Can Altitude Hypothesis Challenge the Second Law of Thermodynamics?

[D H]

Apologies to all.

I've done the nasty math and programming, and the constant temperature wins.A system in hydrostatic equilibrium that follows an adiabatic lapse rate does represent a local max in entropy, but a local max is not the same as a global max. A system in hydrostatic equilibrium with a constant temperature profile also represents a local max in entropy. For a vertical gas system with a fixed total energy and mass that is in hydrostatic equilibrium, the constant temperature configuration has a higher entropy than the adiabatic lapse rate configuration.

So why do we see a lapse rate in the atmosphere? First off, this state is a local max in entropy, and a rather strong one at that. That the troposphere is heated from the bottom / radiates out into space (real gases radiate; ideal gases don't) means that this local max is in many cases the favored state. Getting to the global max from this local max requires going through states that are disfavored entropically.

 

[striphe]

I've had a bit of a rethink about this and I'm not getting something.

T= (mv^2)/k

Naturally a particle will be affected by gravity, so as its velocity moves against gravity, it will slow and if ones velocity has it moving with gravity it will speed up. T= (mv^2)/k so changes in velocity result in changes of temperature.

How is the velocity even throughout the vessel if the particles are affected by gravity?

 

[klimatos]

I've had a bit of a rethink about this and I'm not getting something.

T= (mv^2)/k

Naturally a particle will be affected by gravity, so as its velocity moves against gravity, it will slow and if ones velocity has it moving with gravity it will speed up. T= (mv^2)/k so changes in velocity result in changes of temperature.

How is the velocity even throughout the vessel if the particles are affected by gravity?

Gas temperatures measure mean molecular kinetic energies of translation and not individual molecular energies. They are proportional to the mean of the squares of the individual axial velocities. These individual molecular energies follow a Maxwell-Boltzmann distribution. As long as conditions are isothermal, the mean values in any portion of the vessel will be the same as in any other portion. This isothermal condition requires that the incessant downward pull of gravity be matched by the average of the highly intermittent upward forces of thermal agitation.

By the way, in your formula m is the mean molecular impulse mass (not quite the same as the mean molecular mass for atmospheric air) and v is the root-mean-square of the axial molecular velocity, not the true-path velocity.

Since, under conditions of equilibrium, the mean axial velocity is zero, this makes v the standard distribution of the axial velocity distribution.

 

[striphe]

What are these thermal agitations?

 

[klimatos]

What are these thermal agitations?

The various molecular motions (translation, rotation, vibration, libration) usually subsumed under the concept of molecular kinetic energies.

Generally speaking, however, gas temperatures are usually considered to be functions only of the molecular kinetic energies of translation; i. e., T = mv^2/k. Here, Boltzmann's Constant (k) is the constant of proportionality that relates Kelvins to the KMS system of units.

 

[striphe]

Now to help me understand this, i will need a time based description of how this occurs in a particular instance.

Imagine a planet covered in an atmosphere of ideal gas. On this planet we have our super-long insulated upright tube, but at this moment, its a vacuum; nothing is inside of it.

To fill it with gas, the top cap is opened and gas flows into it. Once the pressure inside equalizes, it is re-sealed.

Now the fact that we have had gas move with gravity, the particles that are lower down, have gained more heat energy due to the conversion of potential energy, into kinetic energy and are hotter 'at the moment at least'.

now that it is left to become static how does temperature even out?

I find it hard to understand how, when a particle from a lower position, will loose energy to gravity when it changes location, to collide with a particle at a higher position.

 

[klimatos]

Now to help me understand this, i will need a time based description of how this occurs in a particular instance.

Imagine a planet covered in an atmosphere of ideal gas. On this planet we have our super-long insulated upright tube, but at this moment, its a vacuum; nothing is inside of it.

To fill it with gas, the top cap is opened and gas flows into it. Once the pressure inside equalizes, it is re-sealed.

Now the fact that we have had gas move with gravity, the particles that are lower down, have gained more heat energy due to the conversion of potential energy, into kinetic energy and are hotter 'at the moment at least'.

now that it is left to become static how does temperature even out?

I find it hard to understand how, when a particle from a lower position, will loose energy to gravity when it changes location, to collide with a particle at a higher position.

Through simple conduction of enthalpy (kinetic energy) and through thermal radiation exchange (if your ideal gas allows for it). Put two masses of air at different temperatures in physical contact and heat will flow from the warmer to the cooler. It will continue to do so until thermal equilibrium is reached.

Forget about individual molecular movements. It is the means that are significant.

 

[striphe]

every particle that moves up loses heat energy to gravity; as the mean velocity of a particular level has been determined by its fall, the average velocity of particles entering a particular level are going to be the same as the particles in that level.

em radiation would allow heat energy to move from one level to another without being affected by gravity in a classical sense, but the general theory of relativity says otherwise. Even light has gravitational potential energy.

 

[Delta Kilo]

I find it hard to understand how, when a particle from a lower position, will loose energy to gravity when it changes location, to collide with a particle at a higher position.

At equilibrium, the molecule will collide with more molecules on average when it is lower than when it is higher due to density gradient but the energy exchanged per collision will be the same on average.

Imagine a column of gas where thin layers are separated by weightless membranes. The molecules in each layer hit harder on the lower membrane than on the upper, but it is compensated by more molecules in the layer below so the net force on the membrane is zero.

 

[striphe]

Does anyone agree with the above statement?

 

[klimatos]

every particle that moves up loses heat energy to gravity; as the mean velocity of a particular level has been determined by its fall, the average velocity of particles entering a particular level are going to be the same as the particles in that level.

Since you seem to be hung up on gravitational changes to molecular velocities, let us approach the problem from that aspect. I am going to use examples from our own atmosphere rather than a hypothetical one because I am more comfortable there and that is where I have data at my fingertips.

At a temperature of 25° C and a pressure of 1000 hPa, the average air molecule will undergo some 3.18 billion collisions per second. The mean molecular speed along the vertical axis will be around 234 meters per second. A temperature differential of only 1° between two adjacent layers means an axial speed difference of some 40 cm per second. The gravitational acceleration or deceleration in one billionth of a second is infinitesimal.

Thus, thermal changes in molecular speeds of ascents and descents will far, far outpace gravitational ones.

 

[klimatos]

Does anyone agree with the above statement?

Delta Kilo properly stated the conditions for an ideal gas under conditions of equilibrium (except for gravity, of course). Keep in mind that both ideal gases and conditions of true equilibrium only exist in computer simulations and in the imaginations of scientists. That is what makes the study of atmospheric phenomena so interesting.

 

[RonL]

Does anyone agree with the above statement?

http://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics

http://en.wikipedia.org/wiki/Equilibrium_(thermodynamics )

There seems to be a big divide between these two categories ? in how people look at system evaluation.

I think I see where you are in this quest for understanding, I am still trying to get things clear in my mind and just in the recent past found this wiki page on non-equilibrium, I have often used Tornado's, hurricane's, and different temperature air masses to try and make some point, only to have some rebuttal based on an equilibrium condition ?

I'm still trying to absorb so much, with still a long way to go.

Ron

 

[striphe]

At a temperature of 25° C and a pressure of 1000 hPa, the average air molecule will undergo some 3.18 billion collisions per second. The mean molecular speed along the vertical axis will be around 234 meters per second. A temperature differential of only 1° between two adjacent layers means an axial speed difference of some 40 cm per second. The gravitational acceleration or deceleration in one billionth of a second is infinitesimal.
.

Now I would absolutely agree that, if two bodies of gas come into contact with each other (one on top of the other) and at their point of contact, that a 1C temperature gradient existed, that heat would transfer from the hotter to the colder.

But when it it comes to the example I there is going to be some distance between a level that is a particular temperature and a level that is one degree less. The dry adiabatic lapse rate according to wiki for air is 9.8C/Km.

 

[Delta Kilo]

You just have to choose whether you talk about real Earth atmosphere or some idealized model, and if it is the latter, then which one in particular.

My remark was for system in static equilibrium which is obviously quite different from adiabatic lapse rate condition. Real Earth athmosphere is yet another different thing.

 

[stevmg]

Simple explanation -

Air particles (N2, O2) move at whatever speed they do on the ground. That is reflected in the air temperature on the surface.. In order for them to get to a higher altitude, their kinetic energy is converted (somewhat) into potential energy. By reducing their kinetic energy the temperature of the gas mixture would drop somewhat (5 degrees Fahrenheit for every 1000 feet elevation.) Analogous to throwing a ball into the air. The kinetic energy which is its speed will drop as the ball goes up and will pick up as the ball descends.

Some of our idiot friends who shoot off their guns on New Year's Eve so-called harmlessly into the air forget "whatever goes up, must come down" and it will come down at almost the same speed it went up at (less air drag) and that bullet can still kill. We have numerous deaths in Miami every New Year's Eve because of that.

 

[stevmg]

Oh, by the way, what do they call the inter-molecular gravitational forces which would theoretically slow down the expansion of a gas into a pure vacuum? With an ideal gas there would be no loss in kinetic energy of the expanding molecules (because they would do no work) and the temperature would remain the same.

I do understand that radiant ("electromagnetic" energy) would deplete as to the square of the radius of expansion (like light does) but not the knietic energy of motion. Although, Einstein did relate kinetic energy to mass m = m0/[sq rt (1 - v^2/c^2)] which would alter things a bit.

Are these called "Newtonian forces?"

 

[striphe]

Stevmg, I can not see how the velocity of particles could be the same high up as they are down low within this closed system. Like a ball or bullet, a gas particle is affected by gravity.

When a gas particle moves by its own velocity from a lower position to a higher position within the closed system, kinetic energy is converted to gravitational potential energy and hence the velocity of the particle is reduced.

When a gas particle moves by its own velocity from a higher position to a lower position within the closed system, gravitational potential energy is converted to kinetic energy and hence the velocity of the particle has increased.

As I have said the only possibility of the closed system having no temperature gradient lies with em radiation.

As the the kinetic/gravity interactions would create a temperature gradient, a greater quantity of em radiation will be emitted from the lower particles than the higher particles due to their temperature difference, this will work against the highlighted effects of gravity.

I see it as being unlikely that the em radiation could neutralise the effects of gravity.

When we add upon a modern understanding of physics, photons that move from a lower position to a higher position, in a gravity field undergo red shift, which reduces their energy. Photons that move from a higher position to a lower position, in a gravity field undergo blue shift, which increases their energy.