem to bring more existing bubbles into contact with one another.

 

[haruspex]

What's the point of shaking the bottle to make it explode then, if liquid and gas produce the same effect?

You mean, because they are at the same pressure? On releasing the pressure, the liquid will only expand a tiny amount for its pressure to drop to atmospheric, so little work is done. The gas expands much more.

 

[haruspex]

why shaking the bottle makes it more 'fizzy'

It is because the shaking creates bubbles from the gas that had been above the liquid. Gas that had been dissolved in the liquid enters these and expands them.
See https://www.scientificamerican.com/article/why-does-a-shaken-soda-fi/.

 

[aspodkfpo]

You mean, because they are at the same pressure? On releasing the pressure, the liquid will only expand a tiny amount for its pressure to drop to atmospheric, so little work is done. The gas expands much more.

So ultimately this action of liquid going forth is due to a difference in pressure. Is there any specific law with regard to things going from high pressure to low pressure?
Why does shaking a bottle cause such a dramatic difference in the movement of the liquid? Is it due to the added force outwards as stated below in this comment, due to the compression of the liquid by the gas?

The gas just above the liquid in a sealed bottle of a fizzy drink is, in equilibrium, at the same pressure as the liquid at the surface. Both phases exert equal forces on each other!

Also, it is Force = Pressure x Area

For the bottle going forwards given that it is filled with liquid, I'm concluding that more force is acting towards the lid of the bottle due to the larger surface area at the bottom. As such the water goes forwards. If given a cylinder, nothing would happen.

For the gas and liquid part, the gas "presses" the liquid against the bottom, causing the liquid to exert a force on the bottom, resulting in a reaction force sending the liquid forwards when open?

 

[etotheipi]

For the bottle going forwards given that it is filled with liquid, I'm concluding that more force is acting towards the lid of the bottle due to the larger surface area at the bottom. As such the water goes forwards. If given a cylinder, nothing would happen.

I don't understand what you mean here; what does it mean for the bottle to be 'going forwards'? What is the 'force', of which there is more towards the lid, acting on, and what is its origin? What would change if the bottle is a cylinder?

 

[aspodkfpo]

I don't understand what you mean here; what does it mean for the bottle to be 'going forwards'? What is the 'force', of which there is more towards the lid, acting on, and what is its origin? What would change if the bottle is a cylinder?

Arrows on soda are force by bottle.

Their conclusion seems to be based off the soda leaving and the uneven force of the bottle which is in a similar situation as with the soda in the 1st image causing the bottle to go backwards.

For the cylinder scenario, the textbooks theory seems to also work properly.

First bottle is without lid open.

 

[etotheipi]

Arrows on soda are force by bottle.

If the lid is open, then the arrows at the top of the bottle are the force due to the atmosphere, on the liquid inside.

If you consider a bottle completely full of liquid, floating in a vacuum (for simplicity), then when you remove the lid the net force on the liquid will be one way and the net force on the bottle the other way, equal and opposite (think conservation of momentum). In the absence of gravitation the pressure in the liquid would be initially uniform (with lid on), so if you remove the lid, the pressure force on the liquid as a whole is now non-zero.

I don't know if that was what you had in mind...?

 

[aspodkfpo]

If the lid is open, then the arrows at the top of the bottle are the force due to the atmosphere, on the liquid inside.

If you consider a bottle completely full of liquid, floating in a vacuum (for simplicity), then when you remove the lid the net force on the liquid will be one way and the net force on the bottle the other way, equal and opposite (think conservation of momentum). In the absence of gravitation the pressure in the liquid would be initially uniform (with lid on), so if you remove the lid, the pressure force on the liquid as a whole is now non-zero.

I don't know if that was what you had in mind...?

Do you understand what the point of shaking the bottle is? And why doing so, causes the liquid to go out?

 

[etotheipi]

Do you understand what the point of shaking the bottle is?

See:

It is because the shaking creates bubbles from the gas that had been above the liquid. Gas that had been dissolved in the liquid enters these and expands them.
See https://www.scientificamerican.com/article/why-does-a-shaken-soda-fi/.

 

[haruspex]

It might help to start by understanding why water alone doesn't do this. Water molecules are polar, so they attract each other electrostatically. For those to separate enough to go their separate ways requires energy.

Dissolved gas molecules are bouncing around individually within the liquid matrix, not as bubbles. Some occasionally make it to the surface and do escape, but this is generally a slow process.

If a few gas molecules happen to be adjacent, i.e. an extremely small bubble, the surface tension in the bubble wall (due to polar attraction) can exert a very high pressure on it, stopping the bubble from expanding, and the gas molecules wander apart again. Only when the bubble reaches a critical size, depending on ambient pressure in the liquid, surface tension and temperature, can it keep growing.

For some of any gas, such as oxygen, to be dissolved in the water is normal. Although some leaks out, other molecules wander in, and the system stays in balance.
At high pressure you can get more of the gas to dissolve. When that pressure is released, it only leaks out slowly, as mentioned.

If there is an internal surface, e.g. a bubble that is tiny but large enough to stay around a while, some molecules leak into that, raising its pressure. That makes the bubble expand, increasing its surface area and hence the rate at which molecules leak into it.
Other things in the container, such as dust or surface cracks in the sides, can also act as nucleation sites. If you look at a glass of fizzy water you will see streams of bubbles rising from tiny flaws in the glass.

Shaking the bottle creates a lot of bubbles that are large enough to survive and grow, leading to a rapid escape of gas molecules from their watery matrix.
Now, if we were to somehow magically remove all the water from a (resting) bottle of soda, leaving only the the gas which had been dissolved, we would find it was at a much higher pressure than when the water was present. This is because the water matrix was effectively caging in the gas. So when we shake the bottle, releasing the gas from solution, the pressure in the container goes up.