The German physicist Rudolf Clausius (1822–1888) was concerned about Carnot’s use of the term waste heat. He developed another version of the Second Law that involves the concept of entropy. Entropy can be defined as the dispersal of energy. The greater the dispersal or spreading the larger the entropy. For example, when salt and pepper are in separate piles the two substances are in distinct locations. If you mix them together they are no longer in separate regions of pure salt and pure pepper, but dispersed throughout the combined pile. Further, it is not possible for the salt and pepper grains to separate themselves. Thus the mixing of salt and pepper increases the entropy of the system. If the salt and pepper could be shaken in such a way that they would separate, then entropy would be decreased, but such an event has never been observed.

Suppose you place ice in water. Ice and water are separate. The water has higher thermal energy, the ice lower, and so the system has low entropy. When the ice melts the two can no longer be separated. The thermal energy is dispersed throughout the system, so the entropy has increased. But wait, you might say, the ice water (the system) has cooled the air around it (the environment), decreasing its entropy because hot gas has greater entropy than cold gas. Calculations, however, show that the increase in the ice/water mixture is greater than the decrease in the air. So a statement of the Second Law is that the entropy of the system and the environment can never decrease.

The increase in entropy suggests a direction of time, sometimes called the “Arrow of Time.” The “forward” direction of time is the one in which entropy increases or remains the same.

Clausius had given Carnot’s work a firmer foundation. Later work by Ludwig Boltzmann, Josiah Willard Gibbs, and James Clerk Maxwell developed the statistical basis for entropy.