---
title: "Laws of thermodynamics"
sort_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.