I can't get this simple question out of my head.

Does tap water have higher entropy since its temperature is almost in equilibrium with the surrounding? Or does hot coffee have higher entropy since it has a higher temperature? Or does hot coffee have lower entropy since heat energy is localized in a small area?

The original question was which has the highest entropy?

  • Ice
  • tap water
  • cold water
  • hot coffee

As a rule, for a single substance (and I'll idealize coffee as water since you don't specifically compare water and coffee at the same temperature), the higher the temperature, the greater the entropy. Put another way, entropy increases monotonically with temperature, reflecting the fact that more potential microstates exist for a hotter macrostate.

Therefore, your second rule of thumb is correct: the hot coffee has the greatest entropy because it's hottest.

Regarding your first rule of thumb, if you were to then allow the coffee to cool down, then the total entropy in the universe would increase because a spontaneous process would have occurred. However, the coffee would lose entropy during the cooling process, and you're asking only about the temperature of the liquid.

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