---
title: "Laws of thermodynamics"
date: 2021-07-07
categories:
- Physics
- Thermodynamics
layout: "concept"
---

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"](/know/concept/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.