Heat

Heat is a form of energy that is transferred between two bodies that are in thermal contact and have different temperatures. For instance, consider two compartments of a vessel separated by a heat-conducting wall and let there be fluids of different temperatures on either side of the wall. Heat will flow from the hotter to the colder compartment.

Also the energy that flows from, or to, the surroundings of an object is called heat. The surroundings may be hot radiating gas that heats a kettle with cold water, as shown in the picture. Heat flows spontaneously from the high-temperature "body" (the burning gas) to  the low-temperature kettle.

The effect of transfer of energy is usually, but not always, a raising of the temperature of the low-temperature object and a decrease in the temperature of the high-temperature object.

Change of aggregation state
A vessel containing a fluid may lose or gain energy without a change in temperature. This happens when the fluid changes from one aggregation state to another.

For instance, a gas of constant pressure condenses (goes over into liquid) at constant temperature, which is the boiling point of the liquid. While condensing, the gas releases the kind of heat that is usually referred to as  condensation energy. Hence, a vessel that contains a condensing gas loses heat, for instance to its environment or to a colder body. Nevertheless the temperature of the vessel remains constant, provided there is still gas left in it. The temperature stays at the boiling point of the liquid, even though the vessel is losing heat.

As another example: consider a vessel containing a melting solid. Its temperature will remain constant while it is receiving heat, as long as not all solid has molten. Only after all of the solid is turned into liquid and the heat transport continues, the temperature of the colder vessel  will rise.

As a last example: the temperature of the water in the kettle shown in the figure will rise quickly to the boiling point of water (100 °C). Then, assuming the gas flame is not switched off, the temperature inside the kettle remains constant at 100 °C for quite some time, even though heat keeps on flowing from flame to kettle. When all liquid water has evaporated&mdash;when the kettle has boiled dry&mdash;the temperature of the kettle will quickly rise again until it obtains the temperature of the burning gas, then the heat flow will finally stop. (Most likely, though, the handle and maybe the metal of the kettle, too, will have melted before that).

Units
At present the unit for the amount of heat is the same as for any form of energy. Before the equivalence of mechanical work and heat was clearly recognized, two units were used. The calorie was the amount of heat necessary to raise the temperature of one gram of water from 14.5 to 15.5 °C and the unit of mechanical work was basically defined by force times path length (in the old cgs system of units this is erg). Now there is one unit for all forms of energy, including heat. In the International System of Units (SI) it is the joule, but the British Thermal Unit and calorie are still occasionally used. The unit for the rate of heat transfer is the watt (J/s).

Equivalence of heat and work
Although both heat and work obey the law of conservation of energy, the two forms of energy are not completely equivalent. Work can be completely (100%) converted into heat, but the converse is not true. When heat is converted into work, part of the heat is not&mdash;and cannot be&mdash;be converted. The non-converted part of the heat flows into a body of lower temperature that must be present to cause a heat flow.

Heat and temperature
The important distinction between heat and temperature (heat being a form of energy and temperature a measure of the amount of that energy present in a body) was clarified by Count Rumford, James Prescott Joule, Julius Robert Mayer,  Rudolf Clausius,  and others during the late-18th century and first half of the 19th century. In that period it became clear that heat is not an invisible and weightless fluid, named  caloric, as was thought by many 18th century scientists, but a form of motion. The molecules of the hotter body are (on the average) in more rapid motion than those of the colder body. The first law of thermodynamics, discovered around the middle of the 19th century, states that the (flow of) heat is a transfer of part of the internal energy of the bodies. In the case of ideal gases, internal energy consists only of kinetic energy and it is indeed only this motional energy that is transferred when heat is exchanged between two containers with ideal gases. In the case of non-ideal gases, liquids and solids, internal energy also contains the averaged inter-particle potential energy (attraction and repulsion between molecules), which depends on temperature. So, for non-ideal gases, liquids and solids, also potential energy is transferred when exchange of heat occurs.

Forms of heat
The actual transport of heat may proceed by electromagnetic radiation (as an example one may think of an electric heater where usually heat is transferred to its surroundings by infrared radiation, or of a microwave oven where heat is given off to food by microwaves), conduction (for instance through a metal wall; metals conduct heat by the aid of their almost free electrons), and convection (for instance by air flow or water circulation).

Entropy
If two systems, 1 (cold) and 2 (hot), are isolated from the rest of the universe (i.e., no other heat flows than from 2 to 1 and no work is performed on the two systems) then the entropy Stot = S1 + S2 of the total system 1 + 2 increases upon the spontaneous flow of heat. This is in accordance with the second law of thermodynamics that states that spontaneous thermodynamic processes are associated with entropy increase. In general, the entropy S of a system at absolute temperature T increases with
 * $$ \Delta S = \frac{Q}{T} $$

when it receives an amount of heat Q > 0. Entropy is an additive (size-extensive) property.

The hotter system 2 loses an amount of heat to the colder system 1. In absolute value the exchanged amounts of heat are the same by the law of conservation of energy (no energy escapes to the rest of the universe), hence
 * $$ \Delta S_\mathrm{tot} = \Delta S_1 + \Delta S_2 = Q\left( \frac{1}{T_1} - \frac{1}{T_2} \right) > 0 \quad\hbox{because}\quad T_2 > T_1. $$

Here it is assumed that the amount of heat Q is so small that the temperatures of the two systems are constant. One can achieve this by considering a small time interval of heat exchange and/or very large systems.

Remark: the expression &Delta;S = Q/T is only strictly valid for a reversible (also known as quasistatic) flow of energy. It is possible to define:
 * $$ \Delta S \equiv \Delta S_\mathrm{ext} + \Delta S_\mathrm{int}\quad\hbox{with}\quad \Delta S_\mathrm{ext}\equiv \frac{Q}{T}\quad\hbox{and}\quad \Delta S_\mathrm{int} \ge 0 $$

It is assumed that &Delta;Sint is  much smaller than  &Delta;Sext, so that it can be neglected.

Semantic caveats
It is strictly speaking not correct to say that a hot object "possesses much heat". On the other hand, it is correct to say that a hot object possesses high internal energy. The word "heat" is reserved to describe the transfer of energy from a high temperature object to a cold one (in short called "heating the cold object"). The reason that the word "heat" is to be avoided for the internal energy of an object is that the latter can have been acquired either by heating or by work (or by both). When internal energy is measured, there is no way of deciding how the object acquired it&mdash;by work or by heat. In the same way as one does not say that a hot object "possesses much work", one ought not to say that it "possesses much heat". Yet, terms as "heat reservoir" (a system  that for all practical purposes is infinite and hence of constant temperature) and "heat content" (a synonym for enthalpy) are commonly used and are strictly speaking incorrect because there is no flow of energy involved.

The molecules of a hot body are in agitated motion and, as said, it cannot be measured how they became agitated, by work or by heat. Often, especially outside physics, the random molecular motion is referred to as "thermal energy". In classical (phenomenological) thermodynamics this is an intuitive, but undefined, concept. In statistical thermodynamics, thermal energy could be defined (but rarely ever is) as the average kinetic energy of the molecules constituting the body. Kinetic and potential energy of molecules are concepts that are foreign to phenomenological thermodynamics that predates the general acceptance of the existence of molecules. Classical thermodynamics can be formulated independent of the constitution of the systems under study.

Quotation
As a result Carathéodory was able to obtain the laws of thermodynamics without recourse to fictitious machines or objectionable concepts as the flow of heat.