A container is divided into two parts by a partition containing a small hole of diameter D. Heli

A container is divided into two parts by a partition...

Key Concepts

Mean Free Path

The mean free path is the average distance a molecule travels between collisions with other molecules. It depends on factors such as the number density of the gas and the effective collision cross?section of the molecules. This concept is fundamental in kinetic theory and determines whether molecular interactions are more collision?dominated or influenced by boundary effects.

Knudsen Number and Transport Regimes

The Knudsen number is the ratio of the mean free path to a characteristic length scale (such as the diameter of an opening). It is a key parameter in determining the gas transport regime. When the mean free path is large compared to the characteristic length, the flow is in the effusive (or Knudsen) regime, while a small mean free path relative to the opening indicates a continuum (or hydrodynamic) flow.

Effusive Flow

Effusive flow occurs when the dimensions of an opening (such as a hole) are much smaller than the mean free path. In this regime, gas molecules pass through the opening without significant intermolecular collisions, and the net flux is determined by the molecular speed distribution and number density. This behavior follows directly from kinetic theory and is characterized by the individual, random passage of molecules.

Hydrodynamic (Continuum) Flow

Hydrodynamic flow, also known as continuum flow, takes place when the characteristic dimensions of the system are much larger than the mean free path. In this regime, the gas behaves as a continuous medium, and the transport of mass, momentum, and energy is governed by macroscopic conservation laws and diffusive or viscous dynamics rather than by individual molecular trajectories.

Transcript

00:01 So this is a fun little problem, combining some ideas from the previous chapters.

00:05 We have helium in two tanks.

00:08 We have 0 .06 moles over here, 0 .03 moles over here.

00:14 This helium is at 600 kelvin, and this stuff is at 300 kelvin.

00:19 And we're assuming that this is all adebatics, so there's no heat transfer outside of this system.

00:25 This is all inflated.

00:27 But we have heat transfer across this barrier here.

00:31 Let's see here.

00:33 This plunger here has a circular cross section with the diameter of 10 centimeters and a weight of 2 kilograms.

00:42 So this is the little piston there.

00:45 So that would tell us what the pressure is because if it's in equilibrium, we figure out the pressure.

00:51 So we know that because this whole system is insulated, we know that qa, the heat from here and going to the heat from here, have to be the same.

01:02 The heat in here and the heat in here have to be the same.

01:05 So that is, in this case, we have a constant volume, right? we have no, there's no change in volume, but there will be a change in pressure.

01:19 So we have the heat transfer from there is, and the number of moles in there, is the heat capacity constant volume times the change in temperature.

01:29 And again, we're assuming that this all equilibrates to the same temperature.

01:32 So we just have t2.

01:34 So we have t -a -1 here, which is 600 kelvin.

01:39 So in this chamber, we actually have a constant pressure process because this weight is just maintaining whatever pressure it is or whatever, how much this way is divided by the cross -sectional area.

01:54 So it can move, so we maintain a constant pressure.

01:59 So in this case, the heat coming in is the number of moles times the heat capacity.

02:04 You can't pressure times the change in temperature.

02:08 So we can have this equation and we know the heat coming out of here has to be the heat coming into here.

02:15 So we can then use this and we know everything in here except t2.

02:19 So we can solve for t2 and symbolically we get this and if we plug in the values we get that the equilibrated temperature of this stuff is 464 kelvin.

02:33 So that's the first question, final temperature.

02:36 So how much heat is transferred from the left side to the right side.

02:40 Well, we can just go back here and either look at one of these or one of these.

02:47 They'll obviously be the negatives of each other.

02:50 And so the heat transferred from the left to the right.

02:52 So what's coming in to be is this value.

02:57 And that is just plugging in numbers, now that we know the final temperature, that's 102 joules.

03:03 We have 102 joules of heat going in from there, from here to the hot to the cold...