What Effect Does Opening a Valve in a Vacuum Have on Air Molecules?

[PainterGuy]

Hi

Suppose there is a 1 m³ cylinder containing 1.3 kg air at 25 C. The cylinder is sitting in a vacuum place. The air inside the cylinder will exert a pressure of almost 1 atm. The cylinder also has a valve which can be opened to let the air out into surrounding vacuum.

The average speed of air molecules at 1 atm and 25 C is almost 450 m/s, and the speed of sound in air is 350 m/s.

When the valve is open, what would be the speed of escaping of air molecules? 340 m/s or 450 m/s?

Thank you for the help.

 

[PainterGuy]

Hi

About the previous post, the speed of escaping air molecules would be 340 m/s - the speed of sound.

Please have a look on the attachment. There are two thermally insulated chambers connected by an orifice and a valve separating them. Chamber 1 contains an ideal gas and chamber 2 is empty, i.e. vacuum. The speed of sound in this ideal gas is, say, 340 m/s.

I think of thermal energy which is also called heat as a random kinetic energy of individual molecules. I hope that my thinking right.

This is going to be my main question. When the valve is opened, the gas flows toward chamber 2 thru the orifice at the speed of sound. I remember that once my teacher said something like as follows. Before opening the orifice the molecules in chamber 1 have some thermal energy, E. As the molecules pass thru the orifice, their energy is unchanged but some of that thermal energy has been converted into the concerted kinetic energy of the gas flowing toward chamber 2, and this results into cooling of an ideal gas. Since the total energy is unchanged, the gas flowing through the orifice is cooled because of the concerned kinetic energy. And once the pressure is balanced in both chambers, the kinetic energy is dissipated, which means turned back into heat and the system ends up with the same temperature it started with.

I'm sure that my teacher was right but I don't see what happens at atomic scale. Let's suppose that the horizontal motion from left to right represents motion along x-axis. It means that when the valve is opened, the x-component of molecules make them to flow into chamber 2 and y- and z-components play no role and hence thermal energy along those directions is unaffected. In short, what really happens at atomic scale which lowers the temperature of an ideal gas as it passes thru the orifice. Could you please help me? Thank you.

 

[256bits]

When the valve is open, what would be the speed of escaping of air molecules? 340 m/s or 450 m/s?

For an individual molecule, neither.
One is the average speed of ALL the molecules in a region ie the 450 ms^-1
The other is the speed of sound in air at a particular temperature. ie the 340 ms^-1
The two speeds can be considered as being a bulk property of many molecules - too many to count - and not of individual molecules.

If one looks at an orifice and the molecules passing through, in very,very small time slices, one would see that the molecules passing through are only those with a trajectory in line with the orifice. ( The orifice could be thin so as to not have any wall friction effects and thus not itself be heated for example during the transition - we will go with that. ) The molecules passing through could be fast or slow ones with the fast overtaking the slow on the other side, or within the orifice depending upon the cross section. Eventually there will be enough molecules bouncing around on the vacuum side and colliding with each other.

The problem is that we do not have the time to count the molecules individually in very short time slices, so we revert to statistical mechanics or continuum mechanics to tell us what is happening.

See:
this is probably what you are interested in with an ideal gas in your OP without intra molecular forces.
https://en.wikipedia.org/wiki/Joule–Thomson_effect
Description
In a free expansion, on the other hand, the gas does no work and absorbs no heat, so the internal energy is conserved. Expanded in this manner, the temperature of an ideal gas would remain constant, but the temperature of a real gas decreases, except at very high temperature.[8

If one wants to go to individual molecular flow,
https://en.wikipedia.org/wiki/Knudsen_number
http://authors.library.caltech.edu/8988/1/NARpof60.pdf
( gets quite difficult )

If it is bulk flow, I would recommend studying the continuum mechanics explanations with formulas such as .

.

 

[Lord Jestocost]

a) The Joule-Thomson effect describes the temperature change of real gases owing to expansion.

b) The speed distribution of the effusing particles is not the Maxwellian equilibrium distribution, viz. faster particles get out with greater probability and are thus "favoured"; that temporarily leads to a temperature imbalance. EDIT: In case the orifice diameter is much smaller the the particle's mean free path.

 

[jartsa]

Hi

Suppose there is a 1 m³ cylinder containing 1.3 kg air at 25 C. The cylinder is sitting in a vacuum place. The air inside the cylinder will exert a pressure of almost 1 atm. The cylinder also has a valve which can be opened to let the air out into surrounding vacuum.

The average speed of air molecules at 1 atm and 25 C is almost 450 m/s, and the speed of sound in air is 350 m/s.

When the valve is open, what would be the speed of escaping of air molecules? 340 m/s or 450 m/s?

Average speed of escaping molecules? That must be 450 m/s, otherwise energy is not conserved.

Well, energy is conserved if one molecule gets the energy of all the other molecules, in which case average speed would decrease, but that is against the law that says that entropy must not decrease.

 

[PainterGuy]

Thank you, everyone.

b) The speed distribution of the effusing particles is not the Maxwellian equilibrium distribution, viz. faster particles get out with greater probability and are thus "favoured"; that temporarily leads to a temperature imbalance. EDIT: In case the orifice diameter is much smaller the the particle's mean free path.

So, you mean that temporary temperature imbalance would be created only if the orific diameter is much smaller than the molecules' mean free path which could be on nanometer scale.

Anyway, I'm still struggling to understand that why an ideal gas would temporarily cool down as passing thru an orifice. Could someone please guide me what really happens at atomic scale? I'm trying to get a conceptual understanding.

 

[Chestermiller]

Thank you, everyone.
So, you mean that temporary temperature imbalance would be created only if the orific diameter is much smaller than the molecules' mean free path which could be on nanometer scale.

Anyway, I'm still struggling to understand that why an ideal gas would temporarily cool down as passing thru an orifice. Could someone please guide me what really happens at atomic scale? I'm trying to get a conceptual understanding.

You're struggling because an ideal gas does not cool down in passing thru an orifice. In the scenario you describe in your original post, the gas cools down in the high pressure chamber, and is already cooler when it reaches the orifice. Then, if the orifice is insulated, it passes through at constant (cooler) temperature.

 

[Lord Jestocost]

Anyway, I'm still struggling to understand that why an ideal gas would temporarily cool down as passing thru an orifice. Could someone please guide me what really happens at atomic scale? I'm trying to get a conceptual understanding.

When considering gas effusion, there could be a slight, temporary effect regarding the reservoir (EDIT: a different case is supersonic expansion).

"In an effusive beam, since molecules effectively ‘wander’ out of the hole whenever they ‘collide’ with it, the Maxwell-Boltzmann distribution of the molecular speeds in the source is more or less maintained in the molecular beam. The distribution is actually somewhat skewed towards higher velocities, since molecules with higher speeds undergo more collisions with the walls and are therefore more likely to exit the hole. The velocity components are conserved in the molecular
beam, with the result that the beam has a broad cos2θ angular distribution, where θ is the angle between the molecular velocity and the beam axis (i.e. the direction normal to the wall of the chamber containing the hole)." Quoted from [PDF]Properties of Gases - Claire Vallance

 

[jartsa]

Thank you, everyone.Anyway, I'm still struggling to understand that why an ideal gas would temporarily cool down as passing thru an orifice. Could someone please guide me what really happens at atomic scale? I'm trying to get a conceptual understanding.

When one molecule goes through the small hole, a (under)pressure wave is emitted into the chamber.

Thermal energy is the only internal energy ideal gas has, so pressure energy of ideal gas is thermal energy of ideal gas.

So the aforementioned wave of low pressure is also a wave of low temperature. So the gas in the chamber gets cooler as molecules are escaping.

EDIT:
Hey don't believe the above, it's wrong Like for example decrease of thermal energy does not mean decrease of temperature, if the number of molecules decreases.

 

[Chestermiller]

When one molecule goes through the small hole, a (under)pressure wave is emitted into the chamber.

Thermal energy is the only internal energy ideal gas has, so pressure energy of ideal gas is thermal energy of ideal gas.

So the aforementioned wave of low pressure is also a wave of low temperature. So the gas in the chamber gets cooler as molecules are escaping.

I stand by what I said. For an ideal gas, there is no change in the gas temperature in passing through the valve.

 

[jartsa]

I stand by what I said. For an ideal gas, there is no change in the gas temperature in passing through the valve.

Well, I though I was saying the same thing as you. The gas in the chamber cools. When exactly does it cool according to you?If there was a filled balloon in the chamber, the balloon would expand when pressure in the chamber decreases. The gas inside the balloon would be doing work, and the gas in the balloon would cool, right?

 

[Chestermiller]

Well, I though I was saying the same thing as you. The gas in the chamber cools. When exactly does it cool according to you?If there was a filled balloon in the chamber, the balloon would expand when pressure in the chamber decreases. The gas inside the balloon would be doing work, and the gas in the balloon would cool, right?

Yes. I thought you were saying the opposite.

The gas within the high pressure chamber cools as it does work by expanding to push the gas ahead of it out of the chamber through the valve. The is basically an adiabatic reversible expansion.

Then, in passing through the valve, its temperature does not change. This is basically the net effect of expansion cooling and viscous heating in the valve. For an ideal gas, these two effects exactly cancel one another.

 

[PainterGuy]

Thank you, everyone. Your help is really appreciated.

First of all, I still think that the gas would enter chamber 2 at the speed of sound.

Please have a look on the attachment. With all due respect, I beg to differ where it is said that there is no temperature decrease as the molecules pass thru the opening or orifice. Yes, I would agree that the temporary temperature decrease affect all of chamber 1 including the opening section but the opening is where the effect is more prominent. For example, the molecules which are close to the left face of chamber 1 would be least affected as the valve is opened compared to the molecules which are close to the opening. As the valve is opened the molecules which happen to be in vicinity of the valve are free to rush toward chamber 2, and as they rush more molecules come to take their place and so on.

Please have a look on the attachment. Before the valve is opened, the molecules per unit volume, on average, have zero velocity. In other words, there is no preferred direction. But, as the valve is opened, all the molecules in opening section which connects chamber 1 with chamber 2 and which lies along x-axis have to have x-component of velocity. I don't know how to explain it but I believe that while passing thru the opening collision frequency (number of collisions per unit area per second) with the surface of opening section would decrease compared to what is was before the valve was opened. Although the equilibrium in entire chamber 1 will be disturbed temporarily until the pressure is balanced in both chambers.

Also the size of the opening (it's diameter or how wide the opening is) matters. As the size of opening decreases, the temporary temperature decrease effect would become more pronounced because the kinetic energy of molecules would need to be more concerted and this kinetic energy is gained at the expense of random thermal energy.

Thank you.

 

[Chestermiller]

Sorry, I can't relate to this at all. I'm a continuum mechanics guy. One thing I do know however is that a gas, in passing from the high-pressure side to the low-pressure side of an insulated valve or porous plug, experiences no change in enthalpy. For an ideal gas, this means no change in temperature. I also know that, for gas in a high pressure insulated chamber that is slowly escaping through a valve, the temperature of the gas remaining in the chamber decreases as a result of adiabatic (nearly) reversible expansion (to push the gas ahead of it through the valve).

 

[jartsa]

First of all, I still think that the gas would enter chamber 2 at the speed of sound.

Yeah. But speed of molecules would be higher. We have:

1: The speed of a molecule
2: The speed at which a molecule proceeds into a specific direction
3: The speed of gas

Speed in statement 1 is higher than speed in statement 2, if 1 and 2 are talking about the same molecule.

Speed in statement 2 and speed in statement 3 are the same, if the molecule in 2 is an average molecule from the gas in 3.

 

[jartsa]

With all due respect, I beg to differ where it is said that there is no temperature decrease as the molecules pass thru the opening or orifice.

I'll call the thing the molecules pass through "pipe".

So well yeah, molecules passing through the pipe are experiencing extra mild collisions with each other. Which causes the gas to emit same kind of thermal radiation as cool gas emits.

But molecules hit the pipe wall normally, so pipe does not cool.

 

[jartsa]

I'll call the thing the molecules pass through "pipe".

So well yeah, molecules passing through the pipe are experiencing extra mild collisions with each other. Which causes the gas to emit same kind of thermal radiation as cool gas emits.

But molecules hit the pipe wall normally, so pipe does not cool.

If we pick among gas molecules one molecule that has a large x-velocity, what is its probable y-velocity? Let's see ... x- y- and z- velocities are called "degrees of freedom", which probably means that x- z- and y-velocities are independent, so there's nothing special about the y-velocity.

The above reasoning is supposed to show that the molecules going through the pipe, which is aligned with the x-axis, have normal y-velocities.

But after a few collisions of molecules with each other the above is not true anymore, the average energies in the x- and y-degrees of freedom have become the same, and the gas cools the "pipe" that it goes through.

 

[PainterGuy]

Thank you, everyone.

First, let me clarify this that I'm playing a role of student and you of an instructor where your guidance means a lot to me. If I seem to be contradicting what you say, it doesn't mean that I'm trying to undermine your input. We all learn in different ways and this just happens to be my way of learning. Thank you.

@Chestermiller:
I believe that 'enthalpy' stands for the internal energy of gas. In case of an ideal gas, all the internal energy is in form of random kinetic energy. While passing thru the pipe which connects chamber 1 with chamber 2, there is no change in enthalpy or internal energy. It's just that some of the random kinetic energy is converted into concerted or directional kinetic energy.

I'm going to repeat what I said earlier. In my humble opinion I don't think that there is no temperature decrease as the molecules pass thru the opening or orifice. Yes, I would agree that the temporary temperature decrease affect all of chamber 1 including the pipe but the pipe where the effect is more prominent. For example, the molecules which are close to the left face of chamber 1 would be least affected as the valve is opened compared to the molecules which are close to the pipe. As the valve is opened the molecules which happen to be in vicinity of the pipe are free to rush toward chamber 2, and as they rush more molecules come to take their place and so on.

I'm sure that what you are saying would be correct too in a 'formal' way but I'm trying to look at the problem from an informal viewpoint where my focus is to observe what is happening at atomic scale in an intuitive way.

When the valve is opened, the molecules close to left face of chamber 1 won't be disturbed that much as compared to the ones which are closer to the pipe because the molecules close to left face have no sensory perception that they need to push the molecules forward on the other side close to the pipe. It's a kind of domino effect. As the dominoes close to the pipe start falling, the movement is triggered toward the left face of chamber 1.

@jartsa:
To some extent, I could see that there is a loophole in what I was saying earlier that the number of collisions per unit area per unit time would decrease. Anyway, let's look at the problem from a different view. Please have a look on the attachment. Suppose that there is a pressure gauge on the pipe. I believe that as the valve is opened, the pressure reading would drop.

Thank you.

 

[Chestermiller]

Thank you, everyone.

First, let me clarify this that I'm playing a role of student and you of an instructor where your guidance means a lot to me. If I seem to be contradicting what you say, it doesn't mean that I'm trying to undermine your input. We all learn in different ways and this just happens to be my way of learning. Thank you.

@Chestermiller:
I believe that 'enthalpy' stands for the internal energy of gas. In case of an ideal gas, all the internal energy is in form of random kinetic energy. While passing thru the pipe which connects chamber 1 with chamber 2, there is no change in enthalpy or internal energy. It's just that some of the random kinetic energy is converted into concerted or directional kinetic energy.

In typical flow through a valve, orifice, or porous plug, there is lots of flow resistance (even if the valve is in the open position), and the kinetic energy of the exit stream can usually be neglected. However, in the case of a nozzle that is aerodynamically designed to provide thrust (and, thus, with an average directional velocity on the order of the speed of sound), the increase in kinetic energy can be significant in reducing the temperature of the gas.

 

[jartsa]

To some extent, I could see that there is a loophole in what I was saying earlier that the number of collisions per unit area per unit time would decrease. Anyway, let's look at the problem from a different view. Please have a look on the attachment. Suppose that there is a pressure gauge on the pipe. I believe that as the valve is opened, the pressure reading would drop

Yes pressure would drop. In case of a very small hole and not very dense gas pressure would drop to half.

When the valve is closed, near the valve there are molecules moving towards the valve and the molecules that have been reflected away from the valve.

When the valve is open, near the valve there are molecules moving towards the valve but no molecules that have been reflected away from the valve.

The pressure is halved because the number of molecules is halved.

None of the molecules have done any work, so they move normally. Of course 'normally' should be understood correctly. The molecules hit the pressure meter normally.If we have dense gas and not very small hole, then a wind is blowing past the pressure gauge, or on it. I would guess the wind would have some effect on the reading on the gauge.