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Astrophysics uses physics to explain stars, planets, galaxies, and their electromagnetic radiation; cosmology extends the questions to the universe as a whole. Much of its evidence arrives as light. The challenge is to infer the source's physical properties from what reaches an observer: a bright star might be nearby, unusually luminous, or both. Stellar parallax and observed radiative flux make this distinction quantitative.

This is an inverse problem. A forward problem predicts observations from a source model and the intervening radiative transfer; the inverse problem works backwards from observations to possible models. A spectral line can constrain chemical elements, temperature, and motion through the Doppler effect, but its strength also depends on the stellar atmosphere. Combining different measurements is therefore more informative than interpreting brightness alone. The attraction is that a tiny change in the direction of light can supply a distance, turning received radiative flux into intrinsic luminosity. ^[ref].

The ideal Atkinson cycle considered here has reversible adiabatic compression, isochoric heat addition, a longer reversible adiabatic expansion and isobaric heat rejection. Its expansion ends at the initial pressure rather than at the initial volume. With constant ideal-gas heat capacities, $r=V_1/V_2$ and $r_p=p_3/p_2$, the temperatures are $T_2=T_1{r}^{\gamma -1}$, $T_3=r_p{T}_2$, and $T_{4^{\prime }}=T_1{r}_p^{1/\gamma }$. This ideal cycle is distinguished from the detailed valve timing and losses of a practical engine.

A gravitationally retained gaseous envelope of a planet. Its pressure, composition and temperature determine its radiative transfer and circulation. A pure saturated water atmosphere follows the saturation pressure along a phase coexistence curve when condensate is locally available.

The lowest part of the terrestrial atmosphere that responds directly to surface fluxes on relatively short timescales through turbulence. Its depth changes with heating, stability and weather; a sharp composition or temperature change can mark its top. A uniform-layer chemical budget converts a surface flux into a volume tendency by dividing by layer depth.

The mass per unit horizontal area above a level in an atmosphere. With constant gravitational acceleration, hydrostatic balance gives $m=(p_s-p_{\mathrm{top}})/g$. This relation connects an observed surface pressure to the total reservoir mass without assuming an isothermal temperature profile.

The fractional excess of water-vapor pressure above equilibrium saturation over a flat pure-water surface at the same temperature: $s=p_v/p_{\mathrm{sat}}(T)-1$. Positive $s$ can drive droplet growth, but activation of a particular aerosol particle requires exceeding its critical supersaturation. Cooling, condensation and mixing jointly determine parcel supersaturation.

Hexokinase provides a concrete connection between sugar chemistry and energy coupling: it transfers a terminal phosphate group from ATP to glucose, forming a phosphorylated sugar and adenosine diphosphate. The enzyme accelerates the transfer, while consuming ATP changes the net chemical reaction being carried out. The donated group reaches the substrate through the catalytic route.^[ref]

For a symbolic teaching model, write $S$ for an unphosphorylated sugar, $S\text{-}P$ for its phosphorylated form, and $P_i$ for inorganic phosphate. These are labels for chemical states, not complete elemental formulas. Using one consistent reference convention at fixed pH and temperature, stipulate the following molar changes in Gibbs free energy:All three numbers are chosen illustrative values, not measured properties of glucose or a particular enzyme. The prime marks a biochemical reference convention with pH held fixed; explicit proton and metal-binding bookkeeping is suppressed consistently. The first two lines are a thermodynamic decomposition: cancelling water and inorganic phosphate gives the net transfer. They do not assert that the reaction mechanism first releases free inorganic phosphate and then recaptures it.

Because Gibbs free energy is a state function, consistently defined reaction changes add. A working reaction mechanism must also connect the desired conversion to ATP consumption; independently allowing ATP hydrolysis beside the sugar does not establish energy coupling. The general distinction is developed under energy coupling. ATP-dependent phosphorylation of a membrane protein can likewise connect a chemical change to active transport.^[ref]

Actual Gibbs free energy changes depend on thermodynamic activities. Even a favourable reference sum can be overcome by accumulation of reaction products or depletion of ATP; it supplies no prediction of the reaction rate. A living biological cell also regenerates ATP. Sustained operation therefore requires a connected metabolism, not an inexhaustible stock of a special molecule.

Define side $P$ as the higher proton electrochemical potential reservoir and side $N$ as the synthesis-side reservoir. For transfer $P\to N$ at common temperature $T$, define the positive downhill proton-motive force magnitude $\Delta p_{\mathrm{drive}}$ byHere $a$ is thermodynamic activity, $\psi$ is electric potential, $R$ is the molar gas constant and $F$ is the Faraday constant. Thus both a higher proton thermodynamic activity and a more positive electric potential on side $P$ contribute to the driving magnitude. This definition fixes the sign; conventions for a signed proton-motive force differ. The calculation under chemiosmosis separates these two contributions numerically.

Assume fixed reservoirs, tight mechanical energy coupling, one transported proton per $c$ subunit per revolution, and three ATP produced per revolution. Neglect leaks, slipping and membrane transport of the chemical reagents. Let $g_{\mathrm{ATP}}>0$ be the local molar Gibbs free energy cost of ATP formation at the stipulated composition, pH and magnesium-binding convention. It is not necessarily the standard state value. Adding changes in the state function gives, per mole of complete turns,For this model net synthesis is thermodynamically favoured when $\Delta G_{\mathrm{turn}}<0$. Equality is a reversible balance, not a promise of a useful reaction rate. A negative value establishes an available driving margin; it does not specify how fast the enzyme runs.

Choose $n=10$, $g_{\mathrm{ATP}}=50.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, $F=96485\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$ and $\Delta p_{\mathrm{drive}}=0.180\mathrm{V}$. These are illustrative inputs, not measurements of the preparation in the experiment below. Thenand $\Delta G_{\mathrm{turn}}=-23.673\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Per ATP, the ideal input is $57.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, leaving a driving margin of $7.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The balance point isAt $0.140\mathrm{V}$ the same synthesis cycle instead has $\Delta G_{\mathrm{turn}}=+14.921\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: the chosen reservoirs favour its reverse.

The machine is reversible: sufficient ATP hydrolysis can drive opposite rotation and pump protons uphill. Actual direction also depends on inhibition and available substrates; a weak proton-motive force need not make every intact enzyme immediately run backward. Respiratory uncoupling supplies a different failure mode: a parallel proton leak dissipates the gradient without productive shaft rotation.^[ref]

The motor budget $10/3$ is not the whole cost of ATP delivered outside a mitochondrion. Import of ADP and inorganic phosphate, and export of ATP, also draw on the electrochemical gradient.^[ref] For a deliberately simplified system budget, assign these transports together one additional downhill-proton equivalent per delivered ATP, valued at the same $F\Delta p_{\mathrm{drive}}$. This is a chosen accounting convention, not a claim of one extra literal proton through $F_0$, nor a universal transport stoichiometry. The total becomes $10/3+1=13/3$ equivalents and $75.258\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of gradient expenditure per delivered ATP at $0.180\mathrm{V}$. The ratio of stored chemical free energy to that expenditure is $50/75.258\approx 66.4\%$, compared with $50/57.891\approx 86.4\%$ for the motor alone. This system budget changes the resource expenditure, not the motor's balance point $\Delta p_{\ast }$. Real transport terms require their own electrochemical potentials, and leaks add further cost.

A steel with an austenitic, face-centered cubic matrix in the specified condition. Alloying and temperature determine its stability.

Equal volumes of ideal gases at the same temperature and pressure contain equal amounts of substance. Named after Amedeo Avogadro. From $pV=nRT$, $V/n=RT/p$ is common to them.

An azeotrope has equal coexisting liquid and vapor compositions: $y=x$. Ordinary fractional distillation cannot cross this composition boundary at fixed pressure; more packing does not remove it. A constant head temperature therefore does not establish a pure chemical substance.^[ref] The stipulated $\alpha =2$ model has $y>x$ throughout $0<x<1$, so it has no interior azeotrope and cannot represent this limit.

Stepwise conductance as transverse propagating modes open in a clean constriction. Each fully transmitted spin-degenerate subband contributes $2e^2/h$; finite temperature, imperfect transmission and unresolved modes round the steps.

For fixed band effective masses and band gap $E_g$, the intrinsic carrier concentration obeys $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. ThusA base-ten plot against $x=1000\mathrm{K}/T$ instead has slope $-E_g/(2000k_B\mathrm{K}\ln 10)$ after the prefactor correction. The intrinsic branch must be used, rather than the extrinsic plateau or dopant freeze-out branch. A changing gap or effective mass makes this an activation-energy estimate, and a small gap uncertainty can produce a much larger multiplicative uncertainty in a room-temperature minority density.

In December 1947, John Bardeen and Walter Houser Brattain obtained transistor action at Bell Telephone Laboratories. William Shockley proposed the distinct bipolar junction transistor within the following month.^[ref]

Their 1948 paper describes ground, etched germanium, initially also anodized and rinsed. Surface contacts separated by $50$–$250\mu \mathrm{m}$ share a broad base contact. Positive emitter biasing injects electron holes; negative collector biasing draws them toward the collector. An input signal changes emitter electric current, and a collector electrical load converts the resulting change into voltage.^[ref]

Early electric field control experiments gave unexpectedly weak responses. John Bardeen proposed that surface states screened the interior. Surface charge could create a barrier and a hole-rich semiconductor inversion layer. Replacing an electrolyte control with a gold contact reversed the response sign, suggesting electron hole carrier injection. Closely spaced contacts made from a gold-coated wedge split with a razor then enabled power gain. The relative importance of surface and bulk transport depended on preparation and collector forming.^[ref]

The 1948 authors found that collector-current increments could exceed emitter-current increments; their current gain depended on biasing.^[ref] A terminal-current ratio alone cannot identify the microscopic path. Surface states, the local barrier, carrier injection and contact preparation prevent a single ideal modern bipolar junction transistor model from explaining every point-contact transistor.

For the conventional electric current directions drawn in the figure, $I_E$ enters the emitter, $I_C$ leaves the collector and signed $I_B$ enters the base. Kirchhoff's current law requiresThus a local current gain $\alpha >1$ requires an accompanying base-current change. The collector increment is not simply a count of injected electron holes, and it does not violate electric charge conservation.

For an experiment following the functional arrangement, first establish each contact's separate current-voltage characteristic relative to the base. Then hold collector biasing fixed, vary a small emitter signal, and compare the collector electrical load voltage with the input. A decisive power gain check measures both input electric current and voltage and the output power in a known electrical load; voltage enlargement alone is insufficient. These are proposed measurement controls, not additional claims about the 1947 laboratory record.

Repeating a setting tests drift; reversing or removing emitter biasing tests whether the response requires carrier injection. Contact motion, surface contamination, temperature, direct electrical coupling and noise can all affect the inference. A small-signal model describes measured local terminal response without deciding every microscopic mechanism.

The 2015 experiment used an air-driven nonmagnetic rotor containing polycrystalline gadolinium, with magnetic shielding and an external fluxgate magnetometer. Air drive reduced motor backgrounds. Measurements at 297–305 K were above the Curie temperature, at rotation frequencies up to 1.5 kHz. Reversing airflow reversed rotation and the magnetic signal; an empty capsule showed no speed-dependent signal. The readout measured stray magnetic field; geometry converted it to magnetization. Sample temperature was checked immediately after stopping. Heating and the conversion model limit inference.^[ref]

An additional analysis check follows from rotation symmetry. For signed sensor output $S$, separateAn ideal linear Barnett effect appears in the odd component; a fixed offset appears in the even component. This is a proposed diagnostic, not a claim about the paper's processing. A background correlated with direction can also be odd: reversal alone is insufficient. Empty-rotor checks, zero-speed readings, calibration of sensor polarity, and controlled temperature changes test different errors. Equatorial stray-field sign must also be converted to magnetization sign: for an axial magnetic dipole moment, the magnetic dipole field at its equator points opposite to that magnetic moment. The apparatus diagram makes this otherwise easy-to-miss sign change visible.

For a barotropic fluid with pressure $P=P(\rho )$, choose an energy density $U$ satisfying $\rho U^{\prime }(\rho )-U=P$. The continuity equation then gives ${\partial }_{t}U+\nabla \cdot (U\mathbf{u})=-P\nabla \cdot \mathbf{u}$. For an isothermal equation of state $P=c_s^2\rho$, one choice is $U=c_s^2\rho \ln (\rho /{\rho }_{\mathrm{ref}})$. A fixed reference density changes $U$ only by a conserved mass term. The same construction holds with surface density and vertically integrated pressure. This closure energy need not equal the microscopic internal energy of a gas maintained at fixed temperature.

For the conjugate base ${\mathrm{A}}^{-}$ accepting a proton from water,This base dissociation constant uses the same water thermodynamic activity approximation and standard state conventions as the acid dissociation constant. Multiplying the two relations cancels the partner thermodynamic activities:Here $K_w$ is the aqueous ion product associated with self-ionization of water. At fixed temperature, a larger acid dissociation constant gives a smaller base dissociation constant for its own conjugate base; this reciprocal rule compares matching conjugate acid-base pairs. A weak acid does not necessarily have a conjugate base that reacts almost completely with water. For example, stipulated $K_a=10^{-5}$ and $K_w=10^{-14}$ give $K_b=10^{-9}$.^[ref]

The value of $K_w$ changes with temperature; $10^{-14}$ is the familiar approximate value near $25{}^{\circ }\mathrm{C}$, not a universal constant. If water is retained explicitly in both full equilibrium constants, their product instead equals $a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}{a}_{{\mathrm{OH}}^{-}}/a_{{\mathrm{H}}_2\mathrm{O}}^2$, the full equilibrium constant for $2{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{H}}_3{\mathrm{O}}^{+}+{\mathrm{OH}}^{-}$. The acid dissociation constant, base dissociation constant and self-ionization of water convention must agree.

Base pairing is the recognition and association of compatible nucleobases. In canonical Watson–Crick geometry, an adenine–thymine base pair has two hydrogen bonds, while a guanine–cytosine base pair has three. These counts refer to those specific geometries and usual chemical forms; they are not a definition of every possible base pair. RNA can also contain other pairings, and canonical adenine–uracil pairing replaces adenine–thymine pairing. See Clark, Douglas and Choi, Biology 2e: DNA structure.

For this illustrative duplex, the six base pairs are A–T, C–G, G–C, T–A, A–T, and C–G. Thus three contribute two hydrogen bonds each and three contribute three each:Each dotted connector in the diagram counts one hydrogen bond; coloured boxes denote whole nucleobases, not individual chemical atoms. The solid rails denote the covalent bond scaffold. Separating the two strands disrupts base pairing and base stacking without requiring the backbone to be cut.

The count of fifteen hydrogen bonds is not fifteen identical contributions to a duplex's Gibbs free energy. Base stacking, interactions with water and ions, temperature, and the order of neighbouring base pairs also matter. Knowing the fraction of guanine–cytosine base pairs alone does not fix a nucleic acid double helix's shape or stability. See Brown, Genomes: pairing and stacking interactions. What makes base pairing especially useful is its selectivity: an exposed DNA strand constrains the partner's nucleic acid sequence, giving DNA replication a chemical recognition rule.

Let a homogeneous plate $0<x<L$ begin at uniform temperature $T_0$, cool with zero-temperature Dirichlet boundary conditions until time $t_b$, then switch to insulated Neumann boundary conditions. For thermal diffusivity $\kappa >0$, the Fourier sine series during cooling is $T=(4T_0/\pi ){\sum }_{n\ge 1,n\text{ odd}}\sin (n\pi x/L)e^{-\kappa (n\pi /L{)}^{2}t}/n$. Integrating the insulated heat equation shows that its integral average is conserved. All nonconstant Fourier cosine series modes then decay, proving that the final uniform temperature is $T_{\infty }=(8T_0/{\pi }^2){\sum }_{n\ge 1,n\text{ odd}}{e}^{-\kappa (n\pi /L{)}^2{t}_b}/n^2$. Its limits are $T_0$ at zero bath time and zero at infinite bath time. Instantaneous changes of boundary condition can create initial-boundary corner incompatibilities without invalidating the positive-time solution.

A useful short-interval approximation replaces a battery by an ideal electromotive force $\mathcal{E}$ in series with a positive internal electrical resistance $r$. This is the Thévenin terminal response with open-circuit voltage $\mathcal{E}$ and Thévenin resistance $r$. The model describes the terminals without asserting that there is a separate physical resistor hidden inside the cell. Both $\mathcal{E}$ and $r$ can change with composition, temperature, electric current and history.^[ref]

For a resistive electrical load $R>0$, take $I>0$ to mean discharge: conventional current leaves the positive terminal. Kirchhoff's voltage law and Ohm's law giveFor the stipulated $\mathcal{E}=3\mathrm{V}$, $r=1\Omega$, $R=4\Omega$,An open-circuit voltage of $3\mathrm{V}$ therefore does not promise $3\mathrm{V}$ at this electrical load. Increasing the load electrical resistance reduces electric current and brings $V_L$ closer to $\mathcal{E}$, as in Thévenin load response.

The signed relation also explains charging within this model. If an external source drives electric current into the positive terminal, $I<0$ and $V_L=\mathcal{E}+|I|r$. The terminal electric potential difference then exceeds $\mathcal{E}$. This algebra is not evidence that every battery is rechargeable.^[ref]