I have somewhat of an understanding for other physical quantities, but as far as entropy goes I only know it to be "disorder". Why is the change in entropy formula an appropriate/useful definition, moreover, why is the equation for entropy an appropriate statement for entropy. With things like volume and pressure, at least, I have a natural inkling as to what they are. What is entropy, more than disorder. Why are there no units? Why are these definitions correct?
Answer
Don't look at entropy as disorder. Thinking of it as disorder has long been a source of confusion. Many texts are moving away from using the disorder description. Macroscopically, it's better to think of entropy as a measure of energy dispersion rather than as a measure of disorder. Microscopically, it's better to think of entropy as a measure of density of states rather than as a measure of disorder. The macroscopic and microscopic concepts come together in the field of statistical mechanics.
Entropy does have units. Entropy has units of energy/temperature, or Joules/Kelvin in the SI system. Entropy would essentially be unitless if temperature was just a measure of energy, but with goofy units. However, temperature is not energy. Temperature is related to but distinct from energy. A gram of ice at 0 °C and a gram of water at 0 °C have the same temperature but different energies.
Because it lets us explain why a couple of grams of liquid water at 5 °C doesn't turn into a gram of ice at 0 °C and a gram of water at ~90 °C. The first law of thermodynamics doesn't complain about this transformation. The second law most certainly does. Entropy puts limits on things and helps answer key questions such as "How much useful energy can this engine extract from a certain amount of fuel?"
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