Categories: Physics, Thermodynamics.

# Laws of 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.

## First law

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.

## Second law

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.

## Third law

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 $$\Delta S$$.

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