What factors contribute to gas pressure in a confined space?

[nhmllr]

In chemistry class today, we learned how to determine atmospheric pressure using a mercury barometer, and the pressure of a confined gas using a mercury manometer. I understood how to used the equations fine, but I did not know where they came from. I thought it would be a relatively simple matter of deriving them (save the constants) but the more I thought about it the more confusing the problem became.

In class we were told that the pressure came from air being forced downwards due to gravity. Makes sense, I figure. However, in the case of a manometer,

The amount of gas pushing down on the mecury in the tube would change based on the size of the spherical container that the gas is contained (if the gas is the same density)

More volume --> More mass --> More force pushing down on the mercury

Pressure = Force/Area
There is now MORE force, but the same amount of area of exposure from the gas to the tube. If we go with this gravity theory, then pressure becomes arbitrary.

It seems as though heat should be involved too.
More heat --> More kinetic energy --> More pressure

In other words, I don't understand how gas pressure works. Intuitively, it seems to be a weird mix of gravity, heat, and properties of the container. My question is this:

If you know volume, density, mass, acceleration due to gravity, surface area of container, and the temperature of a gas, how do you calculate gas pressure, and how do you derive that equation?

 

[Ken G]

The force of gravity on air only applies to the atmospheric pressure, not the pressure in the glass ball. The glass ball protects the gas in the ball from changes in atmospheric pressure. (It would need to be pretty sturdy glass to protect from the average atmospheric pressure, so presumably the gas inside the ball is prepared under average pressure conditions.) So we are really talking about changes in the weight of the air, not the average weight of the air. Those changes manifest in changes in the force on the fluid in the open end of the tube, but not in the gas in the ball. So the fluid moves to compress the gas in the ball if greater force is applied on the open end. So the change in air pressure (due to the weight change) must be balanced by the change in pressure in the ball (due to the volume change) plus the weight of the fluid in the extra height delta h. I'm not really sure which of those two added terms dominates, maybe the glass ball tries to be large enough so that the volume change is insignificant. The fact that you use mercury suggests you want something that is liquid but has a lot of weight, so the weight in the excess height of liquid must be significant.

 

[bbbeard]

In chemistry class today, we learned how to determine atmospheric pressure using a mercury barometer, and the pressure of a confined gas using a mercury manometer. I understood how to used the equations fine, but I did not know where they came from. I thought it would be a relatively simple matter of deriving them (save the constants) but the more I thought about it the more confusing the problem became.

In class we were told that the pressure came from air being forced downwards due to gravity. Makes sense, I figure. However, in the case of a manometer,

The amount of gas pushing down on the mecury in the tube would change based on the size of the spherical container that the gas is contained (if the gas is the same density)

A mercury barometer is just like the system you've drawn except the trapped volume is a vacuum (ignoring for the small vapor pressure of mercury at room temperature).

A mercury manometer allows you to measure the pressure of a system relative to the atmospheric pressure. This is the so-called "gage pressure". (If you want the absolute pressure, you have to find atmospheric pressure with a barometer and add it to the gage pressure.)

In your drawing, the system is at a lower pressure than atmospheric. If you consider the pressure at the surface of the mercury in the tube on the left (open to the atmosphere), it must equal atmospheric pressure. Then if you look at the tube on the right at the same height, the pressure must also be atmospheric. The pressure in the column of mercury decreases as you go up the tube. ("Pressure increases with depth and decreases with height" is a sloppy way to put it.) The rate at which is decreases is ρg where ρ is the mass density and g is the acceleration of gravity. So if the difference in the height of the surfaces of mercury is Δh, the pressure difference is ρgΔh. The gage pressure of the system is -ρgΔh and the absolute pressure is P_atm-ρgΔh. If the column open to the atmosphere were higher than the column open to the system being measured, the sign would be reversed, assuming we always define Δh as the absolute value of the height difference.

BBB

 

[jetwaterluffy]

The amount of gas pushing down on the mecury in the tube would change based on the size of the spherical container that the gas is contained (if the gas is the same density)

Not really. Remember, above the tube, there is 50km of gas above it, pressing down on the gas inside the tube. If you seal the tube off from the outside (which kind of defeats the point of the barometer), and pump air out, the weight of air from above will crush the glass container.