The Theory Of Mixed Gases Related To Specific Volumes Phase To Global Volumes

[Dubbelosix]

The Theory of Mixed Gases Related to Specific Volumes Phase to Global Volumes

 

 

 

 

In previous work, I looked into how gases might behave in the expanding universe.

 

[math]\frac{8 \pi G}{6}[\frac{\rho}{nk_BT} - (\frac{\rho}{nk_BT} + 3P(\frac{1}{nk_BT}))]\frac{\dot{T}}{T} = \frac{8 \pi G}{6}\dot{S}[/math]

 

Replacing

 

[math][\frac{\rho}{nk_BT} - (\frac{\rho}{nk_BT} + 3P(\frac{1}{nk_BT}))]\frac{\dot{T}}{T}[/math]

 

There is no formal difference between this and

 

[math]\frac{\dot{\mu}}{k_BT} = (\frac{p}{nk_BT} - \frac{\rho + p}{nk_BT}) \frac{\dot{T}}{T}[/math]

 

we get

 

[math]\frac{8 \pi G}{6}[\frac{\mu}{k_BT}]\frac{\dot{T}}{T} = \frac{8 \pi G}{6}\dot{S}[/math]

 

We can take small look at the physics of mixed gases in this model as well:

 

[math]\mu_B = -\frac{n_A}{n_B}\mu_A[/math]

 

[math]\frac{8 \pi G}{6}[\frac{\mu}{k_BT}]\frac{\dot{T}}{T} = \frac{8 \pi G}{6}\dot{S}[/math]

 

[math]= -\frac{8 \pi G}{6}[\frac{n_A}{n_B}\frac{\dot{\mu}_A}{k_BT}][/math]

 

This is for a constant entropy and temperature.

 

This term: [math]\frac{n_A}{n_B}[/math] is dimensionless, it is called the mole number fraction of the mixed phases.

 

If the specific volumes are [math]V_A[/math] and [math]V_B[/math] and [math]N_A[/math] and [math]N_B[/math] are the number of molecules are in mixed gas so that the total number of particles in a system is

 

[math]N = N_A + N_B[/math]

 

(see references)

 

Using this one can obtain a relation:

 

[math]V_D = \frac{N_A}{N}V_A + \frac{N_B}{N}V_B = x_AV_A + x_BV_B[/math]

 

This gives rise to a well known Lever Rule; the ratio of mole fractions is equal to the inverse ratio of the distance between the specific volues of a phase to the global specific volumes!

 

[math](x_A + x_B)V_D = x_AV_A + x_B V_B \rightarrow \frac{x_A}{x_B} = \frac{V_B - V_D}{V_D - V_A}[/math]

 

This physics can be implemented into the entropy equation for mixed gases in an expanding universe

 

[math]\mathbf{k}\dot{S} = -\frac{8 \pi G}{6}(\frac{V_B - V_D}{V_D - V_A})\frac{\dot{\mu}_A}{k_BT}[/math]

 

where [math]\mathbf{k}[/math] is the Einstein conversion factor. This entropy equation is now a structure which encodes the inverse ratio of the distances between the specific volumes of the phases to the global specific volumes.

 

But wait a minute! How is [math]\frac{x_A}{x_B}[/math] equal to [math]\frac{n_A}{n_B}[/math]? They are related, the [math](x_A,x_B )[/math] is related to Daltons law. From Daltons law, we have

 

[math]\frac{P_i}{P} = x_A[/math]

 

where [math]P[/math] is the pressure. The law is related to mole number ratio, by the mole ratio relation in the form

 

[math]\frac{P_i}{P_{tot}} = \frac{x_A}{x_B} = \frac{n_A}{n_B}[/math]

 

Being ratio's they are dimensionless and playing similar roles in the physics. I know this seems to be the case because being an ideal gas mixture, it can be expressed in terms of the components partial pressure or the moles of the component:

 

[math]x_i = \frac{P_i}{P} = \frac{n_i}{n}[/math]

 

Just to finish, going back to the entropy equation, the chemical potential divided by the energy [math]kT[/math] can be replaced with the Gibbs free energy for a mixed gas

 

[math](\frac{G}{T})_{P, n} = -S[/math]

 

And when you divide any entropy equation by [math]N[/math] it becomes the entropy per particle [math]-\frac{S}{N} \rightarrow -s[/math]. It is not that entropy is negative, its because [math]G[/math] is negative and so must change the sign of the entropy.

 

 

 

 

 

 

 

 

[ref]

 

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

 

https://www.uam.es/personal_pdi/ciencias/evelasco/master/tema_III.pdf

Edited June 22, 2017 by Dubbelosix

[Super Polymath]

I will admit that there may be a problem with a sign somewhere, because the initial negative sign was obtained

 

 

[math]\mu_B = - \frac{n_A}{n_B} n_A[/math]

 

 

for the theory of mixing gases. Then we obtained a second negative sign from the Gibbs free energy - technically two negative signs cancel out... what is odd is that the left hand side should be negative if [math]G[/math] is positive ie.

 

[math]-S = G[/math]

 

[math]S = -G[/math]

 

But I haven't proven that. I just assumed it. I am not aware of an equation where neither S or G are negative. I'll need to look into it.

That's okay, gravity/matter should take up space & space should be the absence of substance.

 

The negative is the taking away of quantity, it's not assuming a zero or negative quantity. 

 

Remember, the primary notion here is that one can always take another piece from the pie, so to speak. 

 

It's an equilibrium between material & non-material physics. On one hand, you have material in motion, on the other, you never get to the absolute smallest material because the material can always be reduced, so if you never get to the bottom of anything it's not really a material rather than it is simply the structure of reality. Infinite reality is of infinite material in infinite motion, the motion & material themselves are the result of the balancing out of a equation that's trying to solve for existence versus non-existence, what ends up happening is an endless Fibonacci sequence, the golden spiral. That's the inner concept behind the math anyway. 

Edited June 22, 2017 by Super Polymath

[Super Polymath]

Then we obtained a second negative sign from the Gibbs free energy - technically two negative signs cancel out... 

Don't forget about Newtonian Expansion when dealing with gravitational hydrodynamics.

 

They just recently confirmed that the velocity of gravity = c, so I believe the equations (Friedmann, Gibbs, etc) would need to be updated with Newtonian Expansion following that confirmation of gravity's velocity. 

Edited June 23, 2017 by Super Polymath