Configurational entropy (quantitative)

Entropy is the amount of disorder in a system.

The microscopic way to measure the disorder is by looking at the individual pieces of the system, and counting the number of ways a system can be arranged in order to reach its current macrostate. The different possible arrangements are called the microstates.

The macroscopic definition describes the change in the entropy of the system in terms of the amount of heat required to raise the temperature of a system one degree.

The microscopic definition looks at the individual components of a system in order to study the system as a whole.

$S=k_B\ln\Omega$

$k_B =$ Boltzmann constant

$\Omega =$ number of microstates

A box contains two particles. Does it have more entropy when it has 0, 1 or 2 units of energy?

Total Energy	Possible Microstates	Number of Microstates	Entropy
0 units	$E_1=0, E_2=0$	1	$k_Bln(1)=0$
1 unit	$E_1=1, E_2=0$ $$$$ $E_1=0, E_2=1$	2	$k_Bln(2)$
2 units	$E_1=2, E_2=0$ $$$$ $E_1=1, E_2=1$ $$$$ $E_1=0, E_2=2$	3	$k_Bln(3)$

The system has the most entropy when it has 2 units of energy.

The macroscopic definition looks at the overall state of a system in order to study it.

$$\Delta S = \int \frac{Q_{\text{rev}}}{T} .$$

One way of understanding the second law is that the entropy (in an isolated system) always increases.

Consider a partitioned box with one partition filled with a hot gas and another with a relatively cooler gas. The hotter portion has a relatively higher number of molecules with higher velocity compared to the colder one.

When the partition is removed, both come to thermal equilibrium. This is because now the faster and the slower molecules mix up and are evenly distributed.