Zero-point energy question

[5614]

I'm asked to find the ZPE of 1m³ of air, assuming it is all nitrogen.

 

Here's what I did:

 

1) use PV=nkT to find the number of particles in 1m³

P = 1.01 * 10^5

k = Boltzman's consant

T = 298 K

and got n = 2.5 * 10^25 per m³.

 

2) I am given that the "spring constant" (this is all approximate) in nitrogen is s = 2240 N/m. Atomic mass is 14, reduced mass, μ, of N-N system is 7. Converting 7 from AMU to kg and using ω=√(s/m) I get ω=4.39*10^14.

 

3) I'm then told E=ħω/2. I stick in ħ and my value of ω, multiply by the number of molecules in a m³ of air and get the total ZPE in 1m³ of air is ~6*10^5 J.

 

If I made a mistake please let me know where. If I did not make a mistake then I've just said that the ZPE of 1m³ of air is sufficient to "boil" a human, and that doesn't sound right to me! Maybe I'm misunderstanding ZPE, but I can't see a mistake in my calculation. Thanks in advance!

[swansont]

How would you extract the energy to do the boiling, without violating the laws of thermodynamics?

[timo]

Apart from [math] \omega = \sqrt{s/m}[/math] which I cannot verify off my head and the minor inaccuracy that you forgot the units on omega, it looks fine to me. There is nothing wrong with the zero-point energy being big; you only measure offsets from that energy, anyways. Two possibly interesting findings that could be deduced from that result:

1) What would the mass-equivalent of this number be? It's probably little compared to the masses of the molecules.

2) If all of the nitrogen molecules were excited to the 1st excitation level, you'd have an additional energy of 2E. You could compare that to the average energy you expect for the gas (at room temp) and from that draw an estimate to what extent the molecules are vibrationally excited (my guess would be that it's little).

 

EDIT: vv "4.39*10^14/s" doesn't look too bad to me . In fact, I personally favor 1/s over [math]s^{-1}[/math] for frequencies.

[5614]

Ahh, so it is me misunderstanding ZPE.

 

I just looked at 10^5J and thought it were a massively high amount of energy to just be around us. But what you're saying, if I'm correct, is that yes there really is this massive amount of energy around us, and that it must stay around us because, well, it's ZPE, you cannot remove a fundamental lower limit of energy from a system.

 

Am I talking along the right lines? I guess this is what happens when you do questions ahead of when you get taught how to do them!

 

[edit] ah, me/Atheist posted at the same time. I just knew someone would comment on the no units for omega! Grr you! s^-1 looks messy when you're not using Latex!

[Klaynos]

[math]

\omega = \sqrt{s/m}

[/math]

 

Is correct.

 

rads s^-1

 

Don't forget the rads!

 

And urm yeah, can't extract it :'( despite what all the free energy people claim...

[5614]

Radians are dimensionless, you don't have to include it (no?). Besides, I don't lose marks for missing rads, therefore rads aren't required. QED.

 

I've heard all this rubbish about using and abusing ZPE, but I suppose I just didn't realise there was 10^5J of energy around us. Although of course it's ZPE, and not just "energy", and this is the critical difference that I didn't fully appreciate.

 

Cheers guys!

 

 

[edit] ˉ¹, hmm, didn't know there was a superscript minus sign, useful!

[Mr Skeptic]

A good reason to include radians is that it removes ambiguity (eg with cycles or revolutions), and it improves your chances of spotting errors when doing dimensional analysis (the making sure units match).

 

Oh, and don't worry about Klaynos, he's the most nagging member

[Norman Albers]

This is not clear!!! Your term ZPE is not useful to my thinking. Are you asking about the average kinetic energy of an equilibrium ensemble? This would be a fair question of Boltzmann statistics. . . . . . . .OK, I see you are ahead of me into the quantum mode description of ground-state energy. Vis-a-vis what, as others note.

[timo]

What he does is:

1) Assuming the two bound nitrogen atoms can oscillate, as if they were connected by a spring.

2) Describe the oscillation as a harmonic oscillation with V~x², where x is the deviation from the equilibrium state.

3) In a classical harmonic oscillation, you can have any energy state, including E=0 (which you get for x(t)=0).

4) Go to a QM description. In QM, the possible energy states are quantized to [math] E=\hbar \omega (n + 1/2), \ n \in \, \{0,1,\dots \}[/math].

5) Calculate the lowest possible energy for N (independent) of such oscillation (see it as the limit T -> 0 K if you want), where N is the number of nitrogen molecules you expect in 1m³ of air for common conditions.

 

It's supposedly just some homework to get used to do such calculations. I see little to no real point in it, except perhaps comparing the result to other energies (e.g. the energy necessary to boil a human and understanding why we're still not boiled by air or the examples I proposed).

[Norman Albers]

Thank you, Atheist, this is the realm of important confusion. It is clear that one cannot bail out a boat already underwater.

[Klaynos]

Radians are dimensionless, you don't have to include it (no?). Besides, I don't lose marks for missing rads, therefore rads aren't required. QED.

 

I've heard all this rubbish about using and abusing ZPE, but I suppose I just didn't realise there was 10^5J of energy around us. Although of course it's ZPE, and not just "energy", and this is the critical difference that I didn't fully appreciate.

 

Cheers guys!

 

 

[edit] ˉ¹, hmm, didn't know there was a superscript minus sign, useful!

 

try the _{[ sub ] [ /sub ]} and ^{[ sup ] [ /sup ]} tags!

[Norman Albers]

H. Puthoff wrote a paper describing how a presumed quantum fluctuation background of E&M radiation going as [math]\omega^3[/math] will be reflected by electrons and by quarks so that the whole system hangs together and dances as per the zitterbewegung of the electron. I am still puzzling over the significance of his revelations. He admits that the response does not depend upon the power relationship,and this is important to me.