Explanation:
When hot water and cold water are mixed, the entropy of the system increases. This can be explained using the concept of entropy and statistical mechanics.

Entropy:
Entropy is a thermodynamic quantity that measures the disorder or randomness of a system. It is denoted by the symbol "S". In simple terms, entropy can be thought of as a measure of the number of different ways in which the particles of a system can be arranged.

Statistical Mechanics:
Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of a large number of particles in a system. It provides a microscopic understanding of macroscopic phenomena.

Explanation:
When hot water and cold water are mixed, the particles of both substances interact with each other. Initially, the hot water has a higher temperature and the cold water has a lower temperature. As the two substances mix, the thermal energy is transferred from the hot water to the cold water.

Increased Disorder:
This transfer of thermal energy leads to increased disorder in the system. The particles of the hot water gain kinetic energy and move faster, while the particles of the cold water lose kinetic energy and slow down. This increased disorder corresponds to an increase in entropy.

More Microscopic Arrangements:
From a statistical mechanics perspective, the increase in entropy can be explained by considering the number of microscopic arrangements that are possible for the particles. Initially, the particles of the hot water and cold water are separated and have distinct arrangements. When they mix, the particles can arrange themselves in a larger number of ways, leading to an increase in entropy.

Conclusion:
In summary, when hot water and cold water are mixed, the entropy of the system increases. This is because the transfer of thermal energy leads to increased disorder and a larger number of microscopic arrangements for the particles.