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 $\ipdv{S}{U} = 1 / T$ 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 $\Delta S$.

## References

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