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→Second Law of Thermodynamics
Making progress on the theory required the development of the concept of entropy, and the development of statistical mechanics. To see what entropy is about, consider its thermodynamical definition as a differential. Entropy is generally symbolized with a capital S, and heat energy with a capital Q. While defining entropy in terms of its derivative rather than an actual absolute definition may seem to leave something to be desired (it leaves a "constant of integration" unspecified), that generally doesn't matter. This standard definition is used:
Making progress on the theory required the development of the concept of entropy, and the development of statistical mechanics. To see what entropy is about, consider its thermodynamical definition as a differential. Entropy is generally symbolized with a capital S, and heat energy with a capital Q. While defining entropy in terms of its derivative rather than an actual absolute definition may seem to leave something to be desired (it leaves a "constant of integration" unspecified), that generally doesn't matter. This standard definition is used:
::<math>dS = dQ/T</math>
::<math>dS = dQ/T</math>
The change in entropy of some object is the change in heat energy divided by the temperature at which the change takes place. The unit of thermodynamic entropy is the Joule per Kelvin. This is an "extensive" measure. To get the entropy for a given substance, independently of the size of the sample, it has to be divided by the size of the sample. So the unit of entropy for a substance (e.g. ice) is Joules per Kelvin per mole. Or per gram, or per atom, or per liter, or whatever.
Why is this useful? It captures the fact that heat flows downhill. Suppose there is a hot cup of coffee in a cooler room. Heat will flow from the coffee to the room. By conservation of energy, that is, the first law of thermodynamics, the amount of heat energy flowing out of the coffee is equal to the heat energy flowing into the room.
Why is this useful? It captures the fact that heat flows downhill. Suppose there is a hot cup of coffee in a cooler room. Heat will flow from the coffee to the room. By conservation of energy, that is, the first law of thermodynamics, the amount of heat energy flowing out of the coffee is equal to the heat energy flowing into the room.
::<math>dQ_{coffee} = -50\qquad\qquad\qquad\qquad dQ_{room} = 50</math>
::<math>dQ_{coffee} = -50\qquad\qquad\qquad\qquad dQ_{room} = 50</math>