Rudolf Clausius

How to use this course

Rudolf Clausius made few experiments and wrote no popular books. His achievement was conceptual. He showed how two ideas that seemed to contradict each other, Sadi Carnot’s theory of heat engines and the equivalence of heat and work, could both be right. From that reconciliation came the two laws of thermodynamics, internal energy, entropy, and a large part of the kinetic theory of gases. Every course in thermal physics still follows his order of ideas.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04–07 follow his work on thermodynamics (1850–1865); Lessons 08–10 follow his work on the molecular theory of gases (1857–1880). Modern notation is used throughout, with his original symbols mentioned where they help.

Timeline

Köslin and Stettin, 1822–1840

Rudolf Julius Emanuel Clausius was born on 2 January 1822 in Köslin, a small town in the Prussian province of Pomerania, now Koszalin in Poland. His father was a Protestant pastor and a councillor on the regional school board, who ran a small private school. Rudolf was one of a large family; MacTutor counts him as the sixth son.

He first attended his father’s school. In 1838 he moved to the Gymnasium in Stettin, the provincial capital, and finished there in 1840. Little else is recorded of his childhood. Clausius left no autobiography, and most of what we know of his early years comes from obituaries written after 1888.

Berlin and Halle, 1840–1850

In 1840 Clausius entered the University of Berlin. He was drawn to history and attended the lectures of Leopold von Ranke, but chose mathematics and physics. His teachers included the physicist Gustav Magnus, the geometer Jakob Steiner and Peter Gustav Lejeune Dirichlet, who gave him a firm training in analysis. He completed his studies in 1844.

Like most graduates without private means, he needed income. He spent a probationary year teaching at the Friedrich-Werder Gymnasium in Berlin and in 1846 entered August Boeckh’s Royal Seminary for teachers.

The blue sky

His doctorate came from the University of Halle in 1848, with a dissertation on the particles of the atmosphere that reflect light. Clausius tried to explain the blue of the daytime sky and the red of sunrise and sunset, and the polarization of skylight, by reflection and refraction from tiny transparent bubbles of water. The explanation was wrong. In 1871 Lord Rayleigh showed that the colour comes from scattering by particles much smaller than a wavelength, with intensity proportional to \(\lambda^{-4}\). The thesis nevertheless showed Clausius’s method: take a known physical law and follow its mathematical consequences carefully.

In Berlin he belonged to the circle of young scientists around Magnus, where he met the Irish physicist John Tyndall, who later helped make his papers known in English. The circle also produced the Physical Society of Berlin, where Hermann von Helmholtz read his essay on the conservation of force in 1847.

The breakthrough and Zürich, 1850–1867

On 18 February 1850 Clausius read his paper Über die bewegende Kraft der Wärme (On the moving force of heat) to the Berlin Academy. It appeared in the Annalen der Physik the same year and made him known at once. That autumn he was appointed to teach physics at the Royal Artillery and Engineering School in Berlin and became Privatdozent at the university.

The 1850s were a race with William Thomson, later Lord Kelvin, in Glasgow, and William Rankine in Scotland, who were working on the same problems. Clausius published the more systematic formulation. In 1854 he gave his statement of the second law and his “equivalence of transformations”, the germ of entropy.

In 1855 he became professor of mathematical physics at the newly founded Polytechnikum in Zürich, now ETH, and at the University of Zürich. His colleagues included the mathematician Richard Dedekind and the experimental physicist Albert Mousson, who became a close friend. Among the students who heard him was Carl von Linde, later the pioneer of refrigeration. In Zürich he wrote his papers on the kinetic theory of gases (1857, 1858), on disgregation (1862) and on entropy (1865). On 19 November 1859 he married Adelheid Rimpau; they had six children. He declined several calls to other universities before accepting one from Würzburg in 1867.

The moving force of heat (1850)

For the general reader. In 1824 Sadi Carnot explained heat engines by analogy with a water wheel: heat falls from hot to cold, like water, and does work without being used up. By the 1840s James Prescott Joule had shown that work can make heat and heat can make work, at a fixed rate of exchange. The two views seemed incompatible. Clausius showed that each held half of the truth, and the two halves became the first and second laws of thermodynamics.

Two principles

Carnot’s argument used two assumptions: heat (caloric) is conserved, and no engine can produce work from a fall of heat more efficiently than a reversible one. Clausius dropped the first and kept the second. In an engine some heat disappears, converted into work at the rate measured by Joule; the rest passes to the cold body. He wrote the energy balance of a body in the form, in modern notation,

Clausius called \(U\) simply a function of the state, without a name; the term internal energy came later. Thomson reached the same function in 1851.

The ideal-gas Carnot cycle

Take one mole of ideal gas, with \(U=C_VT\) and \(pV=RT\). On an isotherm at \(T\), \(dU=0\) so the heat absorbed equals the work: \(Q=RT\ln(V_b/V_a)\). On an adiabat \(\delta Q=0\) gives \(C_V\,dT=-RT\,dV/V\), so \(TV^{\gamma-1}\) is constant with \(\gamma=C_p/C_V\). A cycle with isotherms at \(T_1>T_2\) joined by adiabats has \(V_b/V_a=V_c/V_d\), hence

The heat \(Q_2\) delivered to the cold body is less than \(Q_1\): the difference has become work. Carnot’s efficiency survives; his caloric does not. From the cycle Clausius also derived that \(C_p-C_V=R\) for an ideal gas, and a relation for vapours that is the subject of Lesson 07.

The second law

For the general reader. A hot cup of coffee cools; the room never heats it up. Clausius made this ordinary fact the foundation of a law of nature: heat never flows by itself from a colder body to a hotter one. From this single principle he derived the limits on every heat engine and refrigerator.

Equivalence

If a Kelvin-violating device existed, it could turn heat from a cold reservoir into work, and the work could heat a hot body by friction: the net result violates Clausius. Conversely, if heat could flow unaided from cold to hot, run a Carnot engine between the reservoirs and send back to the hot reservoir the heat the engine dumps into the cold one. The net result takes heat from the hot reservoir and turns it entirely into work, violating Kelvin.

Proof. Suppose engine \(X\) takes \(Q_1'\) from the hot reservoir and gives work \(W\) with efficiency greater than a reversible engine \(R\). Run \(R\) backwards as a refrigerator, driven by the work \(W\) from \(X\); it delivers heat \(Q_1=W/\eta_R>W/\eta_X=Q_1'\) to the hot reservoir. The combination does no net work and moves heat \(Q_1-Q_1'>0\) from cold to hot, contradicting the Clausius statement. If \(X\) is also reversible, the argument works both ways and the efficiencies are equal.

Absolute temperature

Since the efficiency of a reversible engine depends only on the two temperatures, the ratio \(Q_2/Q_1\) is a universal function \(f(\theta_1,\theta_2)\) of any empirical temperatures. Composing two engines shows \(f(\theta_1,\theta_3)=f(\theta_1,\theta_2)\,f(\theta_2,\theta_3)\), so \(f=\phi(\theta_2)/\phi(\theta_1)\) for some function \(\phi\). Thomson in 1848 and 1854 proposed to define temperature by \(T=\phi(\theta)\). Lesson 04 shows that this agrees with the ideal-gas temperature, and the efficiency is \(1-T_2/T_1\).

Entropy (1854–1865)

For the general reader. Energy tells us how much a system has; Clausius wanted a quantity that tells us which way it will go. He found one, a number that stays the same in ideal reversible processes and grows in every real one. In 1865 he called it entropy, from the Greek for transformation, choosing a word deliberately close to “energy”.

The inequality

In 1854 Clausius described an engine’s cycle as a set of transformations: heat converted into work, and heat passing from one temperature to another. He assigned to each an equivalence value and showed that in a reversible cycle the values cancel. In modern form:

Proof sketch. Supply each heat \(\delta Q_i\), at temperature \(T_i\), from a reversible Carnot engine drawing heat \(\delta Q_i\,T_0/T_i\) from a single reservoir at \(T_0\). After a cycle the system is unchanged, and the net effect is heat \(T_0\sum_i\delta Q_i/T_i\) taken from one reservoir and turned into work. By Kelvin’s statement this cannot be positive. If the cycle is reversible, run it backwards to get the reverse inequality.

The first part follows from equality in the inequality for reversible cycles; the second by closing an arbitrary path with a reversible one. Combining with the first law gives the fundamental relation \(dU=T\,dS-p\,dV\).

Clausius ended the 1865 paper with two sentences that became famous. In paraphrase: the energy of the world is constant, and its entropy tends to a maximum. In a lecture of 1867 he spelled out a consequence: the universe tends toward a state of uniform temperature in which no further change can occur, later called the heat death. In 1862 he had already tried to split entropy into a part due to temperature and a part due to the arrangement of the molecules, which he called disgregation, a step toward the molecular interpretation that Ludwig Boltzmann gave in the 1870s.

Phase changes: the Clausius–Clapeyron relation

For the general reader. Water boils at 100 °C at sea level and at lower temperatures on a mountain. How fast does the boiling point fall with pressure? Émile Clapeyron answered with Carnot’s caloric theory in 1834. Clausius rederived the answer in 1850 from the new thermodynamics and corrected it, and the result now bears both names.

Derivation by a small Carnot cycle

Take one mole through a cycle: evaporate it at \(T+dT\) and pressure \(p+dp\), absorbing \(L\); cool it slightly; condense it at \(T\) and \(p\); warm it back. In the \(p\)–\(V\) plane the cycle is a thin rectangle of width \(v_2-v_1\) and height \(dp\), so the work is \((v_2-v_1)\,dp\). Carnot efficiency gives

Clapeyron had the same structure with an undetermined function of temperature in place of \(1/T\); Clausius identified it. An equivalent derivation sets the molar Gibbs energies of the two phases equal along the curve.

Vapour pressure

If \(v_2\gg v_1\) and the vapour is ideal, \(v_2-v_1\approx RT/p\), and with \(L\) roughly constant

The kind of motion we call heat (1857)

For the general reader. If heat is motion, what moves? In 1857 Clausius answered: the molecules of a gas fly freely in straight lines and bounce off the walls, and their impacts are the pressure. From the measured pressure and density he computed their speed, hundreds of metres per second. He went further than earlier authors by recognizing that molecules also rotate and vibrate.

August Krönig had published a simple kinetic model in 1856. Clausius had been thinking along the same lines for some years and published his own fuller version, Über die Art der Bewegung, welche wir Wärme nennen, in 1857. Daniel Bernoulli had anticipated the pressure formula in 1738, and John Herapath and John James Waterston had developed kinetic ideas in the 1820s and 1840s with little recognition.

Derivation

A molecule with velocity component \(v_x>0\) striking a wall perpendicular to \(x\) transfers momentum \(2mv_x\). The number of such molecules hitting area \(A\) in time \(dt\) is \(n\,v_x A\,dt\) per velocity class, with \(n=N/V\). Summing over molecules with \(v_x>0\), the pressure is \(p=n\,m\,\overline{v_x^2}\). Isotropy gives \(\overline{v_x^2}=\tfrac13\overline{v^2}\).

Comparing with \(pV=nRT\) shows that translational kinetic energy is proportional to absolute temperature. Clausius also argued that equal volumes of different gases at the same temperature and pressure contain equal numbers of molecules, giving a physical argument for Amedeo Avogadro’s hypothesis.

Internal motions

If all the heat energy of a gas were translational, the energy per mole would be \(\tfrac32RT\), and the ratio of specific heats \(\gamma\) would be \(5/3\). For air, \(\gamma\approx1.40\). Clausius concluded that molecules carry rotational and vibrational energy as well, and computed the fraction that is translational.

The mean free path (1858)

For the general reader. If gas molecules move at hundreds of metres per second, why does a smell take minutes to cross a room? The Dutch meteorologist Christophorus Buys Ballot raised this objection to Clausius’s 1857 paper. Clausius replied that molecules collide constantly, travelling only a tiny distance between collisions. He called that distance the mean length of path, now the mean free path.

Derivation

A moving molecule sweeps out a cylinder of cross-section \(\sigma=\pi d^2\) and collides with any molecule whose centre lies inside it. In length \(x\) it meets on average \(n\sigma x\) molecules, so if the others are at rest \(\ell=1/(n\sigma)\). The probability of travelling distance \(x\) without collision is \(e^{-x/\ell}\). When the others move, what matters is the relative speed. If all have the same speed \(v\) in random directions, the mean relative speed is \(\tfrac43v\), which gives Clausius’s factor \(\tfrac34\). With Maxwell’s distribution of speeds, the mean relative speed is \(\sqrt2\,\bar v\), which gives the factor \(1/\sqrt2\). Maxwell’s first paper on gases (1860) began from Clausius’s mean free path.

The concept made transport properties calculable. Viscosity is of order \(\tfrac13nm\bar v\ell\), which is independent of density since \(\ell\propto1/n\). Maxwell predicted this surprising result and confirmed it experimentally. In 1865 Josef Loschmidt combined viscosity data with the volume of the liquid to make the first estimate of molecular size.

The virial theorem and real gases (1870–1880)

For the general reader. Real gases are not ideal: their molecules attract one another and take up space. In 1870 Clausius proved a general theorem of mechanics that relates the average kinetic energy of any bound system to its forces. Applied to a gas, it shows how attractions and repulsions change the pressure. The same theorem now weighs galaxies and stars.

Clausius presented the theorem in 1870 to the Lower Rhine Society for Natural and Medical Sciences in Bonn, under the title Über einen auf die Wärme anwendbaren mechanischen Satz (On a mechanical theorem applicable to heat). He named the quantity \(-\tfrac12\sum_i\mathbf F_i\cdot\mathbf r_i\) the virial, from the Latin vis, force.

Proof. Let \(G=\sum_i m_i\mathbf r_i\cdot\mathbf v_i\). Then \(\dot G=\sum_i m_iv_i^2+\sum_i\mathbf F_i\cdot\mathbf r_i\). Average over time \(\tau\): \(\frac{G(\tau)-G(0)}{\tau}\to0\) because \(G\) is bounded. The second statement follows from Leonhard Euler’s theorem on homogeneous functions, \(\sum_i\mathbf r_i\cdot\nabla_iV=kV\).

Application to gases

For a gas in a container, split the forces into those from the walls and those between molecules. The walls exert pressure \(p\) and contribute \(-\tfrac12\oint p\,\mathbf r\cdot d\mathbf A=-\tfrac32pV\) to the virial. Hence

where \(\mathbf F_{ij}\) is the force on \(i\) due to \(j\) and \(\mathbf r_{ij}=\mathbf r_i-\mathbf r_j\). Without interactions this is the ideal gas law of Lesson 08. Attractions lower the pressure and repulsions raise it. Johannes Diderik van der Waals, in his thesis of 1873, used Clausius’s theorem to justify his equation of state. In 1880 Clausius proposed his own equation for carbon dioxide, \(\big(p+\frac{a}{T(V+c)^2}\big)(V-b)=RT\), with three constants.

Clausius–Mossotti

In his work on electricity, Clausius developed an earlier result of Ottaviano Fabrizio Mossotti relating the dielectric constant \(\varepsilon_r\) of a substance to the polarizability \(\alpha\) of its molecules. In SI units, with number density \(N\),

The optical form, with \(\varepsilon_r\) replaced by the square of the refractive index, is the Lorentz–Lorenz relation.

Würzburg, Bonn and war, 1867–1880

Clausius left Zürich for Würzburg in 1867 and moved two years later to the University of Bonn, where he spent the rest of his life. In the summer of 1870 the Franco-Prussian War broke out. Clausius, then 48, organized an ambulance corps of Bonn students and led it to the battlefields near Metz, where they carried wounded men after the battles of Vionville and Gravelotte. He was wounded in the leg, which left him in pain and with limited mobility for the rest of his life. He received the Iron Cross in 1871. In 1878, on his doctor’s advice, he took up riding to ease his disability.

The war sharpened the national tone of scientific disputes. Clausius had argued over priority with William Thomson in the 1850s, but in good faith. From 1872 he was drawn into a bitter quarrel with the Scottish physicist Peter Guthrie Tait, who credited the founding of thermodynamics largely to British authors and to Joule. Clausius, who had earlier supported the priority of Julius Robert Mayer, replied at length. In 1871 he also answered a priority claim by the young Boltzmann, whose 1866 paper had sought a mechanical meaning for the second law.

In 1875 his wife Adelheid died giving birth to their sixth child, leaving him to bring up the family alone. In the same years he turned to electrodynamics. He proposed a law of force between moving charges (1875–1877) and argued, against Wilhelm Weber, that the force should depend on absolute velocities, not only on relative ones. These papers had less influence than his thermodynamics, as Maxwell’s field theory soon displaced all such force laws. His book Die mechanische Behandlung der Electricität appeared in 1879, the year the Royal Society awarded him the Copley Medal.

Last years and legacy, 1880–1888

In the 1880s Clausius worked on the equation of state of real gases and on new editions of his treatise, recast as Die mechanische Wärmetheorie. He served as rector of the University of Bonn in 1884–85. In 1886 he married again and had one more child. In 1888 he was admitted to the Prussian order Pour le Mérite for Sciences and Arts. He died in Bonn on 24 August 1888, aged 66.

His honours included foreign membership of the Royal Society (1868), the Huygens Medal (1870), the Copley Medal (1879), the Poncelet Prize of the Paris Academy, and membership of the academies of Sweden, the Netherlands, Italy and the Leopoldina. A crater on the Moon is named after him.

Legacy

• Thermodynamics: the first law with internal energy as a state function; the second law in the form taught today; the Clausius inequality; entropy.
• Kinetic theory: internal degrees of freedom, the mean free path and the virial theorem. Maxwell built his kinetic theory on Clausius’s papers, and Boltzmann built on both.
• Followers: Max Planck chose thermodynamics after reading Clausius, and wrote his 1879 doctoral thesis on the second law. Josiah Willard Gibbs placed Clausius’s two sentences on energy and entropy at the head of his great memoir on heterogeneous equilibrium.
• Beyond physics: the virial theorem in astrophysics; entropy in chemistry, information theory and cosmology.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Rudolf Julius Emmanuel Clausius, MacTutor History of Mathematics.
2. Rudolf Clausius, Wikipedia.
3. E. E. Daub, Clausius, Rudolf, Dictionary of Scientific Biography, via Encyclopedia.com.
4. Rudolf Julius Emanuel Clausius, Mathematics Genealogy Project.
5. S. G. Brush, The Kind of Motion We Call Heat, 2 vols., North-Holland, 1976.
6. C. Smith, The Science of Energy: A Cultural History of Energy Physics in Victorian Britain, University of Chicago Press, 1998.

Physics

1. R. Clausius, Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen, Ann. Phys. 79 (1850). English translation in E. Mendoza (ed.), Reflections on the Motive Power of Fire, Dover, 1960.
2. R. Clausius, Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie, Ann. Phys. 93 (1854).
3. R. Clausius, Über die Art der Bewegung, welche wir Wärme nennen, Ann. Phys. 100 (1857).
4. R. Clausius, Über die mittlere Länge der Wege, welche bei der Molecularbewegung gasförmiger Körper von den einzelnen Molecülen zurückgelegt werden, Ann. Phys. 105 (1858).
5. R. Clausius, Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Ann. Phys. 125 (1865). Excerpt in English, Classic Chemistry, Le Moyne College.
6. R. Clausius, Über einen auf die Wärme anwendbaren mechanischen Satz, Ann. Phys. 141 (1870).
7. R. Clausius, The Mechanical Theory of Heat, trans. W. R. Browne, Macmillan, 1879.
8. E. Fermi, Thermodynamics, Dover, 1956: a short modern course in Clausius’s order.
9. Virial theorem and Clausius–Mossotti relation, Wikipedia.