I wonder whether you ever had the chance to see the glass toy called “the drinking bird”. Maybe the slightly older among my readers could remember it. It is a moving toy shaped as a bird, with a round head and pointed beak, fixed above a very long neck. A piece of tissue wraps around the beak and head, apparently just a decoration. At the bottom end of the long neck there is a reservoir containing a colored fluid. A small jar filled with water is placed at the front of the “bird”. To start operation, one gets the bird’s head tissue wet by dipping it in the water jar, after which the bird starts slowly flipping up and down, each time dipping again its beak in the water and jumping back, in what seems a perpetual motion machine of sorts. In this YouTube video somebody took the burden of filming the periodic bird’s motion for 10 hours. The “secret” of the drinking bird lies all in the colored fluid in its abdomen, usually methylene chloride mixed with a dye. This compound boils at a very low temperature, about 39.6ºC. As the water in the head evaporates, the colored fluid moves up into the head, causing the bird to become top-heavy and dip forward into the water. This cools down the methylene chloride in the head, which then moves back into the abdomen, causing the bird to become bottom heavy, and tip up. Except for a very small input of energy – the original head-wetting – the bird seems able to continue to go up and down forever without further energy input. But far from being a perpetual motion machine, the drinking bird is a beautiful example of Carnot’s engine.

The novelty device bears the patent no. 2.402.463, deposited on August 6, 1945 (the very day when the Enola Gay B-29 bomber was dropping the first atomic bomb over Hiroshima) by Miles G. Sullivan, an engineer and chemist at Bell Labs in Murray Hill, NJ. During WWII, Bell Telephone Laboratories worked on a number of war-related subjects, such as developing magnetometers sensitive enough to detect Earth’s magnetic field anomalies caused by submarines. They had a semiconductor department, where silicon and germanium were traditionally studied, alongside with cuprous oxide and other materials. In 1945, Bell Labs reorganized and created a group specifically devoted to do advanced research on semiconductors, led by William Shockley, with John Bardeen and Walter Brattain, who would create the first transistor a couple of years later. Sullivan was an expert in techniques to synthetize semiconductor materials, who had been hired at Bell after a brief stint at Naval Research Laboratories in Washington. He died at age 99, in 2016, after taking his inventions on the Mars surface with the Viking rover, for which he designed the electrical contacts of the solar cells. And as a side hobby, he loved to invent toys such as the “drinking bird”. It is said that United States’ President Herbert Hoover (an engineer himself) had one on his desk in the Oval Office. Albert Einstein (as reported by a 1964 TIME magazine story) stayed up all night at one point, puzzled about how it worked. When asked in the morning whether he had figured it out, he shook his head, but then proceeded to disassemble the bird, looking for the answer.

To understand the functioning of the drinking bird, one can start by looking at the elements one by one. For example, you can remove the water jar after the initial wetting: the bird will continue to dip back and forth for quite a long time, even without getting wet again, but eventually it will stop. So, the motion is not perpetual. It continues until there is enough water to wet the head at each flip, after which it will end. A second test can be done by putting some cool on the bird’s head, like few drops of ice-cold water: the colored liquid immediately starts to rise in the tube, filling the head. When looking closely at the glass construction, it is seen that the abdomen lower sphere is sealed, and communicates with the top head sphere by a tube that reaches almost to the bottom. The methylene chloride inside the bird at room temperature is partly liquid, and partly vapor; however, the vapor inside the lower sphere cannot go into the upper sphere, as it is blocked by the fluid. At rest, the vapor pressure in the head and abdomen are equal. When the top sphere is slightly cooled, the pressure of the vapor in the head decreases, and the higher pressure in the bottom sphere pushes the fluid up in the tube, which slowly fills the upper sphere, making it heavier. At this point, the bird tilts forward up to an almost horizontal position: the liquid in the lower sphere swings, the tube end is open, the two vapor volumes in the head and in the bottom now communicate and get again to the same pressure, a slug of vapor from the bottom raises to the top and dislodges some of the top liquid. This makes the bottom again heavier and the bird tilts up, closing the communication tube, and starting again the cycle.

What is happening when the vapor in the top sphere absorbs heat from some source (for example the ice-cold water), is that some of the methylene vapor starts condensing into liquid. As liquid is about 1,000 times denser, just a small amount of condensation is enough to drastically drop the gas pressure. How the head is cooled when the bird is functioning? Remember that you must start the cycle by dipping the bird’s head in the water jar first. The tissue wrapping the head gets wet, and when the bird is upright this water starts evaporating and removes the heat of vaporization, pretty much like when you get out of the shower and feel the chill of the water evaporating on your skin. The bird’s head gets a little colder than room temperature and makes methylene to condense, thus lowering the pressure. The heat release from vaporization of 1 gram of water is about 2.3 kJ, this is why the fluid inside the bird must be a very volatile species. One can see this by heating the bottom sphere (it is enough to put your hand around it) and looking at the liquid level decreasing, as it evaporates into methylene vapor. As far as the wetting liquid, one can put another liquid with a larger heat of vaporization than water in the jar, and the bird will increase its dipping and drinking because of the faster cooling rate: if you put whiskey in the jar it will release more heat (6 kJ/g) and make your bird getting drunk by drinking faster.

In more technical terms, the bird is a heat engine that converts heat differences into mechanical work. Every time the bird is going to drink, its head gets at a slightly lower temperature than its base, which remains at room temperature. When the tube opens and pressures are equalized, the liquid and vapor are mixed. The temperature of the fluid inside returns to a common value, but slightly lower than room temperature. When the bird returns to upright stance, the liquid in the bottom takes heat from ambient to return at room temperature, while the water around the head releases heat of vaporization in the ambient. This is exactly a heat engine: a device that absorbs heat from a source at higher temperature, converts part of this heat into mechanical work, and rejects the rest into a sink at lower temperature. The water is the fuel for the engine, by only changing phases without undergoing any chemical transformation. In this beautiful paper, Gopala Krishna Vemulapalli, a chemist at University of Arizona, Tucson, provided a nice quantitative analysis of the thermodynamic cycle of the drinking bird. He noticed that the pressure in the upper sphere changes from P(at room temperature) to P1=P2–ρgh, with obvious meaning of the symbols. Since the vapor in the glass bird is a two-phase system, the pressure and temperature of the vapor are related by the Clausius-Clapeyron equation, as ln[(P2–ρgh)/P2] = ΔH/R (1/T2–1/T1). With a simple algebra, the Carnot efficiency of the cycle is η=ρgh*RT1/(P2*ΔH). This shows that η increases with the ratio (ρgh/P2), i.e., a taller bird gets more work done. Also, for a given difference in pressure between head and bottom, the larger is the vaporization enthalpy ΔH, the smaller is the temperature difference, hence η decreases with increasing ΔH (another reason why a highly volatile fluid with small enthalpy is needed).

Of course, the bird is not even close to a real Carnot cycle. Its temperature-entropy diagram (shown in Vemulapalli’s paper) is not quite a rectangle, rather an ellipse, the more rounded as the efficiency η is increased.  Classically, the Carnot engine consists of two sets of alternating adiabatic strokes and isothermal strokes. However, I got recently very much interested in the quantum-mechanical version of thermodynamic systems, and in particular the thermal operation of qubits, which are believed to be perfectly reversible devices. A priori, one may argue that the laws of thermodynamics (with an exception for the First) are defined just for macroscopic systems described by statistical averages, and hence, the question of their validity for microscopic systems consisting of a few particles, or qubits may itself appear meaningless. However, already in 1959 Scovil et al. demonstrated that the working of a quantum three-level maser coupled to two thermal reservoirs resembles that of a heat engine, with an efficiency upper-bounded by the Carnot limit. A quantum analogue of the Carnot engine consists of a working fluid, which could be, e.g., a particle in a box, qubits of various kind, multiple-level atoms, or harmonic oscillators. For the simplest case of a three-level system, the quantum working fluid is the spectrum of energy levels E1<E2<E3; the high-temperature bath can excite transitions hvh=E1-E3, the low-temperature sink induces transitions hvc=E1-E2; and a radiation field is tuned resonantly at the middle frequency hvr=E2-E3. At equilibrium, for each excitation hvh the system loses an energy hvc to the cold sink, and hvr to the radiation, so that the population ratios (n1/n3) and (n1/n2) are maintained steady. The energy exchanged with the two thermal baths can be thought of as “heat”, while the energy exchanged with the radiation field can be identified with “work” extracted from the quantum system (the radiation plays the same role as Carnot’s “piston”). This identification of work and heat implies the energy relation hvh = hvc+hvr, which is the analog of entropy conservation for a reversible cycle. (Reversibility here is within the limit of statistical equilibrium among all the excitations.) A remarkable result appears when the efficiency of this “thermodynamic” system is considered. The quantum system can work as an thermal engine when a population inversion is realized between the levels 2 and 3, n2>n3, which leads to a condition n2/n3=(n2/n1)(n1/n3)>1, that is exp(hvc/kTchvh/kTh)≥1. The efficiency is, as usual, the ratio of the work extracted to the heat supplied by the hot reservoir, η=hvr/hvh=1–hvc/hvh, which – thanks to the previous inequality – gives Carnot’s limit η ≤ 1–Tc/Th.

The proof of existence of Carnot’s limit (a manifestation of the celebrated Second law of thermodynamics) even at such small scales establishes a strong case for the validity of thermodynamics principles down to the microscopic realm, and pushes further the scrutiny of the emergence of thermodynamic laws at the most fundamental level.

Quantum Carnot

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