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The first law of thermodynamics is referred to as a reformulation of the law of conservation of energy.

I am not sure to fully understand this relationship.

My way of picturing it is the following.

A given system $S$ has an internal energy $U_S$. Apart from their mathematical definition, heat $Q$ and work $W$ are often defined in textbooks as "energy in transit". This means, in my mind, that some kind of energy is transferred to/from $S$ from/to its surroundings; let's call its surroundings $\Omega\setminus S$. I'd like to see $\Omega\setminus S$ as another system with its own internal energy $U_{\Omega\setminus S}$; in this view, I think the first law of thermodynamics could be restated as

\begin{equation} \Delta U_S + \Delta U_{\Omega\setminus S} = 0 \end{equation}

i.e.

\begin{equation} U_S+U_{\Omega\setminus S}=\text{constant} \end{equation} and this would clarify (to me) the relationship between the two concepts.

My question is, is this conceptual picture correct or does it have some fundamental flaw? If it's wrong, could someone please specify the relationship between the first law of thermodynamics and energy conservation in a way that a thoroughly obtuse person would understand?

EDIT: I corrected the first equation above.

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I think the first law of thermodynamics could be restated as \begin{equation} \Delta U_S + \Delta U_{\Omega\setminus S} = 0 \end{equation} i.e. \begin{equation} U_S+U_{\Omega\setminus S}=\text{constant} \end{equation} and this would clarify (to me) the relationship between the two concepts.

What you describe is a general law of conservation of energy; it assumes the rest of the universe (surroundings) can be ascribed energy and that the sum is conserved. The problem with this law is that we have no means of controlling the surroundings by definition, so it is (rather well-working) leap of faith.

The First law of thermodynamics is a little bit more restricted and more experimentally grounded. It concerns itself just with the system $S$, the surroundings are left unaccounted for.

It has more formulations which are more or less are equivalent. One of them is

Effect of heat supplied to the system on its state is equivalent to effect of certain equivalent amount of work supplied to it.$^{*}$ If both heat and work are measured in the same units (commonly Joules), a quantity characterizing the equilibrium state $X$, called internal energy, can be defined for all $X$. After the system is supplied heat $\Delta Q$ and work $\Delta W$, internal energy changes by

$$ \Delta U = \Delta Q + \Delta W. $$

(end of the law).

The change of $X$ and $U$ does not necessarily determine values of $\Delta Q$, $\Delta W$; they depend on the way the process is executed. The first law only says whatever the process (reversible, irreversible), change in $U$ is given by sum of heat and work supplied.

$^*$ The work is to be done irreversibly in such a way as to mimic addition of heat, i.e. by stirring the fluid in the system. No amount of reversible work would make the system end up in the same state that addition of heat does (since addition of heat does not conserve system's entropy but reversible work does).

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  • $\begingroup$ Thank you, I know the mathematical formulation of the first law, but actually my question was about explicitating the relationship between this law and the conservation of energy. $\endgroup$ – marco trevi Jul 12 '15 at 12:11
  • $\begingroup$ by the way, why do you write $\Delta Q$ and $\Delta W$? Heat and work are not state variables, rather inexact differentials, or am I wrong? $\endgroup$ – marco trevi Jul 12 '15 at 12:24
  • $\begingroup$ @marcotrevi, I have used $\Delta Q, \Delta W$ to denote any amount of heat and work (not necessarily infinitesimal). $\endgroup$ – Ján Lalinský Jul 12 '15 at 15:59
  • $\begingroup$ @marcotrevi, the relation is described in the first two paragraphs. Conservation of energy is different from 1st law, in that also surroundings can be ascribed energy and that total energy of the universe is constant. $\endgroup$ – Ján Lalinský Jul 12 '15 at 16:03
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Clearly the first law, as stated in Ján Lalinský's answer implies the conservation of energy.

Let us consider a system $S$ and a surrounding $\Omega$. Since energy is additive one can writhe the first law to the universe $T=S+\Omega$ as $$\Delta U_T=\Delta U_S+\Delta U_\Omega=Q_S-W_S+Q_\Omega-W_\Omega.$$ But The system can only exchange heat with the surrounding so that $Q_S=-Q_\Omega$. Similarly, the system can only do work against the surrounding, $W_S=-W_\Omega$. Therefore $$\Delta U_T=0,$$ which means the energy of the universe is constant.

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My question is, is this conceptual picture correct or does it have some fundamental flaw?

Choosing an object as a system, and you must include "incoming" energies - heat and work. Choosing the object as well as whereever the incoming energies come from as the system (and isolating it), you have no incoming energies, but instead internal energy of both. Your picture and rewriting is perfectly fine.

But that is only because I said isolated system. Your rewriting only works under this condition. If not isolated, there might be energy entering, and you must include heat and work again. So in the general case you can't do this rewriting - because then it isn't a general case anymore.

Of course if you consider your "object", defined as your system, to be the entire universe, then you have pretty much given that no energy enters - I believe that is why other answers here talk about the energy of whole universe.

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