Equalization of pressure in heat exchanger

Question

I was solving numericals on Kinetic Theory of Gases when I came across this question

Two closed vessel of equal volume contain air at 105 kPa, 300 K and are connected through narrow tube. If one of the vessel is now maintained at 300 K and other at 400 K what will be the pressure in the vessel?

I don't want the solution of this question but I want to know why the pressure will be same in both the vessel when one is maintained 300 K and other 400 K. The molecules in the vessel at 400 K will have greater kinetic energy and will exert greater pressure on the walls than the molecules of the other vessel.

Now one will say that some of the molecules of vessel at higher temperature will move to the other vessel so that the pressure remains same in both the vessels, but why is it so. Why can't the pressure in both the vessels be different?

Answer 1

Macroscopics

Imagine the tube is initially closed. Then one vessel is heated up. As $V$ and $m$ are constant: $$p=k\,T$$

So if $T$ is increased, $p$ is increased. As $p$ is increased, there will be a net flow of mass, until they are balanced. That's what we observe. Now, we can think of it microscopically.

Microscopics

Initially, there are the same number of particles in each vessel. When heating up there is an increase in velocity. So, in the high-temperature vessel the same number of molecules are with higher velocity. Imagine a plane in the middle of the narrow tube, the number of collisions per time, because of velocity, will be higher in $400\ \mathrm K \to 300\ \mathrm K$ direction. It indicates a net flow of mass. This flow of mass will equal pressures on both sides.

Answer 2

This experiment divides the container so that pressure (a "flow" of gas molecules against the walls) can be equalized thru the narrow tube but net flow of heat from hot molecules passing thru the narrow tube is seriously impeded (essentially does not occur because each section is maintained at 300 K or 400 K).

This non-equilibrium condition is held static by the external devices maintaining the set temperatures and the narrow tube which connects the pressures in the two "systems" but disconnects the heat flow.

To see that the pressures will eventually be the same, imagine connecting the two vessels at 300 K by the narrow tube, then quickly heating one to 400 K. The pressure will increase in the heated tube, and depending on the narrowness of the tube, will decrease as gas flows into the cooler tube (where the pressure will increase because there will be more molecules there, even tho still at 300 K). After some time, the pressures will equalize.

Answer 3

I'm going to answer this question in two parts: First, why will some of the molecules of the vessel at higher temperature move to the other vessel? Pressure is simply the force per unit area that the gas exerts. When a gas is sitting in a vessel, it exerts a force on the walls - and the walls in turn exert an equal an opposite force. Thus, nothing moves (on a macroscopic level) in the system.

If you open a tube to another vessel, then the gas in each vessel will be exerting a force on the gas in the tube. When both vessels are at equal temperatures and pressures, they exert equal and opposite forces, and the gas remains static (on a macroscopic level), and there is no net flow.

Heating up one vessel would increase the pressure due to the ideal gas law: $$pV=Nk_\mathrm BT.$$ Since the higher temperature vessel will have a higher pressure, it will exert more force on the gas in the tube, and the gas has to move away from it.

Secondly, why can't the pressure in both the vessels be different? The answer is that the system is closed, and the number of molecules is conserved - molecules cannot enter from outside the two vessels. Thus, molecules can't move from the hotter vessel to the cooler vessel forever - the hotter vessel would run out of molecules.

The critical part of the ideal gas law to keep in mind is the $N$, the number of molecules. In the hotter vessel, molecules will travel out, and $N$ will be reduced, as long as the pressure is greater than the cooler vessel. Since volume and temperature are constant, pressure must drop until the pressures are equal.