In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate. Entropy is a measure of how disordered a system is. The total entropy in the universe always increases (or, at best, remains constant). Consider a sealed room with a hot cup of tea in it. The tea will tend to lose heat to the surrounding air, making the air molecules move faster. This makes the whole system more disordered, as faster moving molecules are less ordered than slower ones. No energy is lost from the system, but there is an increase in entropy. The more disordered a system is, the more information is needed to describe it. Consider again the room with the cup of tea. When the molecules are heated by the tea, they move faster. It takes more information to describe the exact state of all the faster-moving molecules than it did when they were moving more slowly. This is because each individual molecule, as it can now move faster, has a bigger range of speeds it can be travelling at while the air and tea are in thermodynamic equilibrium. As there are more possible microstates (positions and speeds of individual molecules), it takes more information to describe the system.

It is possible to calculate entropy using the equation:

S = k ln W 

• S is the entropy
• k is Boltzmann constant
• ln is the natural logarithm
• W is the amount of realization possibilities the system has

It can also be referred to Boltzmann-Planck entropy formula.

Notes

The value of W is basically a measure of how likely a system can exist given certain characteristics. Example,imagine you have a deck of cards with 4 identical cards. The deck as a total can be described with parameters such as the number of cards, thickness of the deck, weight and so on. With four cards we have 4x3x2x1 = 24 possible configurations that all lead to the same (in terms of the parameters above) deck of cards. Therefore in this case W = 24.  The Boltzmann constant, k, equals to 1.4*10-²³ J/K and the entropy S is then kln24 = 4.4*10 -²³ J/K.