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---
title: "Laws of thermodynamics"
firstLetter: "L"
publishDate: 2021-07-07
categories:
- Physics
- Thermodynamics
date: 2021-07-05T17:44:53+02:00
draft: false
markup: pandoc
---
# 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"](/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 $\pdv*{S}{U} = 1 / T$ 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 $\Delta S$.
## References
1. H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition,
Princeton.
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