Rudolf Clausius in the context of "Internal energy"

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics, which convey a quantitative description using measurable macroscopic physical quantities but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to various topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering, and mechanical engineering, as well as other complex fields such as meteorology.

Historically, thermodynamics developed out of a desire to increase the efficiency of early steam engines, particularly through the work of French physicist Sadi Carnot (1824) who believed that engine efficiency was the key that could help France win the Napoleonic Wars. Scots-Irish physicist Lord Kelvin was the first to formulate a concise definition of thermodynamics in 1854 which stated, "Thermo-dynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, and the relation of heat to electrical agency." German physicist and mathematician Rudolf Clausius restated Carnot's principle known as the Carnot cycle and gave the theory of heat a truer and sounder basis. His most important paper, "On the Moving Force of Heat", published in 1850, first stated the second law of thermodynamics. In 1865 he introduced the concept of entropy. In 1870 he introduced the virial theorem, which applied to heat.

William John Macquorn Rankine FRSE FRS (/ˈræŋkɪn/; 5 July 1820 – 24 December 1872) was a Scottish mathematician and physicist. He was a founding contributor, with Rudolf Clausius and William Thomson (Lord Kelvin), to the science of thermodynamics, particularly focusing on its First Law. He developed the Rankine scale, a Fahrenheit-based equivalent to the Celsius-based Kelvin scale of temperature.

Rankine developed a complete theory of the steam engine and indeed of all heat engines. His manuals of engineering science and practice were used for many decades after their publication in the 1850s and 1860s. He published several hundred papers and notes on science and engineering topics, from 1840 onwards, and his interests were extremely varied, including, in his youth, botany, music theory and number theory, and, in his mature years, most major branches of science, mathematics and engineering.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

$${\displaystyle pV=nRT}$$where ${\displaystyle p}$, ${\displaystyle V}$ and ${\displaystyle T}$ are the pressure, volume and temperature respectively; ${\displaystyle n}$ is the amount of substance; and ${\displaystyle R}$ is the ideal gas constant.It can also be derived from the microscopic kinetic theory, as was achieved (independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is named after Rudolf Clausius and Benoît Paul Émile Clapeyron. However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later. Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."

Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics. Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.

Nicolas Léonard Sadi Carnot (French: [nikɔla leɔnaʁ sadi kaʁno]; 1 June 1796 – 24 August 1832) was a French military engineer and physicist. A graduate of the École polytechnique, Carnot served as an officer in the Engineering Arm (le génie) of the French Army. He also pursued scientific studies and in June 1824 published an essay titled Reflections on the Motive Power of Fire. In that book, which would be his only publication, Carnot developed the first successful theory of the maximum efficiency of heat engines.

Carnot's scientific work attracted little attention during his lifetime, but in 1834 it became the object of a detailed commentary and explanation by another French engineer, Émile Clapeyron. Clapeyron's commentary in turn attracted the attention of William Thomson (later Lord Kelvin) and Rudolf Clausius. Thomson used Carnot's analysis to develop an absolute thermodynamic temperature scale, while Clausius used it to define the concept of entropy, thus formalizing the second law of thermodynamics.