Entropy is a state function discovered by Clausius in 1854 and defined by the following expression: \begin{equation} dS=\frac{dq}{T} \end{equation} The entropy change between two states 1 and 2 is obtained by integrating the differential expression: \begin{equation} \Delta S=\int_{1}^{2}\frac{dq}{T} \end{equation} Entropy is a measure of the disorder that a system possesses. The more disordered it is, the more entropy it has.

Thus, solid water (ordered molecules) has less entropy than liquid water (molecules with freedom of movement).

Entropy also tells us the direction that spontaneous processes follow. Suppose a container divided by a wall that contains nitrogen on one side and oxygen on the other, we remove the wall and the gases mix spontaneously. Nitrogen and oxygen molecules diffuse through every container, obtaining after a certain time a perfect mixture of both gases.

What would you think if I told you that, spontaneously from this nitrogen-oxygen mixture, the gases begin to separate, reaching a final state in which the nitrogen remains pure in the left half of the container and the pure oxygen in the right half? . You'll rightly think I'm joking. What is the reason why the spontaneous process is the mixture of the two gases and not the separation of them?

It is observed that in any spontaneous process the entropy of the system increases (valid for isolated systems and irreversible processes). The mixture of the gases supposes an increase of the disorder and therefore of the entropy, for that reason it is the spontaneous process. Their separation would imply a decrease in the entropy of the system (increase in order) and cannot occur spontaneously. Entropy is the thermodynamic arrow indicating the direction of spontaneous processes.