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Stream: deprecated: thermodynamics

Topic: flow of molecules

Owen Lynch (Jan 09 2022 at 00:39):

Something that's been bugging me recently is that in thermodynamics, a priori in the formalism there could be a flow of molecules between two systems without a flow of heat. But I don't know what this would correspond to physically; it seems to me that if two systems can exchange molecules than they can of course exchange heat, as molecules carry kinetic energy! Is there a better way to think about this?

Nathaniel Virgo (Jan 09 2022 at 04:01):

I don't know about a flow of molecules, but it's certainly possible to have an exchange of volume without a flow of heat: just picture two systems (gases, say) separated by a piston head that can move but is a good thermal insulator. (But of course, although heat doesn't flow, the energy of the systems changes as they come to equilibrium, because energy is a function of volume.)

In general there seems to be a really strong coupling between energy and everything else - it's generally hard to change a parameter of a thermodynamic system without changing its energy as well. This is probably why energy plays such a central role in thermodynamics the way it's traditionally developed, with everything in terms of $dU$ instead of $dS$. But I also think it's kind of a coincidence in a way - the coupling between energy and everything else is a consequence of mechanics, rather than thermodynamics.

Joshua Meyers (Jan 09 2022 at 05:43):

I think it's important to understand the difference between energy and heat, and heat and work. In thermodynamics, heat and work are the two ways of transferring energy from one system to another, thus the equation $dU=dQ-dW$ (the $-$ is because $dQ$ denotes heat transferred to the system and $dW$ denotes work that the system does to the environment). Work is transfer of energy mechanically, through mass transfer (as you are describing), or any other way -- we can write $dW=PdV - \sum_i \mu_idN_i + XdY+\cdots$. Heat is transfer of energy through any other means.

One example of a flow of molecules between two systems without a flow of heat is if you have two different inert gases (say, ${\sf N}_2$ and ${\sf Ar}$) at the same temperature and pressure in adjacent containers and then you remove the barrier between the containers, allowing the gases to mix.

Joshua Meyers (Jan 09 2022 at 05:52):

Perhaps heat is just change in energy that is not explained by some obvious factor, it's just a catch-all for every other mechanism that we don't fully understand. But there does seem to be something special about it, because it is the only term of the fundamental equation $dU=-PdV+TdS+\sum_i\mu_idN_i+\cdots$ which involves entropy...

Joshua Meyers (Jan 09 2022 at 06:05):

OK so transferring heat to/from a system necessarily increases/decreases the entropy of the system, but transfer of energy into/out of a system through work is possible even while the entropy of the system is constant. That's the difference between heat and work.

Joshua Meyers (Jan 09 2022 at 06:22):

So let's set $dS=0$ and see what happens. Assuming now additional terms $XdY$ (for things like electricity, magnetism, etc.), we get $TdS=dU+PdV-\sum_i\mu_i dN_i=0$. So our system is represented by a point $(U,V,N_i)$ subject to this equation. If we add particles of type $i$ to the system $dN_i>0$, something else has to give to make this equation come out to $0$. If really really all we did was add particles, we can imagine that the particles we added were at absolute zero, not adding any kinetic energy, so all they did was crowd the vessel and reduce the possibilities for movement of the other particles, thus reducing the entropy. One possibility to make $dS$ come out to $0$ is that the new particles did bring some energy with them, exactly the right amount to offset this crowding. We would need $dU=\mu_i dN_i$. Another possibility is that we also remove particles of type $j$ at the same type, but choose particles that have $0$ kinetic energy -- then we would need $-\mu_j dN_j = \mu_i dN_i$. Finally, we could increase the volume to make room for the new particles and offset the crowding -- we would need $PdV = \mu_i dN_i$. Of course we could also do a combination of these effects, as expressed in the entire equation.

Joshua Meyers (Jan 09 2022 at 17:03):

I just realized that adding particles which have zero kinetic energy would actually have two entropic effects: they would lower the "configurational" entropy, as I described, by crowding the other particles; but then they would also increase the "momentum" entropy, by allowing for more combinations of how the momentum is distributed among the particles, provided the sum of the squared momenta is held constant at $2mNU$.

Joshua Meyers (Jan 09 2022 at 17:03):

I think this is why the chemical potential for an ideal gas is negative

Joshua Meyers (Jan 09 2022 at 17:04):

Apparently there's also more terms in the entropy of an ideal gas that I haven't accounted for though... https://en.wikipedia.org/wiki/Sackur%E2%80%93Tetrode_equation#Information-theoretic_interpretation

John Baez (Jan 09 2022 at 18:16):

One interesting thing about the entropy of the ideal gas is that the formula for it involves Planck's constant even though this is a classical ideal gas.

John Baez (Jan 09 2022 at 18:18):

I spent some time thinking about that. Basically without the help of a constant with dimensions of action (like Planck's constant) we cannot get a dimensionless quantity out of the volume in the phase space of a gas, which is required for defining entropy.

Nathaniel Virgo (Jan 10 2022 at 02:30):

I took Owen Lynch to be asking whether you could have a flow of molecules between systems at different temperatures, without an accompanying flow of heat. I guess I was saying that if you asked for an exchange of volume without an exchange of heat, you'd be talking about one system doing $pd V$ work on another. It does make sense to ask about such a work term in the case where the systems have different temperatures, if you imagine them to be separated by an adiabatic piston. So I guess the question is really whether you can have an "adiabatic membrane" that allows an exchange of particles without an accompanying flow of heat.

Owen Lynch (Jan 10 2022 at 02:31):

Yeah, that's exactly my question

Nathaniel Virgo (Jan 10 2022 at 02:57):

If we allow the particles to be electrons instead of molecules then a length of wire seems like it should do the job. Some heat will always flow along a metal wire, but for a long thin conductive wire it's pretty negligible compared to the flow of current. For ions a salt bridge probably works similarly. I don't know if that suggests anything similar that would work for molecules.

Owen Lynch (Jan 10 2022 at 03:56):

OK, neat, I didn't know what a salt bridge was before!

Owen Lynch (Jan 10 2022 at 03:56):

Thanks for giving me these examples!