What is pressure when there are no container walls?

[Edison Bias]

Hi!

I wonder what pressure is when there are no walls.

The reason for me wondering this is that gas pressure actually is defined by change of particle momentum, like

$$p=\frac{F}{S}=\frac{1}{S}\frac{dp}{dt}=\frac{1}{S}\frac{d2mv}{dt}$$

So how can there be pressure without change of momentum?

It's like Heisenberg almost because the pressure is only there when you measure it, when the walls are gone there is no pressure.

In my amateur world I would say that this wall-less pressure is made of intermolecular collissions, it has to be because there can be no pressure without collisions.

This equation can be shown to be derived from particles hitting a surface:

$$p=\frac{2}{3}nEk_p$$

where Ekp is the particle kinetic energy and n the particle density.

It is a very interesting equation, not to mention the fact that it is derived from particles actually hitting a surface.

It says that pressure actually is proportional to kinetic energy density, it is a fascinating description of pressure which's I have never understood but right now I think I understand because pressure is the sum of all the particle's kinetic energy in a certain volume where the quadratic velocities of the different particles is directly reversely proportional to the mass, due to

$$Ek_p=\frac{3}{2}kT$$

While the common gas law may be written

$$pV=n'RT=\frac{N}{N_A}RT=N\frac{R}{N_A}T=NkT$$

This is the equation many physisists rely on but remember these things:
1) Normal gases do interact/collide, Dalton's Law is a joke
2) Boyle's law relies on isothermal changes that are very smal with respect to To
3) The thermal expansion koefficient of volume and pressure may not be the same (at a larger scale).

Edison

 

[anorlunda]

You seem to have an analytical mind and a curious nature. Good for you.

Why be satisfied with a quickie answer on an Internet forum? I think you would enjoy the following and benefit from it. Leonard Susskind's video course on statistical mechanics. It not only answers your question, but probably the question after that and the next few. It shows you how the perfect gas law was derived. In addition, Susskind make it all entertaining.

Only 10 lectures, all available free on youtube. It sounds like you already have the math background needed.

 

[Jamison Lahman]

So how can there be pressure without change of momentum?

You answer this question yourself. There can't be pressure without change of momentum.

In my amateur world I would say that this wall-less pressure is made of intermolecular collissions, it has to be because there can be no pressure without collisions.

This equation can be shown to be derived from particles hitting a surface:

Why does the surface have to be a wall? Why can't it be another molecule? How would you measure pressure without some reference surface?

 

[Edison Bias]

It seems like we can agree on something, there can be no pressure without change of momentum.

This means that particles have to collide with either themselves or with some kind of wall.

If there is no wall, particles then have to collide with themselves.

I have above mentioned one way to calculate pressure using the fact that pressure is kinetic energy density (derived from change of momentum, though)

Personly I would prefere to calculate pressure without walls as being a function of particle collisions, with the assumption that there actually are lots of collisions within a gas of "average" particle density.

Let's try to calculate the the impact number, ns, which gives the number of collisions per time and area unit:

The infinitesimal volume is

$$dV=Svdt\cdot cos\theta...1$$

where theta is the angle with respect to the normal of the surface S.

The number of particles within this volume is

$$n(v, \theta)dvd\theta dV...2$$

but it can be shown (comparing space angles with full space angle) that this is equal to

$$n(v)dv\frac{1}{2}sin\theta dV...3$$

so that the number of molecules within the volume is

$$dn_s=n(v)dv\frac{1}{2}sin\theta Svdt\cdot cos\theta...4$$

Deviding this with dt and S gives the number of particles per time and surface unit, thus

$$n_s=\int_{-\infty}^{\infty}n(v)vdv\int_0^{\pi/2}\frac{1}{2}sin\theta \cdot cos\theta...5$$

where

$$\int_{-\infty}^{\infty}n(v)vdv=n<v>...6$$

where <v> is the mean velocity, it can further be shown that the angle-integral equals 1/4 so that the number of collisions per area and time unit is:

$$n_s=\frac{1}{4}n<v>\propto n<v>...7$$

Now, let's try to calculate mean free length för a particle to pass within a gas

The volume it consumes each second it "flies" is

$$dV=\frac{\pi}{4}d^2vdt...8$$

While distancies is measured centre-to-centre the area of the sweeping volume is actually

$$dV=\pi d^2 vdt...9$$

then the number of molecules wthin dV (and therefore subject to collisions) per unit time is

$$N=\pi n d^2 v...10$$

I've been told that this should be corrected with sqrt(2) due to Maxwell so let's put that in also

$$N=\sqrt(2) \pi n d^2 v...11$$

this is a probability function where N should not exceed unity for no collisions which means that the mean free length is

$$l=\frac{v}{\sqrt(2) \pi n d^2 v}=\frac{1}{\sqrt(2) \pi n d^2}...12$$

where v simply is shorted out due to not being relevant for mean free length.

Now we have two equations: 7 & 12.

7 tells us that the number of collisions on a surface per area and time unit is

$$n_s=\frac{1}{4}n<v>\propto n<v>...7$$

so if we know n and v we can calculate the number of collisions per area and time unit.

n of ordinary air may be calculatet by

$$n=\frac{p}{kT}...13$$

and using p=1,013E5 Pa and T=293K I have recently calculated n as around E25.

v may be calculated as

$$Ek=\frac{3}{2}kT=\frac{mv^2}{2}...14$$

thus

$$v=\sqrt{\frac{3kT}{m}}...15$$

where m can be estimated as 2/10*2*16mp+8/10*2*14mp, reflecting the fact that air consists of 80 nitrogen and neutrons weigh almost equal to protons.

mp=1,67E-27, putting this into the equation gives v=502m/s, thus 7 gives E27... per unit area...

so how wide is a proton?

I have never heard of that kind of data in a physics handbook, let's try to estimate it:

Lets' say that it's mass density corresponds to a known metal in the middle of the periodic table, say copper, the density of copper is 9g/cm^3=900kg/m^3~1000/m^3

We know that it weighs around E-27 kg and its volume is 4pi/3R^3~piR^3.

$$\rho=\frac{m}{V}...16$$

which gives

$$V=\frac{m}{\rho}...17$$

which gives R as E-10, giving a cross sectional area of some E-20, putting this in the equation 7 above gives ns=E7, thus the numer of particle-particle collisions per second is E7 in ordinary air.

The mean free length (equation 12) for air is

$$l=\frac{1}{\sqrt(2) \pi n d^2}...12$$

Using d as approximatelly R (i.e E-10) and n as E25 this gives 1um

Lot's of collisions, indeed :D

Edison

 

[Jamison Lahman]

It seems like we can agree on something, there can be no pressure without change of momentum.

This means that particles have to collide with either themselves or with some kind of wall.

If there is no wall, particles then have to collide with themselves.

I have above mentioned one way to calculate pressure using the fact that pressure is kinetic energy density (derived from change of momentum, though)

Personly I would prefere to calculate pressure without walls as being a function of particle collisions, with the assumption that there actually are lots of collisions within a gas of "average" particle density.

The pressure without walls is the same as with walls. It is easier to calculate with walls because it is an arbitrarily large surface, so we can easily calculate the average force per area of the gas. If you are familiar with Gauss's Law, the wall can be thought of as a Gaussian surface. There does not have to be an actual wall to use a wall to calculate the pressure.

so how wide is a proton?

I have never heard of that kind of data in a physics handbook, let's try to estimate it:

The classical radius of a proton is of the order E-18m (for context, the Bohr radius is is of order E-11m) but that is not the number you want. The protons are tightly bound in the nuclei of the gas particles. It is the gas particles that would be colliding therefore it is the size of the gas particles you would want.

 

[Edison Bias]

I just want to say that a radious for a proton equalling something around E-18 is astonishing!

Just for convenience:

$$R^3\approx \frac{m}{\rho}$$

Putting m=E-27 and rho=E3 gives R=E-10.

Observe that the mass density already is quite high, almost as high as Copper.

To reduce this proton radious to E-18 we will have to increase rho with a factor of close to one thosand.

Thus the proton mass density is almost 1000 times higher than Copper, that is quite dense!

Edison
PS
I did miscalculate the molecular width using one proton only, I should have considered at least a volume of 14 protons, too tired :)

 

[anorlunda]

Thus the proton mass density is almost 1000 times higher than Copper, that is quite dense!

Isn't that what you expect? Atoms (like copper) are made of protons, neutrons, electrons, plus lots of empty volume.

 

[Khashishi]

You are over-analyzing this.
Your title asks "What is the gas pressure when there are no power inputs (Watts)?"
The pressure depends on the temperature and density. The gas still has a temperature if you don't give it a power input.

The next question is "I wonder what pressure is when there are no walls"
Pressure is defined on an imaginary surface element. You don't need a real surface to calculate the pressure. It is a momentum flux passing through some hypothetical surface.

pressure is made of intermolecular collissions

No. Collisions are not necessary to have pressure. A gas of photons, for example, has a pressure and no collisions. If you put a wall in a box of photons, the photons will exert a force on the wall. But, you don't actually have to put the wall in there to define the pressure. It's imaginary.

 

[russ_watters]

You are over-analyzing this.
Your title asks "What is the gas pressure when there are no power inputs (Watts)?"
The pressure depends on the temperature and density. The gas still has a temperature if you don't give it a power input.

The next question is "I wonder what pressure is when there are no walls"
Pressure is defined on an imaginary surface element. You don't need a real surface to calculate the pressure. It is a momentum flux passing through some hypothetical surface.No. Collisions are not necessary to have pressure. A gas of photons, for example, has a pressure and no collisions. If you put a wall in a box of photons, the photons will exert a force on the wall. But, you don't actually have to put the wall in there to define the pressure. It's imaginary.

Minor quibble with that last paragraph; I think you meant collisions with walls are not necessary; collisions between gas molecules are.

But broadly, I agree; to @Edison Bias ; what is it you want to know? You seem to have kind of asked multiple questions that are only sort of related and then gone on some sort of stream of consciousness thought/math exercise. Could you please more concisely state what you are after?

 

[Jamison Lahman]

I just want to say that a radious for a proton equalling something around E-18 is astonishing!

As Rutherford described the gold foil experiment, it was "as if you had fired a 15-inch [artillery] shell at a piece of tissue and it came back and hit you." The atom is very spacious and the nucleus is very dense.

 

[Jamison Lahman]

No. Collisions are not necessary to have pressure. A gas of photons, for example, has a pressure and no collisions. If you put a wall in a box of photons, the photons will exert a force on the wall.

Hmm. I did not know that. The photons are still colliding with the wall, no, or does collision imply a mass interaction?
Edit: Per Wikipedia - Collision, "Collisions in ideal gases approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles which are deflected by the electromagnetic force." This is curious to me.
Edit: Per Wikipedia - Photoelectric Effect, "electron emission when collisions happen with higher energy photons."

 

[Khashishi]

Minor quibble with that last paragraph; I think you meant collisions with walls are not necessary; collisions between gas molecules are.

That's not what I meant, which is why I mentioned a photon "gas". Photons hardly collide with other photons, but still have a pressure.

 

[russ_watters]

That's not what I meant, which is why I mentioned a photon "gas". Photons hardly collide with other photons, but still have a pressure.

I'm not sure that they are equivalent. In either case, the difference between a gas an a really really fine pile of dust on the floor is the fact that the particles bounce around off each other with a certain momentum, causing the collection of particles to have a pressure. Gases need to be constrained by *something*, to do this, which is either a container or gravity, and with gravity it is molecules constraining other molecules.

I suppose you could describe a single molecule in a box as having momentum and therefore a pressure (very unusual), but again in order to have pressure it has to be constrained and bounce off the walls of the box. If not constrained by the walls or other molecules, it will just fly off into space and not have an identifiable control volume.

 

[Edison Bias]

Someone has changed my title, I want it back.

Edison
PS
To russ_watters I just want to say that I am trying very hard to understand gases, pressure and temperature because it interests me.

 

[Edison Bias]

I'm not sure that they are equivalent. In either case, the difference between a gas an a really really fine pile of dust on the floor is the fact that the particles bounce around off each other with a certain momentum, causing the collection of particles to have a pressure. Gases need to be constrained by *something*, to do this, which is either a container or gravity, and with gravity it is molecules constraining other molecules.

I suppose you could describe a single molecule in a box as having momentum and therefore a pressure (very unusual), but again in order to have pressure it has to be constrained and bounce off the walls of the box. If not constrained by the walls or other molecules, it will just fly off into space and not have an identifiable control volume.

I think this answer is interesting and addresses my (original) thread title. I have a really hard time grasping the concept of pressure, it is unclear. One way of vewing it, that is kind of easy to understand, is the fact that pressure is kinetic energy density, it is thus the sum of all particles with thermal speed per volume unit, inside this volume particles flies around with the thermal speed and this speed is dependant on temparature and mass of particle only (due to Ek=3/2kT).

So we have a situation where pressure is made out of particles flying around in a volume at a certain speed and particle density.

This is pressure.

The problem is that the formula used (p=2/3nEk) is derived from the assumption that there are hits on a surface, i.e there has been a change of momentum for the particles, I am questionizing if this formula is correct when there are no walls and therefore no change of momentum but perhaps inter-molecular collisions takes over to such a degree that it "replaces" the hits on a actual wall?

Edison

 

[russ_watters]

Someone has changed my title, I want it back.

I thought I watched the thread from the beginning and didn't see a title change; what title do you want?

 

[russ_watters]

The problem is that the formula used (p=2/3nEk) is derived from the assumption that there are hits on a surface, i.e there has been a change of momentum for the particles, I am questionizing if this formula is correct when there are no walls and therefore no change of momentum but perhaps inter-molecular collisions takes over to such a degree that it "replaces" the hits on a actual wall?

Yes; the math reflects the fact that there is no functional difference between the particles bouncing off each other and bouncing off a wall.

 

[berkeman]

Someone has changed my title, I want it back.

I thought I watched the thread from the beginning and didn't see a title change; what title do you want?

I changed the title soon after the thread was created, I think before there were any replies. The original thread title made no sense to me given the text of the first post, but if the thread has developed far enough now that the original title makes sense, we could change it back.

It was: 'What is pressure when there are no walls?'

 

[Edison Bias]

Yes; the math reflects the fact that there is no functional difference between the particles bouncing off each other and bouncing off a wall.

Interesting, so my statement is correct that the physics of it all changes from wall-impulses to particle-impulses just due to the fact that there are no walls anymore which we would need to measure pressure?

I have learned and above derived an equation that gives the probability for collisions per molecule which I repeat

$$a=\sqrt{2}\pi nd^2<v>$$

the number of molecules that collides per second are then

$$na=\sqrt{2}\pi n^2d^2<v>$$

and devided by two this gives the number of collisions and for ordinary air I have (using p=nkT) calculated n as E25 and v (using Ek=3/2kT) as around E3 and if d=E-10 for an air molecule this equals somewhere around E33 collsions per second.

Quite many collisions, thus there are obviously lots of momentum changes in gases, even without walls :)

On the other hand, if we add walls should not the pressure get even higher?

Edison

 

[Jamison Lahman]

Quite many collisions, thus there are obviously lots of momentum changes in gases, even without walls :)

On the other hand, if we add walls should not the pressure get even higher?

Edison

Calculating the number of collisions allows us to assume that the changes in momentum are frequent enough that the pressure is uniform over the whole volume. Adding walls gives a surface which we can use to find the exact pressure on that surface. The only difference with the walls is we no longer have to rely on the assumption that there a sufficient number of collisions. Furthermore, the pressure on the surface would be known, and then the pressure within the surrounding gas would be assumed given a high number of collisions.

 

[Edison Bias]

Interesting, thanks for that explanation.

Now I wish to try to calculate the pressure without walls knowing particle density and collision density per time unit only.

I will recalculate the number to try to get everything right (please tell me if I'm more than a magnitude wrong)

First we have (for ordinary air)

$$n=\frac{p}{kT}\approx \frac{E5}{E-23\cdot E3}...1$$

which makes n=E25 and this is the particle density in ordinary air at normal pressure and temperature.

The collision density per time unit is

$$n_{ct}=\frac{1}{2}\sqrt{2}\pi n^2d^2...2$$

Here we need to estimate d for an air molecule.

The mass for an air molecule might be estimated by

$$m=(2/10*16*2+8/10*14*2)m_p...3$$

where mp is the mass of a proton and we have 20 percent Oxygen and 80 percent Nitrogen, thus the mass per molecule is 29mp~E2mp.

Knowing from above that the radious of a proton is around E-18 we have a cross sectional area of about E-36.

Taking E2 times E-36 we have E-34 as area for the air molecule , thus E-17 might be considered its diameter.

Getting back to equation 2, we now have

$$n_{ct}=\frac{1}{2}\sqrt{2}\pi n^2d^2\approx E50*E-34=E16$$

So we now suddenly have only E16 collisions per volume and time unit.

The thermal velocity comes from

$$E_k=\frac{3}{2}kT$$

thus

$$v=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3kT}{29m_p}}\approx \sqrt{\frac{E-23*E3}{E-26}}=E6$$

Now to the exciting part, what pressure does this mean?

$$p=\frac{F}{S}=\frac{2mv/dt}{S}=\frac{2n\cdot 29m_p\cdot v}{n_{ct}d^2}\approx \frac{E25 \cdot E-26\cdot E6}{E16\cdot E-17}=E6$$

I can't believe it because this is only one magnitude from the truth (E5 Pa)!

Edison

 

[rcgldr]

As an example of pressure without container walls, consider a sphere of gas in space contained only by gravity.

 

[Edison Bias]

As an example of pressure without container walls, consider a sphere of gas in space contained only by gravity.

Interesting.

Let's take it closer to earth.

I was about to start a new thread about a related thought titled "What makes the atmosphere end?"

First, we may roughly estimate the hight of the atmosphere as

$$h=\frac{p}{\rho g}$$

using rho as 1, g as E1 and p as E5 we get E4 and thus 10km.

I don't know how far from the truth this is but it sounds reasonable, it sounds reasonable in spite of the fact that rho can hardly be the same all the way up to the end of the atmosphere.

So what happens up there?

The mass density (rho) will probably get less and less.

Another way to view the familiar pressure equation p=nkT is actually

$$p=\frac{\rho}{m}kT$$

But here we can see that temperature is mixed into this also so we can't blame a lesser pressure for lesser rho only, temperature is probably changing also.

And way up there pressure should equal zero, I think, because further out we have space and vacuum.

And the only way for pressure to equal zero is if rho (or n) is zero.

In other words, there may still be temperature there but... wait a minute, isn't temperature defined by speed of particles?

So how can there be temperature (I believe that the tempereture at the end of the atmosphere is not 0K) when there are no particles?

Edison
PS
What should end the atmosphere, I believe, is gravity but the way gravity works in this respect is unknown to me. One way of thinking is that, while 10km is so much less than 6370 km as I think the radious of the Earth is, the g-force is almost constant and working hard on the air molecules and while their temperaure and thus speed is low they do not have the momentum to get any higher, please correct me from this pathetic attempt in understanding :)

 

[rcgldr]

So how can there be temperature (I believe that the tempereture at the end of the atmosphere is not 0K) when there are no particles?

As the altitude increases, the temperature decreases part of the way, but then starts increasing as altitude increases beyond some point.

temperaure and thus speed is low they do not have the momentum to get any higher

Due to the randomness of collisions, some of the hydrogen molecules at the edge of the atmosphere achieve escape velocity.

 

[Jamison Lahman]

where mp is the mass of a proton and we have 20 percent Oxygen and 80 percent Nitrogen, thus the mass per molecule is 29mp~E2mp.

Knowing from above that the radious of a proton is around E-18 we have a cross sectional area of about E-36.

Taking E2 times E-36 we have E-34 as area for the air molecule , thus E-17 might be considered its diameter.

Molecules are most certainly not 10^-17m meters in diameter. Molecules (unless completely ionized) have a cloud of electrons. The molecular collisions we have been discussing are not with the molecules' nuclei. The inter-molecular repulsion is due to the electrons which is also how we define the size of the molecule.

You are certainly mathematically proficient and you have a genuine curiosity which is probably most important. As @anorlunda suggested, it might serve you more to get a textbook and go through that. Understand the underlying principles and then you'll easily be able to apply them to a plethora of scenarios.

 

[Edison Bias]

I am going through a textbook as we speek, what is wrong with thinking and trying to understand by myself along with the studies?

I also think that my estimation of the diameter of the air molecule probably was not the best, but please explain to me why the result was so "perfect".

Instead of telling me to study more by myself it would be nicer if you addressed what I am doing wrong, othewise it would seem like you are some kind of priest knowing it all without any intent of sharing.

I am on this nice forum to ask questions, socialize and get answers.

Edison

 

[anorlunda]

Instead of telling me to study more by myself it would be nicer if you addressed what I am doing wrong, othewise it would seem like you are some kind of priest knowing it all without any intent of sharing.

Actually, PF rules don't allow personal theories. That includes "I think X, what's wrong with that?" This thread is not bad, you're only slightly over the line IMO. But it is better to ask "what's right?" rather than "what's wrong with my reasoning?"

BTW, have you considered the expanding pressure waves of sound? Sound could not propagate in air unless molecules collided a lot.

 

[Edison Bias]

As the altitude increases, the temperature decreases part of the way, but then starts increasing as altitude increases beyond some point.

Very interesting to hear, so you say that temperature decreases part of the way (that is that the speed of the air molecules decreases) but then actually increases as altitude increses (thus the air molecules start to move faster again), this is extremely interesting, why does the air molecules start to move faster again and what is the temperature at the rim of the atmosphere?

At the rim there should be no air molecles, right?

This while temperature at the rim obvioulsy is not 0K, how do you explain that?

Due to the randomness of collisions, some of the hydrogen molecules at the edge of the atmosphere achieve escape velocity.

Firstly, you must mean Nitrogen, right? :)

Then I think what you say is interesting because as I understand it, some air molecules simply escape into space and are lost due to the randomness of the temperature at the rim of the atmosphere, I think I can appreciate that because temperature and momentum is a random process so it is likely that some air molycules reach escape velocity.

But what is escape velocity? Can the random rate of momentum change of an air mulecule at the rim of the atmosphere be higher than Earth can pull back? While the air molecule is then pushed by this force due to a random particle collision in the "right" way, gravitational force may not be able to pull it back, I think I understand this but please tell me if I am wrong.

Edison

 

[rcgldr]

Very interesting to hear, so you say that temperature decreases part of the way (that is that the speed of the air molecules decreases) but then actually increases as altitude increases ...

You can start with this wiki article, then follow the links.

https://en.wikipedia.org/wiki/Atmospheric_temperature

hydrogen ... escape velocity

Firstly, you must mean Nitrogen, right?

Another wiki article:

https://en.wikipedia.org/wiki/Atmospheric_escape

 

[Jamison Lahman]

I am going through a textbook as we speek, what is wrong with thinking and trying to understand by myself along with the studies?

It is not worth the time to create nor critique proofs void a basic understanding of chemistry, no matter how "perfect" the result. The cross-sectional area is off by 14 orders of magnitude.

I also think that my estimation of the diameter of the air molecule probably was not the best, but please explain to me why the result was so "perfect".

I have explained twice how to estimate the diameter of a molecule. There must be another poor estimation to offset your error calculating the area of a molecule.

Instead of telling me to study more by myself it would be nicer if you addressed what I am doing wrong, othewise it would seem like you are some kind of priest knowing it all without any intent of sharing.

Everyone should study more. I know I should. Don't mistake ignorance for stupidity. Besides, I think I have been quite nice trying to help you in my own, personal free-time. My point was only that it may prove more beneficial to learn the fundamental concepts before making elaborate proofs.

 

[russ_watters]

Interesting, so my statement is correct that the physics of it all changes from wall-impulses to particle-impulses just due to the fact that there are no walls anymore which we would need to measure pressure?

You don't need walls to have or measure pressure and having them or not having them does not change the pressure -- it is what it is (yes, that is circular).

On the other hand, if we add walls should not the pressure get even higher?

Why? Take a fish tank full of water and drop a plate down into it. Does anything change? Why or why not?

 

[Nidum]

On the other hand, if we add walls should not the pressure get even higher?

Interesting concept though . You could not only make a perpetual motion machine but you could actually build a machine that generated output power from fresh air .

 

[Dadface]

From my understanding of the question a wall is something, apart from other particles, that the particles can collide with. If walls are excluded from the system the question seems to be asking for the analysis of a thought experiment and not a real experiment for which the laws of physics apply. Even if an expression for pressure is somehow derived it would be impossible to verify this pressure by measurement because the introduction of any type of barometer into the system necessitates the introduction of walls into the system. In fact it's an extension of the thought experiment to assume that anything that enables observations to be made can be bought into the system.

 

[Edison Bias]

So it seems like the atmosphere ends due to gravity and thus air molecules having too little momentum to move any further, right?

Some seems to have it though because someone told me of escape velocity but I recon that it is a rare process (otherwise the atmosphere would be depleted after a while).

But how about the temperature?

Using

$$p=nkT$$

it tells us that when there are no particles, pressure is zero but it could also be zero if temperature is zero but we all know that that ain't true, so what is temperature at the rim of the atmosphere?

I think that temperature differs from 0K due to thermal radiation from the Sun (during daytime) and from the Earth (and it's atmosphere) during night, right?

So the temperature just outside the rim of the atmosphere, where there are no air molcules (obs), consists of thermal radiation, right?

But then there is a difference between temperature and temperature, so to speak.

Edison
PS
Please don't give me any links to read or tell me to go and read a textbook (which I am already doing, by the way), let's just dicuss this like grown and passionate human beings. By the way, I am a post graduate M.Sc studying physics on my own spare time because I am very interested, this means that I have no one (other than you skilled guys) to ask if I have problems understanding or wish to elaborate.

 

[Jamison Lahman]

But how about the temperature?

Using

$$p=nkT$$

Per https://en.wikipedia.org/wiki/Ideal_gas, "the ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size become important."