# Work with thermodynamical equilibrium condition

A thermodynamic system being in thermodynamic equilibrium is characterized by the property that for every thermodynamic potential $$F$$ which describes the system, its differential $$dF$$ is zero. Let consider for example the internal energy $$U(S, V, N_i)$$ for now. If the system is in thermodynamic equilibrium then

$$dU= \frac{\partial U}{\partial S}dS + \frac{\partial U}{\partial V}dV + \sum_i^n \frac{\partial U}{\partial N_i}dN_i = TdS + pdV + \sum_i^n \mu_i dN_i = 0$$

Note that $$\frac{\partial U}{\partial S} =T, \frac{\partial U}{\partial V} =p, \frac{\partial U}{\partial N_i} =\mu_i$$.

Question: How this condition $$dU$$ helps "in practice" when one works with concrete systems and want to find out in which $$(S_0, V_0, (N_i)_0)$$ the system has its "equilibrium"?

When I try to apply it I obtain something nonsensical and I want to understand which mistake I make here. Back to our condition $$dU=0$$ implies that $$\frac{\partial U}{\partial S} =T, \frac{\partial U}{\partial V} =p, \frac{\partial U}{\partial N_i} =\mu_i$$ should be all zero, because $$S, V$$ and $$N_i$$ are independent variables, therefore the differentials $$dS, dV$$ and $$dN_i$$ as well. So I obtain $$n+2$$ conditions $$\frac{\partial U}{\partial S} =0, \frac{\partial U}{\partial V} =0, \frac{\partial U}{\partial N_i} =0$$

But this not make any sense to me simply because this would imply that if the state is equilibrium state, then always its $$T, p$$ and $$\mu$$ are all zeroes. But certainly there are thermodynamical systems which are in equilibrium but their $$T, p, \mu_i$$ are not zero.

I'm confused now, what I'm doing wrong? could anybody explain to me how to "read" and "work" with the condition $$dU=0$$ correctly? sorry, if my question is too easy for people with elementary knowledge on this topic but also after long search I nowhere found an answer.

In practice, this means that if you have an isolated system characterized by fixed values of entropy, volume, and number of molecules, its internal energy $$U(S,V,N)$$ at equilibrium is minimum with respect to any other variable different from those determining the thermodynamic state. For example, if the system is in a container and a fixed, impenetrable and insulating wall is separating subsystem $$1$$ from subsystem $$2$$, this is equivalent to have two separate subsystems with energies $$U_1(S_1,V_1,N_1)$$ and $$U_2(S_2,V_2,N_2)$$. If the constraint on thermal insulation is relaxed, and heat can flow between the two subsystems varying $$S_1$$ and $$S_2$$ but without entropy production, $$S_1+S_2=S$$, then only one additional independent variable, say $$S_1$$, represents the constraint. The vanishing of the first order variation of the total energy with respect to $$S_1$$ $$\frac{\partial{U(S,V,N,S_1)}}{\partial{S_1}}=0,$$ at fixed $$S,V,N$$, provides the condition for thermal equilibrium after removal of the constraint, i.e. the equality of the temperatures of the two subsystems.
• Assume we have an isolated system about which we only know that it has fixed macroparameters $S, V ,N$ and we can always measure $U$ (I purposly not wrote $U(S, V,N)$ because as far as I know the internal energy can be only expressed as $U(S, V,N)$ if the system already is in td equilibrium, but that's what we are going here to check; see my comment on it below in $**$). Before we start we measure $U$. What we do now is we try to vary artificially all possible physical parameters of the system but with respect only one rule: everything what we do should not change $S, V, N$. – Eddy Wa Feb 13 at 2:37
• That is we can try to change the system in every way we want as far as the rule is not violated. And when we do it we continuously measure $U$. Assume we are able somehow (that's a thought experiment) to can out all possible manipulations of this system which respect to rule not to change $S, V, N$. If the minimal value of $U$ coinsides with $U$ at the beginning then our system was in equilibrium state. Does this approach make sense? – Eddy Wa Feb 13 at 2:37
• Motivation: by the procedure in the thought experiment to change the system in all possible ways as far as $S, V, N$ not change I intended to imitate the mathematical procedure to vary any variable in the system different from $S, V, N$ – Eddy Wa Feb 13 at 2:37
• Note on $**$: As far as I understand the ussue the function $U$ (or any other td potential) exists always but the only case when it depends only on three parameters (as here $S, V, N$) is only the case when the system is in equilibrium. In case of non equilib $U$ in general should exist but is too complicated (maybe depend on infinity many parameters) to work with, right? – Eddy Wa Feb 13 at 2:39