- Joined
- May 26, 2019
Time for Thermodynamics Sperging (a personal favorite.) I am assuming you want an at-least realistic description of what happens when these two volumes are suddenly contacted across a discrete plane made up of one face a piece of these cubes described in the problem.
I wrote a python script that does these calculations out for me because I am a lazy engineer who is shit at actual Computer coding but uses it to get his mathematics done for him.
Assumptions: Using Van Der Waal's Equation will account for molecular forces in a manner sufficient to assume ideality in the behavior of chemical potential.
no reactions
final Volume is fixed
Cube A, Cube B, and the equilibrium state are well mixed.
So, in short, it's about 16 second before thermal equilibrium, thus all that happens with regard to the questions in OP is that heat and mass are exchanged. Within that kind of time frame, not much really gets accomplished, with the molar fraction of Cl2 reaching something on the order of 0.003, on its way to 0.04 It will then probably take something like a few hours to reach compositional equilibrium
There might be a reason you've been waiting 6 months, @Null.
Perfectly understandable.
3 things happen right off the bat
1. because of the greater mass, temperature, and thus pressure, there is immediately a movement towards momentum equilibrium. With a difference in Pressure of 0.05 bar, this kicks off the process of the system going to equilibrium with an impulse of momentum from Cube B to Cube A as these discrete volumes contact. This is done within microseconds and due to the chemical nature of diatomic gases, is really a minor event. Some work is done on Cube A's volume by cube B, but as we are assuming that the volume remains constant in the combined system the overall this work is negligible. Much of the Pressure change which describes the work done to equilibrium, however, is more the result of temperature change and mass transfer than the adiabatic compression comprising the work term resulting from the higher pressure. This is justified by the change in molar volume from intial state to equilibrium state being about 4% in total, indicating that compression of the gases is not a significant contributor. Thus, I am focusing on the heat transfer, which is coupled with the mass transfer.
2. Going along with the theme of a system moving towards equilibrium, heat and mass then immediately migrate from cube B to cube A. This happens simultaneously and the motion of both the heat and mass are linked together. This regime dominates everything that the problem cares about until thermal equilibrium is achieved in the system.
3. After thermal equilibrium is established, there will be some more mass diffusion that will then occur, soon achieving equilibrium with regards to mass as well
Papa Adolfo done sperged out calculating out what happens with 2 & 3 because event one is pretty minor. The
TL : DR answer to the question is that primarily, heat and mass are exchanged between the discrete volumes A&B until equilibrium is achieved. Chlorine, being the only component unique to B will be the primary species whose mass transfer actually matters, evrerything else is essentially just a transfer of heat and momentum. This particular sperg thus mostly focused his calculations on Chlorine and how it acts in the system.
3 things happen right off the bat
1. because of the greater mass, temperature, and thus pressure, there is immediately a movement towards momentum equilibrium. With a difference in Pressure of 0.05 bar, this kicks off the process of the system going to equilibrium with an impulse of momentum from Cube B to Cube A as these discrete volumes contact. This is done within microseconds and due to the chemical nature of diatomic gases, is really a minor event. Some work is done on Cube A's volume by cube B, but as we are assuming that the volume remains constant in the combined system the overall this work is negligible. Much of the Pressure change which describes the work done to equilibrium, however, is more the result of temperature change and mass transfer than the adiabatic compression comprising the work term resulting from the higher pressure. This is justified by the change in molar volume from intial state to equilibrium state being about 4% in total, indicating that compression of the gases is not a significant contributor. Thus, I am focusing on the heat transfer, which is coupled with the mass transfer.
2. Going along with the theme of a system moving towards equilibrium, heat and mass then immediately migrate from cube B to cube A. This happens simultaneously and the motion of both the heat and mass are linked together. This regime dominates everything that the problem cares about until thermal equilibrium is achieved in the system.
3. After thermal equilibrium is established, there will be some more mass diffusion that will then occur, soon achieving equilibrium with regards to mass as well
Papa Adolfo done sperged out calculating out what happens with 2 & 3 because event one is pretty minor. The
TL : DR answer to the question is that primarily, heat and mass are exchanged between the discrete volumes A&B until equilibrium is achieved. Chlorine, being the only component unique to B will be the primary species whose mass transfer actually matters, evrerything else is essentially just a transfer of heat and momentum. This particular sperg thus mostly focused his calculations on Chlorine and how it acts in the system.
I wrote a python script that does these calculations out for me because I am a lazy engineer who is shit at actual Computer coding but uses it to get his mathematics done for him.
Assumptions: Using Van Der Waal's Equation will account for molecular forces in a manner sufficient to assume ideality in the behavior of chemical potential.
no reactions
final Volume is fixed
Cube A, Cube B, and the equilibrium state are well mixed.
Any even halfway decent thermodyunamics calculation has a stupid number of inputs. this is no exception. This part of the code defines basic state properties of the system, including Van Der Waal's EOS coefficients, number of moles, mole fractions, Volume (in Liters because that makes a lot of the chemical potential calcs easier,) and, of course, Temperature in Kelvin. Also, minor calculations were performed here to determine specific heat capacities of the components and the mixtures A and B.
The calculations done herein seem petty but will be important for later discussion: Calculation of the mixture VDW coefficients. I used Van Der Waal's EOS because this accounts for the majority of intermolecular interactions that we need to care about in a scenario like this one where we have diatomic gases with reasonably similar dipole moments, and reactivities. This allows for the core of this calculation to say that it treats the gasses in a realistic and not ideal manner.
Next, The heat characteristics that I give a shit about are calculated in the script. Namely, these include system enthalpies, which will define the heat transport occurring in this conceptual process. After Enthalpy, it is relatively straight forward to calculate out the equilibrium temperature. This temperature is, in effect the end point for the more complicated process of coupled transport of heat and mass that is calculated out and discussed in section 1. Also calculated is a rough number for heat transfer coefficient assuming that the plane across which this transport occurs is a discrete square that can be thought of as a Plate of infinitesimal thickness and is not an impediment to mass transfer.
After I define what I care about in heat quantification, I need to calculate out the basics of chemical potential of the system. This is in particular because, despite the many MANY other implications of chemical potential, it represents the thermodynamic change in energy from the addition of some discrete mass to a system at constant volume and Entropy. This means it is an intrinsic property that is not accounted for in the heat or mass balances and thus plays a role in the diffusion question central to the problem. (Author's note; Don't think this is all Chemical Potential entails. People have written many PhD theses on what the fuck chemical potential actually is and what it represents. This is just for entry level calculating coupled transport by a bachelor degree chemical engineer with some time on his hands.) In this section, I alter the literature values for chemical potential with regard to temperature. The difference in Pressure is assumed to be adequately quantified in the VDW EOS calculated out earlier because it is at a lower Pressure than the literature value
from the python script comments:
"
## Calculations focus only on the Chlorine, for time and illustration purposes
## this is, in part because Cl2 is the only unique species undergoing mass
## diffusion
## the part where PDE's would come in, are herein. for my first iteration,
## I will show an approximation for the del(mu)/del(x) term in the soret effect
## the primary justification for this is that the system is low pressure, and
## High Volume in a medium temp range. These conditions mean that the Van Der
## Waal's EOS used initially can reasonably be assumed to account for most of
## the deviation from ideality of the real gas
"
Thus the real crazy shit hits the fan. the Quantitative method I will use for the Soret Calculation includes a particularly nasty little term for its calculation. In the denominator of the Thermal diffusivity factor, is the partial derivative of the function between chemical potential and molar fraction. This being done in full would require the derivation of what would undoubtedly be a multi-variate, non-linear partial differential equation using the Gibbs-Duhem equation. I'm not going to do that herein because that would honestly be the better part of a month to do properly, and it is not necessary. The reason it is not necessary herein is because we can use ideal approximations from here on out and get a reasonable result that would demonstrate conceptually what happens with this problem. Thus, I initially have done the full-retard thing of assuming a linear/step change situation that discretely homogenizes to a value after mixing. One can also very likely fit an exponential decay relationship to get a more realistic relationship between the chemical potential and mol fraction but I am lazy and have spent enough time on this to be perfectly honest.
The calculations done herein seem petty but will be important for later discussion: Calculation of the mixture VDW coefficients. I used Van Der Waal's EOS because this accounts for the majority of intermolecular interactions that we need to care about in a scenario like this one where we have diatomic gases with reasonably similar dipole moments, and reactivities. This allows for the core of this calculation to say that it treats the gasses in a realistic and not ideal manner.
Next, The heat characteristics that I give a shit about are calculated in the script. Namely, these include system enthalpies, which will define the heat transport occurring in this conceptual process. After Enthalpy, it is relatively straight forward to calculate out the equilibrium temperature. This temperature is, in effect the end point for the more complicated process of coupled transport of heat and mass that is calculated out and discussed in section 1. Also calculated is a rough number for heat transfer coefficient assuming that the plane across which this transport occurs is a discrete square that can be thought of as a Plate of infinitesimal thickness and is not an impediment to mass transfer.
After I define what I care about in heat quantification, I need to calculate out the basics of chemical potential of the system. This is in particular because, despite the many MANY other implications of chemical potential, it represents the thermodynamic change in energy from the addition of some discrete mass to a system at constant volume and Entropy. This means it is an intrinsic property that is not accounted for in the heat or mass balances and thus plays a role in the diffusion question central to the problem. (Author's note; Don't think this is all Chemical Potential entails. People have written many PhD theses on what the fuck chemical potential actually is and what it represents. This is just for entry level calculating coupled transport by a bachelor degree chemical engineer with some time on his hands.) In this section, I alter the literature values for chemical potential with regard to temperature. The difference in Pressure is assumed to be adequately quantified in the VDW EOS calculated out earlier because it is at a lower Pressure than the literature value
from the python script comments:
"
## Calculations focus only on the Chlorine, for time and illustration purposes
## this is, in part because Cl2 is the only unique species undergoing mass
## diffusion
## the part where PDE's would come in, are herein. for my first iteration,
## I will show an approximation for the del(mu)/del(x) term in the soret effect
## the primary justification for this is that the system is low pressure, and
## High Volume in a medium temp range. These conditions mean that the Van Der
## Waal's EOS used initially can reasonably be assumed to account for most of
## the deviation from ideality of the real gas
"
Thus the real crazy shit hits the fan. the Quantitative method I will use for the Soret Calculation includes a particularly nasty little term for its calculation. In the denominator of the Thermal diffusivity factor, is the partial derivative of the function between chemical potential and molar fraction. This being done in full would require the derivation of what would undoubtedly be a multi-variate, non-linear partial differential equation using the Gibbs-Duhem equation. I'm not going to do that herein because that would honestly be the better part of a month to do properly, and it is not necessary. The reason it is not necessary herein is because we can use ideal approximations from here on out and get a reasonable result that would demonstrate conceptually what happens with this problem. Thus, I initially have done the full-retard thing of assuming a linear/step change situation that discretely homogenizes to a value after mixing. One can also very likely fit an exponential decay relationship to get a more realistic relationship between the chemical potential and mol fraction but I am lazy and have spent enough time on this to be perfectly honest.
The method used herein is outlined by L.J.T.M. Kempers (RIP in Power) in "A comprehensive thermodynamic theory of the Soret effect in a multicomponent gas, liquid, or solid" 2001
The spooky part of this method is calculating a thermal diffusivity constant used in a simple relation which describes the change in molar fraction of a component (herein, Chlorine) which is resulting from the Soret Effect. The Soret effect terminates at thermal equilibrium, thus we can assume it dominates up until the system reaches relaxation time with regard to thermal equilibrium. Given that we are talking about elemental, diatomic gases, the components can be assumed to be Newtonian fluids whose relaxation time is described by Newton's Law of cooling.
Going along with the "This is just sperging, not grad school level work" theme, I am treating the N2/O2 mixture of Cube A as a binary component in the Soret effect calculations
The Results indicate that the Soret effect dominates the mass transport for ~7% of the diffusive action towards equilibrium.
This takes approximately, according to newton's law, 16 seconds
The spooky part of this method is calculating a thermal diffusivity constant used in a simple relation which describes the change in molar fraction of a component (herein, Chlorine) which is resulting from the Soret Effect. The Soret effect terminates at thermal equilibrium, thus we can assume it dominates up until the system reaches relaxation time with regard to thermal equilibrium. Given that we are talking about elemental, diatomic gases, the components can be assumed to be Newtonian fluids whose relaxation time is described by Newton's Law of cooling.
Going along with the "This is just sperging, not grad school level work" theme, I am treating the N2/O2 mixture of Cube A as a binary component in the Soret effect calculations
The Results indicate that the Soret effect dominates the mass transport for ~7% of the diffusive action towards equilibrium.
This takes approximately, according to newton's law, 16 seconds
So, in short, it's about 16 second before thermal equilibrium, thus all that happens with regard to the questions in OP is that heat and mass are exchanged. Within that kind of time frame, not much really gets accomplished, with the molar fraction of Cl2 reaching something on the order of 0.003, on its way to 0.04 It will then probably take something like a few hours to reach compositional equilibrium
There might be a reason you've been waiting 6 months, @Null.
