Categories: Physics, Thermodynamics.

The **laws of thermodynamics** are of great importance to physics, chemistry and engineering, since they restrict what a device or process can physically achieve. For example, the impossibility of *perpetual motion* is a consequence of these laws.

The **first law of thermodynamics** states that energy is conserved. When a system goes from one equilibrium to another, the change \(\Delta U\) of its energy \(U\) is equal to the work \(\Delta W\) done by external forces, plus the energy transferred by heating (\(\Delta Q > 0\)) or cooling (\(\Delta Q < 0\)):

\[\begin{aligned} \boxed{ \Delta U = \Delta W + \Delta Q } \end{aligned}\]

The internal energy \(U\) is a state variable, so is independent of the path taken between equilibria. However, the work \(\Delta W\) and heating \(\Delta Q\) do depend on the path, so the first law means that the act of transferring energy is path-dependent, but the result has no “memory” of that path.

The **second law of thermodynamics** states that the total entropy never decreases. An important consequence is that no machine can convert energy into work with 100% efficiency.

It is possible for the local entropy \(S_{\mathrm{loc}}\) of a system to decrease, but doing so requires work, and therefore the entropy of the surroundings \(S_{\mathrm{sur}}\) must increase accordingly, such that:

\[\begin{aligned} \boxed{ \Delta S_{\mathrm{tot}} = \Delta S_{\mathrm{loc}} + \Delta S_{\mathrm{sur}} \ge 0 } \end{aligned}\]

Since the total entropy never decreases, the equilibrium state of a system must be a maximum of its entropy \(S\), and therefore \(S\) can be used as a thermodynamic “potential”.

The only situation where \(\Delta S = 0\) is a reversible process, since then it must be possible to return to the previous equilibrium state by doing the same work in the opposite direction.

According to the first law, if a process is reversible, or if it is only heating/cooling, then (after one reversible cycle) the energy change is simply the heat transfer \(\dd{U} = \dd{Q}\). An entropy change \(\dd{S}\) is then expressed as follows (since \(\pdv*{S}{U} = 1 / T\) by definition):

\[\begin{aligned} \boxed{ \dd{S} = \Big( \pdv{S}{U} \Big)_{V, N} \dd{U} = \frac{\dd{Q}}{T} } \end{aligned}\]

Confusingly, this equation is sometimes also called the second law of thermodynamics.

The **third law of thermodynamics** states that the entropy \(S\) of a system goes to zero when the temperature reaches absolute zero:

\[\begin{aligned} \boxed{ \lim_{T \to 0} S = 0 } \end{aligned}\]

From this, the absolute quantity of \(S\) is defined, otherwise we would only be able to speak of entropy differences \(\Delta S\).

- H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition, Princeton.

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