Conservation of energy is the first law of thermodynamics, and one of several conservation laws.

It is stated as follows:

The total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system.

Although ancient philosophers as far back as Thales of Miletus may have had inklings of the First Law, it was first stated in its modern form by the German surgeon Julius Robert von Mayer (1814-1878) in his "Remarks On the Forces of Inorganic Nature" in Annalen der Chemie und Pharmacie, 43, 233 (1842). Mayer reached his conclusion on a voyage to the Dutch East Indies (now Indonesia), where he found that his patients' blood was a deeper red, because they were using consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He had discovered that heat and work were both forms of energy, and later, after improving his knowledge of physics, he calculated a quantitative relationship between them.

Meanwhile, in 1843 James Prescott Joule independently discovered the law by an experiment, now called the "Joule apparatus", in which a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the thermal energy (heat) gained by the water by friction with the paddle.

Unfortunately for Mayer, his work was overlooked in favour of Joule's, and Mayer attempted to commit suicide. Later, Mayer's reputation was restored by a sympathetic account in John Tyndall's Heat: A Mode of Motion (1863).

A similar law was written in the privately published Die Erhaltung der Kraft (1847) by Hermann von Helmholtz.

With the discovery of special relativity by Albert Einstein, conservation of energy was shown to be a special case of a more general rule. According to special relativity, mass and energy are interchangable with the famous equation E = mc2.

Conservation of energy can be shown through Noether's theorem to be the result of the time-invariance of the laws of physics.


One formulation for the first law of thermodynamics is
where Q is heat transferred into the system from the surroundings, W is work done by the system, and E is the internal energy of the system. This energy is mostly kinetic energy: the potential energy can be assumed to be negligible. Pressure-volume work (e.g. done by a gas on a piston) is defined to be
Equation (1) can be interpreted as follows: Q is heat energy being input into the system. The system the can use this incoming energy to do two things: (1) do work, or (2) increase its own internal energy. Here is an analogy: Q is income, which can then be spent to buy things (W), or it can be saved in a bank account ().

If all the heat is used to do work ( and ) then the system is undergoing an isothermal process, which means that its temperature remains constant. This is because the system's internal energy is proportional to its temperature.

If all the heat is used to increase internal energy, ( and ) then the system is undergoing an isochoric process, also called isometric process. This is a process in which the system's volume is constant:

so that, according to equation (2),
W = 0.

It is also possible for the heat energy to be used up partially by doing work and partially by increasing internal energy. Examples of such processes are the isobaric process and the adiabatic process.

Equation (1) is the one preferred by engineers. Another convention preferred by chemists is

where W is work done on the system by the surroundings. In this case pressure-volume work is defined to be
Equation (3) can be interpreted to mean thus: that heat Q and work W are energies being transferred into or out of the system. The system then responds by increasing or lowering its internal energy accordingly. Equation (3) is more symmetric in the sense that internal energy E is a state function (it is conservative; independent of the chosen path (process) between initial and final thermal states) whereas neither Q nor W are state functions: they do depend on which particular (thermodynamic) process is chosen to connect the initial and final thermostatic states.

The law of conservation of energy excludes the possibility of perpetuum mobile of the first kind.


  • Engines of Our Ingenuity, episode 722 - radio broadcast by John Lienhard, produced by KUHF-FM Houston)
  • Peter J. Nolan, Fundamentals of College Physics, 2nd edition, William C. Brown Publishers, 1995.
    • Chapter 17, Thermodynamics.
  • Oxtoby & Nachtrieb, Principles of Modern Chemistry, 3rd edition, Saunders College Publishing, 1996.
    • Chapter 8, Thermodynamic Processes and Thermochemistry.