Re: Again with the Force
Chris,
PV = nRT, the ideal gas equation states that if you have 2 samples of the same gas, and the samples are of equal volume and equal number of moles, then the sample with the higher pressure must have the higher temperature, and vice versa. It's just an equality,
PV/nR = T, let V/nR = C be a constant
C*P = T, so pressure varies directly with temperature.
My original statement implied that V/nR was a constant between the 2 canisters of gas, otherwise my statement was useless (and I explicitly stated I meant they were constant in later posts). Again, I was just trying to show a relation between the internal temperature (and pressure) of a gas and its kinetic energy, leading to an increased force exerted on the walls of the canister.
uDevil,
That's the thing (about pressure). A gas will ALWAYS exert a pressure, and therefore a force, against the walls of any container that it is held in. Even if it is a huge room like in your perfume example. If there is always a pressure, there is always a force. And since they are unquestionably exerting some force on the walls of its "container", they must be exerting force on each other as well. This simply occurs during the collisions where momentum is transferred. You don't need a pressure *difference* just pressure (which you have). My analogy to a highly pressurized canister was used simply to "see" that there must be some force. Decrease the pressure by making the "canister" a large room doesn't change the fundamental idea, it just means there is less pressure and less force acting between the molecules.
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I don't see how the gas does work as it is not expanding against an applied force.
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There IS an applied force though! When those molecules are redistributing around the room, they are constantly being bumped into by the other air molecules in the room, whatever they may be. Even if you imagine the rest of the room as a vacuum, the molecules must move in a *net* direction to get from their tiny little space after being squirted out of the perfume bottle to the rest of the room. If they stayed put (their "natural" state) they would simply be vibrating in place (like they were in the canister or bottle) and you could say there is zero net force. However, when they are put out in the open, the momentum transfer between the molecules causes a net motion towards the unoccupied regions of the room. Remember, Newton's original formulation of his first law was F = dp/dt (p is momentum), so since we are clearly changing the momentum of the molecules as they are going from being stationary (vibrating in place means no net motion and it can thus be considered stationary) to moving across the room -- there must be a force.
For the expansion of a gas, Work = (integral) p dV , where you are integrating from the initial volume the gas occupies to the final volume. This follows directly from the definition of work = F.ds (I could show you the derivation if you like). The point is that ANYTIME a gas goes from a region of smaller volume to larger in the real world, it is doing work and therefore there must be some force. As I said above, there will always be some pressure exerted. And in your perfume example, the molecules are doing work on each other (and the other molecules in the room), as it redistributes throughout the room when its volume is expanding. Work is also equal to kinetic energy change, and when the molecules go from being stationary in the bottle to having a definite non-zero net direction, there must be a kinetic energy change, which means there is work done, and to do work you must apply a force.
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