Change in internal energy and reversible paths between states

Question

As I understand it, the relation (1) $dU = TdS - pdV + \sum_{i} \mu_idN_i$ always holds, even for irreversible paths. The justification seems to be that (2) "$U, S, V, N_1, N_2, ...$ are all state variables." I sort of intuitively see how (1) follows from (2) but I would like a more rigorous explanation of it. I was thinking that, IF it is possible to find a reversible path with the same starting point (a) and endpoint (b) as any arbitrary (possibly non-reversible) path, then we have $\int_a^b TdS - pdV + \sum_{i} \mu_idN_i = \int_a^bdQ +dW = \int_a^b dU$. In the limit, for infinitesimal processes, we should have $TdS - pdV + \sum_{i} \mu_idN_i = dU$.

The problem is that I'm not sure that there always is a reversible path corresponding to any arbitrary non-reversible path (this related question doesn’t seem to have a satisfying answer). For instance, if $\Delta G < 0$ between point (a) and (b), then every path between (a) and (b) has a negative Gibbs free energy and is presumably not reversible (I may be wrong about this since $\Delta G < 0$ implies non-reversibly under the assumption of constant p and T).

A. Does there always exist a reversible path between two states? And if so, does my reasoning about why (2) implies (1) seem correct?

B. If not are there any other (rigorous) ways of explaining why (2) implies (1)?

Answer 1

I think of it differently. I think of it as describing the mutual changes in these state parameters (U, S, V , N's) between two closely neighboring (i.e., differentially separated) thermodynamic equilibrium states, one at (U,S,V, Nj) and the other at (U+dU, S+dS,V+dV,Nj+dNj). This mutual change in the variables is independent of how tortuous and/or irreversible the process was for transitioning the system between the two neighboring thermodynamic equilibrium states, as long as, in the end, they end up differentially separated from one another. Thus, the system may have taken a very long and irreversible path between the two closely neighboring end states. Note that the relationship does not apply at every location along the long irreversible process path, only to the initial and final closely neighboring thermodynamic equilibrium end states.

Answer 2

I think the logic is a bit different than that. Saying that the change in the internal energy is the same along an irreversible path as along a reversible path with the same endpoints doesn't imply that both paths exist, it's just meant to say that if they both exist they'll both yield the same change in any state variable, because they'll go to and from the same pair of states.

That being said, does a reversible path always exist? It's kind of subjective. You could say that any physical process is always slightly irreversible because processes move forward in time and forward in time is by definition the direction in spacetime in which entropy is increasing. Or you could say that purely formal, non-physical processes can be reversible.

I think perhaps though the confusion between these two arguments rests on something that we under-emphasize: it's crucial to specify whether we're talking about the system or the universe. You say "every path between (a) and (b) has a negative Gibbs free energy and is presumably not reversible". You're right to express hesitancy: whether the system's Gibbs free energy is increasing or decreasing doesn't tell us for sure whether the process is irreversible or reversible. What tells us that is whether the universe's entropy is increasing or decreasing. For example, if we put hot coffee in a cool room heat will flow out of the coffee, so its entropy will decrease. But far from being a violation of the 2nd Law, this is in fact an irreversible process because the entropy of the room increases more.

For this reason, if you draw a path from point A to point B and ask me whether it's reversible or irreversible, I'll tell you I don't know. Before I can answer that, I need to know how much heat (or whatever else might affect entropy) is flowing into the environment. If the change in entropy of the system along the path is equal and opposite to the change in entropy of the environment the path is reversible.