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Connect one physical resistor $R$ across an ideal lossless capacitor $C$, with no applied source and an ideal readout that draws no electric current. Represent the resistor by its noisy Thévenin equivalent. The resulting RC low-pass frequency response gives the stationary capacitor voltage power spectral densityIts mean square follows by the effective noise bandwidth calculation:This is also the equipartition theorem: the mean capacitor energy is $\frac{1}{2}C⟨v_C^2⟩=\frac{1}{2}{k}_{\mathrm{B}}T$. Raising $R$ makes the low-frequency spectrum taller and its width narrower in exactly compensating proportions. It changes the equilibration time $RC$, not the final variance.

The integral to infinity belongs to the ideal classical model. For a physical electrical circuit, assume thermal equilibrium, linear components with negligible extra dissipation, and a corner far below the frequencies where parasitic effects, frequency-dependent electrical resistance, or $hf\sim k_{\mathrm{B}}T$ invalidate the model; the omitted high-frequency tail must be negligible. A finite-bandwidth instrument measures only part of this variance. Its input capacitance contributes to the total $C$, and its finite input electrical resistance introduces both loading and its own thermal electrical noise.

For $T=300\mathrm{K}$ and $C=1\mathrm{nF}$, the predicted equilibrium variance is $4.141947\times 10^{-12}{\mathrm{V}}^2$ and the root mean square voltage is $2.0352\mu \mathrm{V}$. At $R=10\mathrm{k}\Omega$, $RC=10\mu \mathrm{s}$ and $B_{\mathrm{n}}=25\mathrm{kHz}$. At $R=100\mathrm{k}\Omega$, $RC=100\mu \mathrm{s}$ and $B_{\mathrm{n}}=2.5\mathrm{kHz}$; the equilibrium root mean square voltage is identical.

To obtain the growth curve, prepare $v_C(0)=0$ exactly and connect the resistor at $t=0$ while maintaining its temperature. The RC low-pass filter's causal impulse response is $h(t)=e^{-t/(RC)}/(RC)$ for $t\ge 0$. Classical ideal white noise with two-sided power spectral density $2k_{\mathrm{B}}TR$ givesThe factor $2k_{\mathrm{B}}TR$ here is the two-sided time-domain noise intensity; the equilibrium spectral integral above used the one-sided $4k_{\mathrm{B}}TR$. They give the same long-time limit. The variance approaches equilibrium with time constant $RC/2$, although a deterministic initial voltage decays with time constant $RC$. If the capacitor already starts in thermal equilibrium, its initial variance supplies the missing exponentially decaying term and the total variance stays constant.

Whether increasing electrical resistance increases Joule heating depends on what the supply holds fixed. Consider separate ideal resistor experiments with $R=5\Omega$ and $R=10\Omega$, each maintained at its specified temperature. At fixed electric current $I_0=1\mathrm{A}$, doubling $R$ changes the required electric potential difference from $5$ to $10\mathrm{V}$ and doubles power from $5$ to $10\mathrm{W}$. At fixed electric potential difference $V_0=5\mathrm{V}$, doubling $R$ halves electric current from $1$ to $0.5\mathrm{A}$ and halves power from $5$ to $2.5\mathrm{W}$. These are stipulated model values, not measured data.

The comparison also predicts feedback when a particular resistor's electrical resistance rises with temperature: a supply maintaining electric current increases its heating power, whereas one maintaining terminal electric potential difference reduces it. Predicting the actual temperature still needs the material response $R(T)$ and its heat transfer to the surroundings; a nominal supply setting does not establish which drive is actually constant at the resistor.

In a dilute thermal helium gas with rapid formation–decay balance of beryllium-8, write its small equilibrium abundance schematically as $n_8\propto n_{\alpha }^2{T}^{-3/2}\exp [-{\Delta }_8/(k_{B}T)]$. A narrow isolated Hoyle state gives a second-step coefficient proportional to $T^{-3/2}\exp [-{\Delta }_C/(k_{B}T)]$. Multiplication yieldswith the stabilizing decay strength absorbed into the coefficient. Here $k_B$ is the Boltzmann constant. The two positive offsets count energy relative to their respective entrance thresholds, not relative to the carbon-12 ground state. A resonance enhances capture at the relevant energy; the Boltzmann factor still selects how many encounters reach it. This is a sequential narrow-level approximation with fixed nuclear widths, excluding plasma screening and changes of composition or mass density.

William Alfred Fowler lists the Hoyle state $7.654\mathrm{MeV}$ above the carbon-12 ground state, $0.2875\mathrm{MeV}$ above beryllium-8 plus an alpha particle, and $0.3794\mathrm{MeV}$ above three alpha particles.^[ref] Subtraction gives ${\Delta }_8=0.0919\mathrm{MeV}$ for the diagram.

Normalize the calculated rate at $T_0=10^8\mathrm{K}$ and $E_r=0.3794\mathrm{MeV}$. Holding alpha particle number density and the decay strength fixed gives the original sensitivity calculationAt $T_0$, a deliberately chosen shift $\delta E=+100\mathrm{keV}$ multiplies the rate by $9.12\times 10^{-6}$; $-100\mathrm{keV}$ gives $1.10\times 10^5$. These offsets are hypothetical perturbations, not alternative measured nuclear levels. The reference curve has local temperature exponent of a thermonuclear reaction $-3+E_r/(k_B{T}_0)=41.03$: raising temperature by $10\%$ gives a factor $41.12$ in this fixed-environment calculation. A real star changes its stellar structure and composition as it burns, so these factors do not directly predict final carbon abundance or a constraint on life.

For an original idealized mathematical model, let $n_{{\mathrm{O}}_2}(t)$ be the amount of substance of molecular oxygen consumed since the initial reading and $n_{{\mathrm{CO}}_2}(t)$ the carbon dioxide produced. Suppose a fraction $f$ of that carbon dioxide is removed from the gas, all other gas inventories remain fixed, and temperature $T$ and gas pressure $P$ stay constant. The ideal gas law gives the positive gas contractionwhere $R$ is the molar gas constant and $V$ the gas volume. With complete absorption, $f=1$, the contraction reflects molecular oxygen uptake regardless of the amount of carbon dioxide made. For a uniform pipette bore of area $A$, an inward boundary displacement $x$ gives $D=Ax$. A graduated pipette can instead supply the displaced volume directly.

Let $D_s$ denote sample contraction and $D_b$ the matched blank contraction. Define $D_{\mathrm{corr}}=D_s-D_b$ with both taken relative to their own initial readings. Under the preceding assumptions,The slope of corrected volume against time estimates uptake over the chosen interval. The approximate sign acknowledges that a liquid boundary also responds to hydrostatic and capillary pressure, while the gas can exchange water vapour. A fixed-volume apparatus would instead show a pressure decrease; its ideal conversion is $n_{{\mathrm{O}}_2}\simeq V(P(0)-P(t))/(RT)$ after complete absorption and appropriate blank correction.

The ideal gas law also exposes drift without any biology: for fixed gas inventory, $V\propto T/P$. A one-percent rise in absolute temperature at fixed pressure produces a one-percent expansion. Matching the initial gas spaces makes such common changes easier to subtract. Subtracting unmatched blanks can leave a physical signal that looks biological.

Here is an explicitly stipulated elementary kinetic model, not experimental data. A well-mixed ideal closed system has constant temperature and volume. Alongside an uncatalysed elementary reaction, allow an illustrative catalytic encounter that returns C immediately:Both channels obey the law of mass action. In the second, the forward reaction rate is ${\alpha }_f[\mathrm{A}][\mathrm{C}]$ and the reverse reaction rate is ${\alpha }_r[\mathrm{B}][\mathrm{C}]$. C cancels in each encounter, so its free concentration $c_C$ stays constant without an external supply. This toy elementary reaction encodes regeneration without binding; it is not a claim that real catalytic cycles universally consist of one encounter. A binding cycle can accumulate a reaction intermediate, in which case constant total C does not imply constant free $c_C$.

Adding the parallel reaction rates gives effective first-order reaction rate constantsBecause $k_f^u/k_r^u={\alpha }_f/{\alpha }_r=3$, each channel balances at the same $[\mathrm{B}]/[\mathrm{A}]=3$, and so does their sum. At $c_C=0$ the pair is $(0.030,0.010){\mathrm{s}}^{-1}$; at $c_C=0.001\mathrm{mol}{\mathrm{L}}^{-1}$ it is $(0.300,0.100){\mathrm{s}}^{-1}$. Both give $K=3$ with ideal thermodynamic activities and a common concentration standard state. At $T=300\mathrm{K}$ their compatible standard reaction change in Gibbs free energy is ${\Delta }_r{G}^{\circ }=-RT\ln 3\approx -2.7403\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Choosing an unrelated reverse rate constant would break this consistency.

Let $S=[\mathrm{A}]+[\mathrm{B}]$ be the conserved concentration, and define the dimensionless quantity $x=[\mathrm{B}]/S$. ThenSolving this first-order linear differential equation givesThe chemical relaxation time is $25\mathrm{s}$ without C and $2.5\mathrm{s}$ with the chosen C. From either $x_0=0$ or $x_0=1$, the exponential decay of the deviation leads to the same chemical equilibrium $x=3/4$. Reducing that deviation to one percent takes $\ln (100)\tau$: about $115.13\mathrm{s}$ versus $11.51\mathrm{s}$.

At this chemical equilibrium, conversion continues in both directions. For $S=1\mathrm{mol}{\mathrm{L}}^{-1}$, the uncatalysed channel has equal forward and reverse reaction rates $0.0075\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$. Adding C gives total directional reaction rates $0.075\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ each, yet zero net reaction rate. Faster molecular exchange and unchanged composition coexist. This exact calculation explains how catalysis can matter enormously within an observation window while leaving the final equilibrium constant intact.

An ideal cyclic heat engine producing no entropy overall. Between two equilibrium reservoirs its efficiency reaches the Carnot bound; finite reservoir temperatures can be treated as varying through successive reversible cycles.

At fixed temperature, a photon gas has constant energy density $u$ and radiation pressure $p=u/3$. An increase $\Delta V$ in volume requires both $\Delta U=u\Delta V$ and expansion work $p\Delta V$, giving absorbed heat $4u\Delta V/3$.

A thermodynamically reversible process can be reversed without a net change of the system and surroundings. Its entropy change equals reversible heat transfer divided by temperature. A reversible adiabatic process conserves entropy; adiabaticity alone does not forbid internal entropy production in an irreversible evolution.

In his 1883 paper, Osborne Reynolds investigated how the character of pipe flow depends jointly on velocity, pipe diameter and viscosity. His final dye apparatus drew water from a large glass tank through a horizontal glass tube with a smooth trumpet-shaped entrance. A separate reservoir supplied strongly colored water just upstream of the entrance; a valve below the tank controlled the outflow. The glass tube was immersed in the tank, so its internal dye band could be watched. Reynolds repeated the trials with different tube diameters and water temperatures. ^[ref]

The volumetric flow rate was measured independently of the appearance of the dye. Reynolds timed a one-inch fall of the tank's water level, corresponding to a calibrated discharged volume. In modern notation, if this volume is $V$ and the elapsed time is $\Delta t$, then $Q=V/\Delta t$ and $U=4V/(\pi D^2\Delta t)$. He determined the glass tubes' internal diameters by filling them with water. For a uniform bore, a known filled length $\ell$ and measured water volume $V_{\mathrm{tube}}$ determine $D=\sqrt{4V_{\mathrm{tube}}/(\pi \ell )}$; checking bore uniformity matters because the same volume would only give an average cross-section in a tapered tube. Thermometers at different tank levels checked both the temperature and its uniformity. Reynolds compared the viscosity at different temperatures using Poiseuille's results and checked his own water by separate experiments of that type. Thus the Reynolds number estimate did not come from judging the dye band or from assuming that transition occurred at a preset value. ^[ref]

A practical repetition can instead collect a known outflow volume in a measuring vessel and time its collection, or weigh the collected water and divide its mass by mass density. Measure the internal diameter, rather than the external diameter, and obtain kinematic viscosity at the measured water temperature from independently established properties or a separate viscosity measurement. Record the dye appearance and observation position alongside $Q$, $D$, temperature and the resulting Reynolds number. These are alternative measurement procedures following $Q=V/\Delta t$, not additional historical claims about Reynolds' dye apparatus.

Begin with settled water and a small dye supply, then open the outlet valve in small steps. Reynolds saw a thin straight band at sufficiently low speeds. With a tank that had not fully settled, the band could wander while retaining its coherence. At higher discharge it could abruptly spread into the surrounding water downstream of the entrance. Short illumination by an electric spark resolved curls within what otherwise looked like a colored cloud. This connected the visible breakup with internal irregular motion rather than merely the appearance of an opaque pipe. ^[ref]

The controls are part of the result. Reynolds used a smooth entrance and let the tank rest to reduce inlet disturbances. He found that a heated tank could retain slow circulation as it cooled, so measuring one temperature was insufficient: thermal uniformity and residual motion also mattered. His wire-coil trial broke up the band below the previous onset speed, but he explicitly cautioned that once the dye was mixed he could not tell from that dye alone whether the motion became orderly again farther along the tube. In a repetition, keep the dye supply small enough to act approximately as a passive scalar, avoid a strong injection jet, and record the inlet condition and downstream observation distance. If also testing the Hagen-Poiseuille equation, allow the profile to develop and measure pressure drop between positions within the developed region; do not treat the entrance and outlet losses as part of its uniform pressure gradient. The dye experiment by itself does not establish fully developed flow. ^[ref]

The experiment's usefulness is that it compares a visible change in the motion with independently measured scales. Repeating it with several diameters and temperatures, while controlling disturbances, tests the organizing role of the Reynolds number. Its limitations are equally instructive: dye is a tracer of transport, diffusion still acts in laminar flow, and a local dye observation cannot by itself prove that turbulence persists throughout an arbitrarily long pipe.

A rheometer imposes a controlled deformation or stress and measures the material response to determine a constitutive equation.

Measurement of stress–deformation response using a calibrated rheometer. Geometry, temperature, sample history and possible wall slip must be controlled before interpreting a measured force or torque as a bulk constitutive response.

Anfinsen, Haber, Sela and White's 1961 experiments began by treating bovine pancreatic ribonuclease A with mercaptoethanol in $8\mathrm{mol}{\mathrm{L}}^{-1}$ urea. Urea promotes unfolding; mercaptoethanol breaks protein disulfide bonds. The peptide bonds along the chain remain intact. After separating the chain from those reagents, the authors allowed its reactive sulfur groups to form protein disulfide bonds in the presence of molecular oxygen, at pH $8.2$ and temperature $23$–$24{}^{\circ }\mathrm{C}$. They examined protein concentrations of $0.1$ and $2.0\mathrm{mg}\mathrm{m}{\mathrm{L}}^{-1}$. Measurements showed recovery of native properties under these conditions; the isolated major component had normal activity toward both RNA and a synthetic substrate. See Anfinsen, Haber, Sela and White, The Kinetics of Formation of Native Ribonuclease during Oxidation of the Reduced Polypeptide Chain, PNAS 47, 1309–1314 (1961), pp. 1309–1311.

The observation supports the sufficiency of that intact sequence to recover native protein structure in the tested surroundings without a folded template. It does not establish that every protein refolds, that the original living biological cell needed no assistance, or that activity always returns immediately. Protein folding, forming protein disulfide bonds and recovering an active site are related processes rather than interchangeable measurements.

Anfinsen's lecture also discusses forming protein disulfide bonds while the chain remains in concentrated urea. This gave mostly inactive material; after removing urea, a small amount of mercaptoethanol permitted rearrangement toward the native form. As an exact combinatorial check, eight labeled reactive sulfur groups have $7\times 5\times 3\times 1=105$ ways to be paired into four protein disulfide bonds: choose the partner of one group, then repeat with the unpaired groups. Only one pairing is the native one. Equal counting does not mean that all pairings are equally likely experimentally, nor that actual protein folding searches them uniformly. See Anfinsen's Nobel lecture, 1973 publication: scrambled ribonuclease and the 105 pairings, p. 223.

Ribonucleic acid, or RNA, is a nucleotide polymer whose nucleotide sequence can both carry information and organize a working molecular shape. Its backbone contains alternating ribose sugars and phosphate groups connected through covalent bonds. Its usual nucleobases are adenine A, uracil U, guanine G, and cytosine C. Thus RNA uses U where DNA uses T; its ribose also differs chemically from DNA's deoxyribose. See NHGRI: RNA.

In transcription, an enzyme uses one DNA strand as a template for RNA. For example, the DNA template $3^{\prime }\text{-}\mathrm{TACGG}\text{-}{5}^{\prime }$ specifies $5^{\prime }\text{-}\mathrm{AUGCC}\text{-}{3}^{\prime }$ in RNA. The base pairing rules now include A–U rather than A–T, while G–C remains. This RNA matches the other DNA strand's $5^{\prime }$-to-$3^{\prime }$ lettering after T is replaced by U. Identifying the template strand matters: replacing T by U in an arbitrarily chosen DNA strand is not a general method for deriving the RNA product. See Alberts et al., Molecular Biology of the Cell: from DNA to RNA.

RNA is often a single strand, but single-stranded does not mean straight or entirely unpaired. Complementary regions within one RNA molecule can undergo base pairing, bringing distant positions together. Consider the illustrative twelve-nucleotide nucleotide sequence $5^{\prime }\text{-}\mathrm{GCAUUUUUAUGC}\text{-}{3}^{\prime }$. Pair positions 1 with 12, 2 with 11, 3 with 10, and 4 with 9. The resulting stem-loop has four base pairs: G–C, C–G, A–U, U–A. Positions 5–8 remain unpaired in the loop. Both arms belong to the same continuous backbone, and the paired arms have the directions of antiparallel nucleic acid strands.

The diagram proposes a possible pairing pattern, not a prediction that this short RNA must adopt that shape under every condition. Actual RNA folding depends on additional interactions, temperature, ions, and associated proteins; the full three-dimensional arrangement contains more information than the drawn base pairs. The example's purpose is to show how a linear nucleotide sequence can specify contacts within itself.

Different RNA roles make this flexibility useful. Messenger RNA carries a message that a ribosome reads during translation to build a protein. Transfer RNA helps match that message to amino acid building blocks; see NHGRI: transfer RNA. Ribosomal RNA forms structural and catalytic parts of the ribosome. Other RNA participates in gene regulatory networks, changing how strongly genes are expressed. Many RNA molecules therefore function without being translated into proteins; for some genes, functional RNA is the product itself.

Most strikingly, some folded RNA acts as an enzyme: these catalytic RNA molecules are ribozymes. The ribosome's bond-forming active site is built from RNA, although the complete ribosome also contains proteins. Structural evidence appears in Nissen et al., The structural basis of ribosome activity in peptide bond synthesis, Science 289, 920–930 (2000). RNA therefore connects a readable record with physical recognition, regulation, and catalysis: the same kind of polymer can be a message, an adaptor, or part of molecular machinery.

A ribozyme is an RNA molecule whose folded structure provides catalysis of a chemical reaction. A protein enzyme builds its active site from an amino acid chain; a ribozyme builds a working active site from RNA, sometimes with metal ions and associated proteins. Base pairing can help recognize a substrate, but recognition alone does not establish catalysis: the RNA must promote chemical conversion. A passive messenger RNA template being read by a ribosome is not thereby a ribozyme. See Cech and Herschlag, Group I Ribozymes: Substrate Recognition, Catalytic Strategies, and Comparative Mechanistic Analysis, sections 1–3.

The surprise is that an information-bearing polymer can also perform chemistry. Examples include the self-splicing group I intron of Tetrahymena thermophila and the catalytic RNA component of bacterial RNase P, discussed below. The ribosome supplies an everyday example: its peptide bond-forming active site is made of ribosomal RNA, although the complete machine also contains proteins. Nissen and colleagues examined structures of a large ribosome subunit with substrate analogues and found RNA contacting those analogues at the catalytic centre. This supports an RNA active site, not a claim that isolated ribosomal RNA can carry out all of biological translation. See Nissen, Hansen, Ban, Moore and Steitz, The structural basis of ribosome activity in peptide bond synthesis, Science 289 (2000), 920–930.

For the Tetrahymena group I intron, RNA splicing removes the intron and joins its flanking exons. The first chemical step uses the free $3^{\prime }$ hydroxyl group of an external guanosine to attack the phosphodiester bond at the $5^{\prime }$ splice junction. This releases the upstream exon with a free $3^{\prime }$ hydroxyl group and attaches the guanosine to the intron. That newly freed hydroxyl group then attacks the phosphodiester bond at the other splice junction, joining the exons and releasing the linear intron. Each step exchanges one backbone connection for another; it is not simply separation of base pairs. See Cech, Zaug and Grabowski, 1981: the phosphoester-transfer mechanism.

The motivating experiment began with an unexpected control. As Cech recounts, Zaug purified a ribosomal RNA precursor from Tetrahymena using a detergent and phenol extraction, then compared test tubes with and without added cell nucleus extract. Incubation in salts and nucleotides released the same amount of small intron RNA in both cases, detected by gel electrophoresis. Repetition made a missing ingredient or accidental omission less plausible. Activity also survived treatments intended to destroy proteins. Nevertheless, absence of added extract did not by itself exclude a tightly attached, resistant protein, or prove that the small RNA had been newly excised rather than released from an already processed complex. See Thomas R. Cech, 1989 Nobel lecture, pp. 655–659: the unexpected control and alternative explanations.

An independent chemical readout addressed the latter problem. Cech mixed precursor RNA with phosphorus-32-labelled guanosine bearing three phosphate groups (GTP), and used gel electrophoresis followed by exposure to X-ray film to locate the label. Label appeared in the excised intron with magnesium present; it was absent when magnesium was omitted or labelled ATP replaced the labelled guanosine reagent. This experiment is described in Cech, 1989 Nobel lecture, pp. 657–658: labelled-guanosine controls. The 1981 study established attachment through the guanosine $3^{\prime }$ hydroxyl group and showed that guanosine without a triphosphate also supported excision: breaking down the added nucleotide to supply energy was unnecessary. See Cech, Zaug and Grabowski, In vitro splicing of the ribosomal RNA precursor of Tetrahymena, Cell 27 (1981), 487–496: primary abstract. Separately, sequencing and terminal analysis found an extra $5^{\prime }$ guanosine absent from the corresponding DNA sequence. See Zaug and Cech, The intervening sequence excised from the ribosomal RNA precursor of Tetrahymena contains a 5-terminal guanosine residue not encoded by the DNA, Nucleic Acids Research 10 (1982), 2823–2838. Together, incorporation and sequence evidence support a new covalent bond, beyond a change in how RNA associates.

Kruger, Grabowski, Zaug, Sands, Gottschling and Cech's 1982 test removed dependence on a precursor purified from Tetrahymena. They cloned the intron and adjacent exons into a DNA template, made precursor RNA with purified bacterial RNA polymerase under conditions suppressing RNA splicing, and removed proteins by detergent–phenol extraction. Denaturing gel electrophoresis separated precursor species. Purified precursor was then incubated with the guanosine reagent, magnesium and salt; matched samples without this incubation supplied a precursor baseline. Their standard assay used $10\mathrm{mM}$ magnesium chloride, $0.1\mathrm{mM}$ of that guanosine reagent, $100\mathrm{mM}$ ammonium sulfate and a buffer solution at pH $7.5$, for $30$ minutes at temperature $37{}^{\circ }\mathrm{C}$, with specified variations in individual experiments. Gel electrophoresis and radioactive-film detection showed excised intron RNA; sequencing verified its $5^{\prime }$ junction. See Kruger et al., Self-Splicing RNA, Cell 31 (1982), 147–157, figures 3–4 and Experimental Procedures.

Excision alone could leave disconnected exons. To test joining, the authors paired labelled RNA with DNA spanning the joined-exon sequence, removed exposed RNA with enzymes, and separated the protected pieces by denaturing gel electrophoresis. Spliced samples yielded protected RNA long enough to span the junction. See Kruger et al., 1982, figure 6: independent evidence for exon ligation. The experimental logic matters: an intron-sized band identifies a candidate product, while sequence and junction assays test its chemical identity.

These controls support sufficiency of the prepared RNA under the assay conditions. They do not show that every RNA self-splices, or that proteins are dispensable inside a biological cell. Many group I introns use proteins to help attain an active fold. See Cech and Herschlag, Group I Ribozymes, section 6: protein facilitation. Nor does a failed assay necessarily identify an inactive nucleotide sequence: unsuitable ions, pH, or temperature can prevent a working active site from forming.

The distinction between self-processing and repeated use became especially clear with bacterial RNase P. In 1983 Guerrier-Takada, Gardiner, Marsh, Pace and Altman separated the RNA and protein components of RNase P from two bacteria. Purified RNA cleaved precursor transfer RNA under conditions containing $60\mathrm{mM}$ magnesium ions, or $10\mathrm{mM}$ magnesium ions plus $1\mathrm{mM}$ spermidine. The protein components alone had no catalytic activity. At $5$–$10\mathrm{mM}$ magnesium without spermidine, both components were needed. Thus the comparison demonstrated an RNA catalyst while also explaining why protein assistance could matter under other conditions. RNase P produces the mature $5^{\prime }$ end of transfer RNA by removing its leader; it acts on a separate substrate and can be used again. See Guerrier-Takada et al., The RNA moiety of ribonuclease P is the catalytic subunit of the enzyme, Cell 35 (1983), 849–857: primary abstract.

Self-splicing demonstrates RNA chemistry, but one precursor completing one rearrangement is not yet evidence for catalytic turnover: repeated conversion of separate substrates by a regenerated catalyst. The self-splicing precursor changes its own connectivity and incorporates external guanosine. In 1986 Zaug and Cech showed that a shortened Tetrahymena intron RNA could cleave and rejoin separate short RNA substrates in a sequence-dependent reaction. This demonstrated an RNA enzyme acting beyond its own self-processing. See Zaug and Cech, The intervening sequence RNA of Tetrahymena is an enzyme, Science 231 (1986), 470–475: primary abstract.

Read the second figure as a counting argument. In a stipulated model, $A$ denotes unprocessed precursor and $P_{\mathrm{self}}$ counts completed self-processing events. Set $A(0)=R_0$, $P_{\mathrm{self}}(0)=0$ and $dA/d\tau =-A$. The exponential function solution gives $P_{\mathrm{self}}/R_0=1-e^{-\tau }$, never exceeding one. Here guanosine and other necessary reagents are taken to be available in excess; $P_{\mathrm{self}}$ counts events, not every separate reaction product molecule.

For a second stipulated model, let an unchanged ribozyme $R$ perform $R+S⟶R+P$ on a separate substrate $S$. Assume the effective law of mass action $dS/dt=-kRS$, no reverse reaction, no ribozyme loss and no inhibited or long-lived bound states. Set $R=R_0$, $S(0)=5R_0$, $P(0)=0$ and $\tau =k{R}_{0}t$. Then $S/R_0=5e^{-\tau }$ and $P/R_0=5(1-e^{-\tau })$, with $S+P=5R_0$. At $\tau =1$, self-processing gives about $0.632$ completed events per initial RNA, whereas the regenerated ribozyme gives about $3.161$ reaction product molecules per initial ribozyme. More than one substrate conversion per initial ribozyme, with appropriate background and identity controls, is evidence for turnover. Product accumulation below that threshold alone cannot distinguish a reusable catalyst from one-shot chemistry. These models illustrate that distinction; actual enzyme kinetics can include folding delays, binding, saturation and product release.

A physicist in the NIST Photon Physics Group.^[ref]

The Richardson-Dushman law models the magnitude of emitted electric current density asHere $T$ is absolute thermodynamic temperature, $W$ is the work function, $k_B$ is the Boltzmann constant, and $A$ is the Thermionic Richardson constant. The signed electric current density points opposite to the outgoing electron motion; $J$ here is positive. For uniform emission over area $S$, the emitted electric current magnitude is $I=SJ$.

The Richardson-Dushman law assumes an emitter close to thermal equilibrium, a surface with a well-defined and approximately constant work function over the fitted temperature range, and predominantly thermal escape over the barrier. Interpreting a measured collector electric current with this law additionally requires an emission-limited thermionic current, known emitting area, and negligible return emission from the collector. Strong electric fields can lower the surface barrier or enable quantum tunnelling; then this zero-field expression is insufficient.

For a deliberately chosen teaching model, take $W=2.0\mathrm{eV}$ and $A=120\mathrm{A}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{K}}^{-2}$, expressing the barrier in electronvolts. These are illustrative parameters, not measured properties of a named material. Using $k_B=8.617333262\times 10^{-5}\mathrm{eV}{\mathrm{K}}^{-1}$, at $1000\mathrm{K}$ the Boltzmann factor is about $8.33\times 10^{-11}$, yetA 20 percent increase in temperature raises the predicted electric current density by about 69 times. For a chosen emitting area $S=0.010\mathrm{c}{\mathrm{m}}^2$, these correspond to approximately $0.100\mathrm{mA}$ and $6.89\mathrm{mA}$ if collection is complete. The small escaping fraction is replenished by the external electrical circuit.

The sensitivity is especially clear from the logarithmic derivative at fixed $A,W$:In the chosen model at $1000\mathrm{K}$, a $1\mathrm{K}$ rise changes $J$ by about 2.5 percent to first order. Accurate temperature measurement matters even when electric current measurement is easy.

In his Nobel lecture, delivered on 12 December 1929 for the 1928 prize, Owen Willans Richardson recalled heating wires for weeks while hand-pumping away gas released by the apparatus. With platinum, he found a finite emission per unit area that rose rapidly with temperature. His initial 1901 formula used $T^{1/2}$; he preferred the $T^2$ form in 1911.^[ref]

The early apparatus surrounded a heated platinum wire with a metal cylinder connected to earth through a galvanometer. Wire electrical resistance supplied the temperature readout. After checking collection saturation, Richardson measured at a wire potential of $-120\mathrm{V}$, over roughly $1300$–$1600\mathrm{K}$; reversing polarity suppressed the negative discharge.^[ref]

The inference rests on several controls. A temperature sweep probes thermal escape; reversing the collecting electric field tests the sign of the mobile electric charge; varying collection voltage tests whether transport is limiting. Prolonged pumping and heating test whether a transient gas release is responsible. These controls address different alternatives, and none by itself establishes a clean, uniform surface.

For a modern repetition, stabilize each temperature, measure the collector electric current against collecting electric potential difference, and record the pressure and surface preparation. Repeat selected temperatures after cooling to reveal drift. Use the collection plateau and the actual emitting area to construct a Richardson plot. The resulting slope estimates the work function only within the stated surface and transport assumptions; it does not turn an uncontrolled wire into a reference material.

Choose a fixed positive reference $A_{\mathrm{ref}}$ with the same units as $A$. Taking the natural logarithm of a dimensionless ratio givesThus the Richardson plot is a straight line when $A$ and $W$ are constant, with slope $-W/k_B$ and intercept $\ln (A/A_{\mathrm{ref}})$. The familiar notation $\ln (J/T^2)$ silently chooses units; the reference makes that choice explicit. Changing $A_{\mathrm{ref}}$ shifts the intercept and leaves the inferred work function unchanged. If the horizontal axis instead uses $1000\mathrm{K}/T$, its slope is $-W/(1000\mathrm{K}{k}_B)$.

For the chosen $2.0\mathrm{eV}$ example, choose $A_{\mathrm{ref}}=120\mathrm{A}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{K}}^{-2}$. Then $y(1000\mathrm{K})=-23.2090$ and $y(1200\mathrm{K})=-19.3409$. Their slope is approximately $-23209\mathrm{K}$, giving $W=2.0\mathrm{eV}$. Two temperatures determine a line algebraically; several temperatures allow curvature and drift to be detected. These values are calculated, rather than experimental observations.

A straight trace alone does not establish a unique surface work function. Adsorption, changing surface coverage, or a mixture of regions with different barriers can change the inferred slope. For independent emitting patches with area fractions $f_i$, constant $A_i$ and barriers $W_i$,Lower barriers contribute disproportionately. Even with fixed patches, the slope of the Richardson plot gives a temperature-dependent weighted mean of the $W_i$, rather than their simple area average. If $A$ or $W$ themselves vary with temperature, the apparent slope-derived barrier becomesCheck collection, temperature calibration, and surface stability before assigning curvature to a new emission mechanism.