The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.

When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.

Consider an infinitesimal step in the expansion, which is a reversible, isothermal process. It can be shown that the percentage increase in the ideal gas’s volume is directly proportional to the amount of heat it receives from its surroundings and inversely proportional to the temperature at which it expands. This observation motivates the quantitative definition of entropy change.

The infinitesimal change in entropy is defined by the infinitesimal heat transferred divided by the temperature at which it is transferred. The definition is valid only for reversible processes. Entropy has the unit of joule per kelvin.

When the infinitesimal entropy change is integrated, a finite change in entropy is obtained for a reversible process. An arbitrary constant can be added because only the change in entropy is important.

For example, the entropy of an ideal gas undergoing reversible, isothermal expansion increases. It can be shown that just like the internal energy of a system, which appears in the first law of thermodynamics, entropy is also a state function. The second law of thermodynamics can be restated with the help of the quantitative definition of disorder via entropy.