A mathematical statement that links the variables to show their interdependence is called an equation of state; the gas laws are simple examples of such equations. Equations of state take on their simplest form when the Kelvin temperature scale is used; on this scale 0° corresponds to the lowest temperature theoretically possible.

When the external conditions are altered, a thermodynamic system will respond by changing its state; the temperature, volume, pressure, and chemical composition will adjust to a new equilibrium. The most important kinds of changes are adiabatic and isothermal changes. An adiabatic change is one that occurs without any flow of heat. The system is thermally insulated from the environment, and the first law of thermodynamics requires that the work done by or on the system be equal to the loss or gain of the system's internal energy. An isothermal change occurs when the system is in contact with a heat reservoir, so that the system remains at the temperature of the reservoir. In the isothermal process, heat flows from the reservoir if the system is expanding and into the reservoir if the system is being compressed. For an ideal gas the internal energy depends only on the temperature; hence the internal energy remains constant during an isothermal change, and the heat absorbed from or by the reservoir is equal to the work done on or by the environment.

Toward the middle of the 19th cent. heat was recognized as a form of energy associated with the motion of the molecules of a body (see kinetic-molecular theory of gases). Speaking more strictly, heat refers only to energy that is being transferred from one body to another. The total energy a body contains as a result of the positions and motions of its molecules is called its internal energy; in general, a body's temperature is a direct measure of its internal energy. All bodies can increase their internal energies by absorbing heat (see heat capacity). However, mechanical work done on a body can also increase its internal energy; e.g., the internal energy of a gas increases when the gas is compressed. Conversely, internal energy can be converted into mechanical energy; e.g., when a gas expands it does work on the external environment. In general, the change in a body's internal energy is equal to the heat absorbed from the environment minus the work done on the environment. This statement constitutes the first law of thermodynamics, which is a general form of the law of conservation of energy (see conservation laws).

A cyclic process is one that returns the system, but not the environment, to its original state. A closed cycle consisting of two isothermal and two adiabatic transformations is called a Carnot cycle after the French physicist Sadi Carnot, who first discussed the implications of such cycles. During the Carnot cycle occurring in the operation of a heat engine, a definite quantity of heat is absorbed from a reservoir at high temperature; part of this heat is converted into useful work, but the balance is expelled into a low-temperature reservoir and thus "wasted." The greater the temperature difference between the two reservoirs, which in a steam engine are represented by the boiler and the condenser, the greater the fraction of absorbed heat that is converted into useful work. It is, however, theoretically impossible to convert all the heat extracted from the reservoir into useful work.

In general it is impossible to perform a transformation whose only final result is to convert into useful work heat extracted from a source that is at the same temperature throughout. This statement is Lord Kelvin's version of the second law of thermodynamics. Another version of this law, formulated by R. J. E. Clausius, states that a transformation is impossible whose only final result is to transfer heat from a body at a given temperature to a body at higher temperature; in other words, the spontaneous flow of heat from hot to cold bodies is reversible only with the expenditure of mechanical or other nonthermal energy. These two versions of the second law of thermodynamics can be shown to be entirely equivalent.

The second law is expressed mathematically in terms of the concept of entropy. When a body absorbs an amount of heat Q from a reservoir at temperature T, the body gains and the reservoir loses an amount of entropy S=Q/T. Thus, in a reversible adiabatic process (no heat change) there is no change in the total entropy. If an amount of heat Q flows from a hot to a cold body, the total entropy increases; because S=Q/T is larger for smaller values of T, the cold body gains more entropy than the hot body loses. The statement that heat never flows from a cold to a hot body can be generalized by saying that in no spontaneous process does the total entropy decrease.

In all real physical processes entropy increases; in ideal reversible processes entropy remains constant. Thus, in the Carnot cycle, which is reversible, there is no change in the total entropy. The engine itself experiences no net change in entropy because it is returned to its original state at the end of the cycle. The entropy gained by the low temperature reservoir is equal to the entropy lost by the high temperature reservoir. However, according to the formula S=Q/T, less heat need be expelled into the low temperature reservoir than is extracted from the high temperature reservoir for equal and opposite changes in entropy. In the Carnot cycle this difference in heat appears as useful mechanical work.

A postulate related to but independent of the second law is that it is impossible to cool a body to absolute zero by any finite process. Although one can approach absolute zero as closely as one desires, one cannot actually reach this limit. The third law of thermodynamics, formulated by Walter Nernst and also known as the Nernst heat theorem, states that if one could reach absolute zero, all bodies would have the same entropy. In other words, a body at absolute zero could exist in only one possible state, which would possess a definite energy, called the zero-point energy. This state is defined as having zero entropy.