# The entropy

## Definition

The probability of a state of a system is caracterised by a state function called
entropy (symbol $S$, measured in $\frac{J}{K\cdot mol}$).

The entropy $S$ of a system is the (logarithm of) the number of possible configurations which a system can have.

$S$ is
- always positive or zero (there exists at least one possible configuration!)
- all the greater as the number of possible configurations (= "the disorder" ) is great.

The entropy $S$ measures the "disorder" of a system.

## The second principle of thermodynamics

What is happening with a children's room when the parents are absent?
Isolated systems tend spontaneously to amaximum of disorder!!

In an isolated system
spontaneous transformations occur always with an increase of entropy:
$\Delta S >0$

For many chemical reactions it is possible to anticipate the variation of entropy remembering that entropy increases always with the "disorder":

### Example

The reaction
$H_2O(l)$ $\longrightarrow$ $H_2O(g)$
will have $\Delta S >0$, because in the gaseous state molecules are more "disordered" than in the liquid state.
The → Table of standard entropies allows to calculate the variations of entropy during chemical reactions in standard conditions ($25^oC$ and $1\;bar$):

### Example

$2NH_4NO_3(s)$ $\longrightarrow$ $2N_2(g)$ $+$ $4H_2O(g)$ $+$ $O_2(g)$
$\Delta S$
$= 2\cdot S(N_2(g))$ $+$ $4\cdot S(H_2O(g))$ $+$ $S(O_2(g))$ $-$ $2\cdot S(NH_4NO_3(s))$
$= 2\cdot 191.61$ $+$ $4\cdot 188.83$ $+$ $205.138$ $-$ $2\cdot 151.08$
$= 1041.52\frac{J}{K\cdot mol}$,
a positive value, because indeed, the "disorder" increases!

### Example

$2H_2(g)$ $+$ $O_2(g)$ $\longrightarrow$ $ 2H_2O(l)$
$\Delta S$
$= 2\cdot S(H_2O(l))$ $-$ $(2\cdot S(H_2(g))$ $+$ $S(O_2(g)))$
$= 2\cdot 69.91$ $-$ $(2\cdot 130.684$ $+$ $205.138)$
$= -326.686\frac{J}{K\cdot mol}$,
a negative value, because indeed, the "disorder" decreases, liquid water is more "orderly" than the mixture of gases $N_2$ and $O_2$!
This reaction is spontaneous (explosion).
However, there is no contradiction with the second principle, as the system is not isolated.
When the system is not isolated, the spontaneous transformations can occur with a decrease in entropy:
You see the museum guard acting on the system (the visitors) to send them back to the entry door, whereas they would have spread spontaneously over the whole museum!