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Entropy is defined as the ratio of infinitesimal heat transferred to the absolute temperature at which it is transferred, expressed as dS = dQ/T. This definition applies only to reversible processes and has units of joules per kelvin. For finite changes, the infinitesimal entropy change is integrated to obtain the total entropy change of a system.

When a cold substance absorbs heat, its molecular constituents become more disordered and move with greater randomness. At higher temperatures, the disorder of any substance increases significantly. Entropy quantifies this increase in disorder, so a cold substance absorbing heat experiences a larger relative increase in entropy than a hot substance absorbing the same amount of heat.

In the Carnot cycle, which consists of reversible processes, the ratio of heat exchanged to the temperature of the heat reservoirs remains constant. This constant ratio defines entropy change. The Carnot cycle demonstrates that entropy is a state function, and the concept of entropy provides a quantitative foundation for understanding the second law of thermodynamics and the directionality of natural processes.

During isothermal expansion of an ideal gas, disorder increases because gas molecules have more volume to move around in. The percentage increase in volume is directly proportional to the heat absorbed and inversely proportional to the absolute temperature. Therefore, entropy increases during reversible isothermal expansion, quantifying the increased molecular disorder.

Entropy is a state function because its value depends only on the current state of the system, not on the path taken to reach that state. Like internal energy, entropy is a property that characterizes a system's condition. Only the change in entropy matters for calculations, allowing an arbitrary constant to be added without affecting physical predictions or comparisons.

The second law of thermodynamics states that work cannot be completely converted back into heat, making natural processes directional and irreversible. Entropy provides a quantitative measure of this irreversibility through the concept of disorder. The second law can be restated using entropy: the entropy of an isolated system always increases or remains constant, never decreases.

When a reversible process is not isothermal, it can be treated as a series of many infinitesimal isothermal processes occurring at different temperatures. The total entropy change is calculated by summing the ratio of heat transferred to temperature for each step. As the infinitesimal heat change approaches zero, this sum becomes an integral that yields the finite entropy change.