Thermodynamics Pixel Sticker Sheet

12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Thermodynamics. Every character below has its own sticker page with the full physics panel.

Dilutra

Below about 0.87 K a mixture of helium-3 and helium-4 separates into a ³He-rich phase floating on a dilute phase. Even at absolute zero the dilute phase keeps a few percent of ³He dissolved. Ifft, Edwards and Sarwinski measured this limit at 6.37 ± 0.05% (Phys. Rev. Lett. 19, 831, 1967); later work gives about 6.4–6.6% depending on pressure. Pumping ³He out of the dilute phase pulls more across the boundary, and that ‘evaporation’ into the dilute phase absorbs heat. It works continuously, down to a few millikelvin.

x₃(T → 0) ≈ 6.4–6.6%
x₃: fraction of ³He atoms in the dilute (⁴He-rich) phase

Boltzbrain

In the 1890s Ludwig Boltzmann suggested that our ordered world might be a rare fluctuation in a universe otherwise in thermal equilibrium. The difficulty is that a small fluctuation is far more likely than a big one. A single brain, complete with false memories, needs a much smaller drop in entropy than a whole ordered universe, so in such a picture most observers would be lone “Boltzmann brains”. Andreas Albrecht and Lorenzo Sorbo (Phys. Rev. D 70, 063528, 2004) discussed it as the so-called “Boltzmann's brain” paradox.

P ∝ exp(ΔS/k_B)
P: relative probability of a fluctuation; ΔS: entropy change of the fluctuation, negative for a fluctuation into order (J/K); k_B: Boltzmann constant, 1.380 649 × 10⁻²³ J/K exactly in the 2019 SI

Nyquista

Any resistor at a temperature above absolute zero hisses with voltage noise, because its electrons jiggle thermally. John B. Johnson measured it at Bell Labs and Harry Nyquist explained it, both in 1928 (Phys. Rev. 32, 97 and 32, 110). The noise power per unit bandwidth depends only on temperature and resistance, not on the material.

⟨V²⟩ = 4k_BTRΔf
⟨V²⟩: mean-square noise voltage (V²); k_B: Boltzmann constant, exactly 1.380 649 × 10⁻²³ J/K since 2019; T: temperature (K); R: resistance (Ω); Δf: measurement bandwidth (Hz)

Seebecka

In 1821 Thomas Johann Seebeck joined bismuth and copper in a loop and heated one junction. A compass needle beside it swung. A temperature difference between the junctions drives a voltage, and in a closed loop a current: thermoelectricity. He reported it to the Prussian Academy (Abh. Akad. Wiss. Berlin, 1822–23).

V = S ΔT
V: thermoelectric voltage (V); S: Seebeck coefficient difference of the two metals (V/K); ΔT: temperature difference between the junctions (K)

Invertia

Temperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.

1/T = ∂S/∂E < 0
T: absolute temperature, in the standard (Boltzmann) definition (K); S: entropy (Boltzmann form, from the number of states at energy E) (J/K); E: total energy of the spin system (J); N_↑, N_↓: number of particles in the upper and the lower level; Δ E: energy gap between the two levels (J); k_B: Boltzmann constant (J/K)

Paddlewheel

James Prescott Joule let weights fall on pulleys, turning a paddle wheel inside an insulated vessel of water. Friction of the paddles warmed the water a tiny amount. Dividing the work done by the falling weights by the heat produced, he reported in 1850 (“On the Mechanical Equivalent of Heat”, Phil. Trans. R. Soc. 140, 61) that 772 foot-pounds of work raise one pound of water by 1 °F.

W = J Q
W: mechanical work (ft·lbf); J: mechanical equivalent of heat (ft·lbf per BTU); Q: heat produced (BTU)

Radiometra

A Crookes radiometer has four vanes, each black on one side and silver on the other, inside a bulb pumped down to a partial vacuum. In light, the vanes spin with the dark sides retreating. If light pressure drove it, the reflective sides would be pushed harder (a reflected photon gives twice the push of an absorbed one) and it would spin the other way. Instead, the dark side gets warmer, and the thin gas does the work. James Clerk Maxwell (Phil. Trans. R. Soc. 170, 231, 1879) and Osborne Reynolds, in his study of thermal transpiration (Phil. Trans. R. Soc. 170, 727, 1879), explained it: gas creeps around each vane's edge from the cold side to the hot side, and the reaction pushes the vane towards its cooler, silver side.

dark side warmer ⇒ vane moves dark side trailing
dark side: absorbs light and heats up; silver side: reflects light and stays cooler; gas: thin enough for molecules to travel far between collisions, but not a hard vacuum

Ratchetta

The ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.

rate forward ∝ e^{−ε/k_BT₁}, rate back ∝ e^{−ε/k_BT₂} ⇒ T₁ = T₂ gives ⟨W⟩ = 0
ε: energy needed to lift the pawl over one tooth (J); k_B: Boltzmann constant, exact in the 2019 SI; T₁: temperature of the gas around the vane (K); T₂: temperature of the ratchet and pawl (K); ⟨W⟩: average work delivered per unit time

Ruchardta

In 1929 Eduard Rüchardt measured the heat-capacity ratio of a gas with a simple trick (Phys. Z. 30, 58). A ball fits snugly in the neck of a large flask. Push it and it bounces up and down on the springy gas below. The squeezes are quick, so no heat flows and the gas acts adiabatically. Timing the oscillation gives γ.

γ = 4π²mV/(A²pT²)
γ: ratio of heat capacities C_p/C_v; m: mass of the ball (kg); V: volume of gas (m³); A: cross-section of the neck (m²); p: gas pressure under the ball (Pa); T: oscillation period (s)

Sortex

Landauer argued that erasing one bit of information must release at least k_B T ln 2 of heat. Because a Maxwell's demon has to erase its records eventually, this cost is what saves the second law of thermodynamics.

Q_erase ≥ k_BT ln 2
Q_(erase): heat released to the surroundings when one bit of memory is erased (a lower bound, not a fixed cost) (J); k_B: Boltzmann constant (J/K); T: temperature of the surroundings that absorb the heat (K); ln 2: natural logarithm of 2: one bit is a choice between two states

Argonix

Lord Rayleigh weighed nitrogen with great care and found that nitrogen taken from air was about half a percent denser than nitrogen made from ammonia, a gap far larger than his errors. With William Ramsay he removed all the nitrogen and oxygen from air and found a stubborn leftover gas, about 1% of the atmosphere, with a new spectrum and no chemistry at all. They named it argon, ‘the lazy one’ (“Argon, a New Constituent of the Atmosphere”, Phil. Trans. A 186, 187, 1895).

ρ_{air N₂} ≈ 1.005 ρ_{chem N₂}
ρ_{air N₂}: density of ‘nitrogen’ from air, which still contains argon; ρ_{chem N₂}: density of nitrogen from chemical sources

Echobel

In acoustic gas thermometry, the resonant frequencies of a gas-filled spherical cavity of known size give the speed of sound. For a dilute monatomic gas the speed of sound fixes k_BT per atom. Measurements of this kind at the triple point of water (NPL, LNE, INRiM and others) gave the most precise values of the Boltzmann constant before the 2019 redefinition.

c² = γk_BT/m
c: speed of sound in the zero-pressure limit (m/s); γ: heat-capacity ratio, 5/3 for an ideal monatomic gas; k_B: Boltzmann constant, 1.380 649 × 10⁻²³ J/K exactly; T: thermodynamic temperature (K); m: mass of one gas atom (kg)