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Let $r=a_{\mathrm{ADP}}/a_{\mathrm{ATP}}$ and use a one-molar standard state, so the inorganic phosphate activity is approximately $a_P=0.006$. With the source's stated standard hydrolysis value,The downhill proton-transfer Gibbs free energy is $F\Delta p$ for the signed driving force $\Delta p=-0.150$ V. Ideal reversible balance of one complete turn requiresSolving yieldsAt 298.15 K the estimate is $3.39\times 10^2$. Temperature is not specified in this paragraph; the first value explicitly adopts physiological temperature rather than pretending it is supplied. The unexpectedly large ADP/ATP ratio follows from using the printed standard value $-50$ kJ/mol with the stated dilute inorganic phosphate. One must not silently replace that value by another convention or treat it as the already concentration-corrected cellular hydrolysis Gibbs free energy.

This estimate assumes ideal thermodynamic activities, consistent pH and magnesium conventions, tight coupling of ten protons to three ATP, no leak or transport overhead, and operation close to reversible balance. A working cell is not in global equilibrium. Net synthesis at this fixed driving force requires $\Delta G_{\mathrm{turn}}<0$, hence $r$ above this reversible-balance threshold; finite chemical kinetics and dissipation prevent the stoichiometry alone from determining an exact physiological ratio.

Use signed ion valence $z$ and define $V={\psi }_1-{\psi }_2$. The molar electrochemical potential in side $i$ of an ideal dilute solution isTransfer of an infinitesimal amount $dn$ from side 1 to side 2 changes the total Gibbs free energy by $({\tilde{\mu }}_2-{\tilde{\mu }}_1)dn$. At equilibrium this derivative is zero, soFor a cell with side 1 inside, this is $E_i=(RT/z_{i}F)\ln (c_{i,\mathrm{out}}/c_{i,\mathrm{in}})$. The concentrations are equilibrium free-ion concentrations, not automatically their exact initial values. For nonideal solutions the ratio must be a ratio of thermodynamic activities. The temperature and solvent must be common to both reservoirs, and an ion must actually be able to exchange for this equilibrium condition to be dynamically established.

Equal compartment volumes are unnecessary for deriving the voltage law, but help specify number conservation if concentration changes are appreciable. For equal volume $\mathcal{V}$ and a monovalent cation transfer $n$ from side 1 to 2, $c_1=c_1^0-n/\mathcal{V}$, $c_2=c_2^0+n/\mathcal{V}$. Combined with the membrane capacitance, this gives an implicit finite-reservoir equilibrium rather than treating both concentrations as externally clamped.

Write $u=x/L$ and $A=k_{B}T{L}_p$. The worm-like chain force-extension interpolation isA fit to an elastic rising branch estimates its contour length $L$ and, if not fixed independently, its persistence length $L_p$. Comparing successive fitted contour lengths gives the length released by a domain and helps distinguish whole-repeat unfolding from partial events. $L_p$ estimates the flexibility of the unfolded chain and the bending rigidity $A$; at a stated temperature, $A=k_{B}T{L}_p$.

For small extension, expansion gives $f\simeq [3k_{B}T/(2L_{p}L)]x$, so the effective entropic spring constant is $3k_{B}T/(2L_{p}L)$. Close to full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$, giving strong strain stiffening. These regimes constrain the fit differently and correlate its parameters if the measured interval is short. The fit does not by itself identify a unique folding barrier or unfolding kinetic rate.

Persistence-length convention. In a three-dimensional isotropic worm-like chain the standard definition is $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Thus at $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})\simeq 68.4^{\circ }$. This is not literally the mean angle $⟨\theta ⟩$. For a simple counterexample, equally probable angles $0^{\circ }$ and $90^{\circ }$ have mean angle $45^{\circ }$ but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The source's angle description is shorthand for the tangent-correlation convention.

Model the target as a perfectly absorbing sphere of reaction radius $r_0$, with a fixed target and a dilute far-field polymerase density $c_{\infty }$. Steady radial diffusion satisfies ${\nabla }^{2}c=0$, $c(r_0)=0$, $c(\infty )=c_{\infty }$, henceTherefore, for a rate law multiplying number densities,For the earliest diffusive target encounter among independent searchers, with $N_p$ independent well-mixed searchers in volume $V$, the approximate first-arrival rate is $N_p{k}_a/V$, giving ${\tau }_3\simeq V/(N_p{k}_a)$. The target's effective reaction radius includes encounter geometry and is not uniquely determined by its contour length.

A numerical estimate needs temperature, cell volume and a capture-geometry convention, none of which is fixed numerically in the source. State representative estimates $T=300\mathrm{K}$ and $V=1\mu {\mathrm{m}}^3$. The Stokes–Einstein estimate for a sphere of radius $15\mathrm{nm}$ in the stipulated viscosity isUsing the target contour scale $40\times 0.34\mathrm{nm}=13.6\mathrm{nm}$ as an order-of-magnitude $r_0$ gives $k_a\simeq 2.50\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ andThis is an idealized $10^{-2}$-second-scale estimate, not a unique geometric prediction. Finite cell shape, crowded motion, accessibility and orientational restrictions can reduce the ideal capture rate.

If nonspecific binding merely sequesters searchers and their free fraction is $0.2$, the three-dimensional capture hazard is reduced fivefold: $\tau \simeq {\tau }_3/0.2\simeq 0.067\mathrm{s}$ for those estimates. If bound molecules instead search productively by sliding, a free-time fraction alone cannot determine the total arrival time or its sign of change. That is the distinct facilitated-search mechanism treated in c.

Usable biological free energy has several interconvertible forms. At approximately fixed temperature and pressure, chemical reactions are described by chemical potentials, ${\mu }_i={\mu }_i^{\circ }+RT\log a_i$, and ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. ATP hydrolysis, for example, has ${\Delta }_{r}G={\Delta }_r{G}^{\circ \prime }+RT\log (a_{\mathrm{ADP}}{a}_{\mathrm{Pi}}/a_{\mathrm{ATP}})$ under the specified biochemical standard-state convention. The activities and proton/ion convention matter. Breaking a bond alone costs energy; the net free-energy change includes the products, solvent and all bond changes, not a store released simply by snapping one bond.

Redox reactions provide free energy when electrons move between donors and acceptors with a favourable potential difference: $\Delta G=-nF\Delta E$. Electron-transfer chains can couple that drop to ion transport, building an electrochemical gradient. For transfer from an outer to an inner reservoir,Both concentration difference and electrical potential contribute. ATP synthase couples downhill ion flow to nucleotide free energy; driven pumps can perform the reverse conversion and maintain gradients used by transporters and excitability.

Mechanical work appears as $Fdx$ or torque times angle, and can be stored elastically, for example $\kappa x^2/2$. Molecular motors convert chemical free energy into movement or force; mechanically driven reactions can conversely reshape chemical transitions. Electrical energy can also be stored in membrane capacitance, while bulk ionic concentration gradients store additional chemical free energy. Light supplies photon energy $h\nu$, which photosynthetic machinery can convert into redox and chemical potential differences.

Coupling is essential: an overall favourable reaction can drive an unfavourable one only when their processes are linked, rather than merely present in the same cell. At fixed conditions the extracted work is bounded by the consumed free-energy decrease. Dissipation produces heat and entropy, so energy conservation does not mean all energy remains available for work. Thermal fluctuations can drive random motion, but a persistent directional cycle requires a nonequilibrium free-energy source; an equilibrium asymmetric structure alone is not a free motor.

If the ideal-chain segment count and segment length remain fixed, the entropic spring constant is proportional to absolute temperature. ThereforeThis is an increase by about $11.7\%$. The factor $k_{B}T$ multiplies the free energy penalty for the loss of chain configurations; a larger force is required to impose the same normalized weak extension at higher temperature.

Not degrading does not guarantee that a real protein's folded state, effective spring constant or solvent interactions remain unchanged. Thermal unfolding and changes in persistence length can occur without covalent degradation. Those changes modify the experimental curve and may invalidate a fixed-$N,b$ comparison. The boxed value is the explicit ideal-FJC prediction under fixed segment geometry, not an unconditional thermal-stability prediction for the protein.

The single-block DNA zipper partition function retains one contiguous paired block and ignores multiple separated bubbles. A block of $j\ge 1$ paired bases can start at any of $N-j+1$ positions. Its first pair has initiation weight $\kappa$, and the remaining $j-1$ pairs each have propagation weight $s$. ThusThe source's symbol $k$ in the displayed expressions denotes the same initiation constant as $\kappa$, not Boltzmann's constant. Association and other reference-state factors are absorbed into that statistical weight; the model treats the propagation weights as uniform despite the actual sequence's different bases.

Put $r=j-1$. Since $N-r$ counts how many terms of a finite geometric series contain $s^r$,ConsequentlyThe apparent singularity is removable: at $s=1$, the original sum gives $q=\kappa N(N+1)/2$, which is $45\kappa$ for these nine bases.

At $s\gg 1$, the longest block dominates:Conditional on at least one pair, the fully paired probability approaches one, with leading correction of order $2/s$. If a fully unpaired reference state of weight one is included, the full partition function is $Q=1+q$, so a large propagation constant alone does not guarantee predominant duplex if $\kappa s^{N-1}$ is still very small.

The given sequence contains five A–T and four G–C pairs when fully complementary. Real duplex stability also depends strongly on stacking, sequence context, salt and temperature; it cannot be calculated just by counting two versus three hydrogen bonds. All else comparable, sequences with greater duplex propagation stability resist bubble opening more strongly, while easier-to-open, often A–T-rich start-site regions facilitate initiation. The homogeneous-$s$ zipper demonstrates that principle but does not resolve a quantitative sequence-specific melting profile or polymerase's active stabilization of the open state.

A stable native conformation is favoured when its Gibbs free energy, including the protein and its surrounding solvent, is lower than that of competing conformational ensembles under the specified conditions:Folding greatly reduces the chain's conformational entropy, an unfavourable contribution. It can nevertheless gain favourable hydrogen-bonding and packing interactions, van der Waals contacts, screened electrostatic interactions and solvent free energy. In aqueous globular proteins, burying nonpolar surface reduces the cost of exposing it to water; the hydrophobic effect is not an extra covalent attraction between all hydrophobic residues. Hydrogen bonds and ionic contacts must be judged relative to the interactions those groups made with solvent before folding, so merely counting bonds in the folded structure does not determine stability.

The protein primary structure determines which contacts and protein secondary structures can cooperate. A free-energy landscape biased towards compatible contacts permits many routes towards the native ensemble, rather than an exhaustive random search through every conformation. Folding funnels describe this bias, while barriers explain metastability and folding rates. Reversible denaturation/refolding experiments show that some proteins can recover native structure and activity when suitable conditions are restored, supporting a sequence-encoded thermodynamic tendency without asserting that every protein folds unaided.

“Stable” means an equilibrium preference and a useful lifetime, not one perfectly rigid shape. Native proteins fluctuate and can have several functional conformations; the unfolded ensemble remains populated with relative weight depending on its full free energy. Changes of temperature, pH, salt, solvent, ligands or pressure alter the balance, and molecular chaperones can prevent protein aggregation or assist kinetic access without supplying a different primary sequence. Membrane proteins require their appropriate environment, and intrinsically disordered proteins need not possess one stable folded state in isolation.

Thus a released stretched chain can gain configurational entropy on first recoiling yet later favour a more ordered native structure because the total free energy balance, including solvent, is different. Equilibrium stability, barriers to unfolding and resistance to irreversible protein aggregation are distinct properties. The existence of a native minimum alone does not guarantee rapid, reversible folding under every experimental condition.

The given sphere has radius $a=5\mathrm{nm}$. The Stokes–Einstein relation at an illustrative $T=300\mathrm{K}$ givesFor $N=30$ well-mixed independent searchers in cell volume $V$, the target-arrival rate is $N{k}_a/V$. The corresponding mean first arrival time isThe source does not fix $V$, the reaction radius or temperature. For an order-of-magnitude estimate, a bacterial volume around $1\mu {\mathrm{m}}^3$ and reaction radius $5$–$10\mathrm{nm}$ give about $0.06$–$0.12\mathrm{s}$. A 20-base-pair B-form site is about $6.8\mathrm{nm}$ long, supporting a nanometre-scale capture radius but not uniquely defining its spherical value. The estimate is the ideal diffusion-limited time, not a measured intracellular search time: nonspecific binding, orientational matching, crowding and limited accessibility are excluded.

Take forward as increasing $x$, and specify the sinusoidal amplitude convention byThus the unforced barrier is $2A$, not $A$. Let $s=Fd/(2\pi A)=0.7051$ and $\theta =\arcsin s$. Stationary points exist because $F<2\pi A/d=7.09\mathrm{pN}$. A minimum is at $x_m=d\theta /(2\pi )$ and its forward saddle at $x_{+}=d(\pi -\theta )/(2\pi )$. The distances and activation barriers areUse an Arrhenius estimate with the supplied distance-squared waiting-time scaling, $\tau \propto {\ell }^2{e}^{\Delta U/(k_{B}T)}$. Calibrate at $F=0$ with ${\ell }_0=d/2$ and rate $k_0=1{\mathrm{s}}^{-1}$ in each direction. At the explicit room-temperature estimate $T=300\mathrm{K}$, $k_{B}T=4.142\mathrm{pN}\mathrm{nm}$, givingApplying the same distance rule separately backwards gives $k_{-}\simeq 0.00241{\mathrm{s}}^{-1}$ and hence the literal scaling-model estimateThermodynamic qualification: for isothermal diffusion in this translated tilted potential, adjacent wells differ by $Fd=35\mathrm{pN}\mathrm{nm}$, so local detailed balance instead requires $k_{+}/k_{-}=e^{Fd/(k_{B}T)}$. The two separate distance prefactors introduce an extra factor $({\ell }_{-}/{\ell }_{+}{)}^2\simeq 8.91$, inconsistent with that requirement. Keeping the forward estimate while enforcing the proper ratio givesBoth interpretations are shown because the supplied distance heuristic is not, by itself, a thermodynamically consistent prescription for both directional prefactors. Additional hidden chemical driving would require its own affinity, not an unexplained prefactor asymmetry. A common transition-state prefactor, as in a consistent Kramers model, preserves the Boltzmann rate ratio. Moreover the forward barrier is only about $0.58k_{B}T$, so rare-escape exponential dwell kinetics and the numerical rate should be regarded as rough estimates. If “amplitude” meant the entire unforced barrier height, the maximum restoring force would be only $\pi A/d=3.55\mathrm{pN}$ and no dwell wells would survive at $5\mathrm{pN}$; the amplitude convention is therefore essential.

At fixed $N$ and $b$, the entropic spring constant is proportional to absolute temperature, because the free-energy penalty for a given loss of conformational entropy is $-T\Delta S$. ThereforeFor the literal printed chain this is $\boxed{1.287\times 10^{-6}\mathrm{N}{\mathrm{m}}^{-1}}$. The same factor multiplies either alternative contour-consistent spring constant from part (d), provided its Kuhn length is held fixed. Real DNA's persistence length, base pairing and solvent interactions can themselves vary with temperature and salt; then $b(T)$ contributes through $k\propto T/b(T{)}^2$ at fixed segment count, or $k\propto T/[L_{c}b(T)]$ if the physical contour is instead held fixed. The simple linear temperature result belongs to the specified ideal-chain model.

After the load is removed, a reversibly stretched DNA molecule relaxes towards a thermally fluctuating coil, releasing stored free energy as heat through the surrounding fluid. There are many more disordered chain conformations than strongly aligned ones, so entropy favours retraction; viscous drag sets how rapidly it occurs. At long times an unconstrained ideal chain has zero mean end-to-end vector but a nonzero typical size, $R_{\mathrm{rms}}=b\sqrt{N}$; it does not contract to a point.

At finite temperature the chain continues fluctuating, and an anchored or otherwise constrained molecule need not regain an isotropic ensemble. If stretching broke attachments, denatured the DNA or caused irreversible structural changes, recovery can be incomplete or hysteretic. Those outcomes cannot be decided from the ideal FJC alone.

A charged biological molecule attracts counterions and repels ions of the same sign. Thermal diffusion opposes complete localization, producing a mobile ionic atmosphere whose charge partly cancels the molecule's electric field at larger distances. Solvent polarization also reduces the field through its permittivity, but that dielectric response is distinct from the mobile-ion screening cloud.

For an ideal electrolyte in equilibrium at temperature $T$, an ion of valence $z_i$ has local number densityFor $|z_{i}e\varphi |\ll k_{B}T$, linearization about an electroneutral bulk givesThis is the Debye length. Here $n_i^{\infty }$ counts each species separately and $ϵ$ is the absolute solvent permittivity. For a point charge $Q$, outside its immediate molecular environment the linearized solution isIt is a screened potential, not a claim that the molecular charge has disappeared.

In terms of ionic strength $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$ with $c_i$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, the density sum is $2N_{A}10^{3}I$, soIn water near $298\mathrm{K}$, ${\lambda }_D\simeq 0.304\mathrm{nm}/\sqrt{I/(1\mathrm{mol}{\mathrm{L}}^{-1})}$: about $9.6\mathrm{nm}$ at $1\mathrm{mM}$ and $0.79\mathrm{nm}$ at $0.15\mathrm{M}$. Increasing ion density or valence shortens the range; temperature and solvent permittivity also matter. The displayed temperature dependence holds at fixed permittivity, which itself changes with temperature in water.

The derivation assumes dilute mobile ions, a continuum uniform dielectric and weak potentials. Near highly charged DNA or protein surfaces, nonlinear Poisson-Boltzmann equation behaviour, counterion condensation, finite ion sizes, specific binding and correlations can matter. At physiological salt the predicted length approaches molecular scales, so the formula is a useful far-field scale with limited microscopic accuracy, rather than a complete account of the first hydration layer.

Treat the target as an absorbing sphere of effective capture radius $r_0$, surrounded by a uniform dilute reservoir of RNA polymerase with number density $C_{\infty }$. The steady diffusion equation, spherical symmetry, and boundary conditions giveBy Fick's laws, the inward capture rate per target is $4\pi r_0^2{D}_A{C}^{\prime }(r_0)=4\pi D_A{r}_0{C}_{\infty }$. Multiplying by the target number density and comparing with the bimolecular rate law therefore yields the Smoluchowski association rateIts units here are ${\mathrm{m}}^3{\mathrm{s}}^{-1}$, since the concentrations count particles rather than moles. The absorbing boundary assumes every encounter is productive; orientation constraints would reduce the effective association rate constant.

The Stokes–Einstein relation, for RNA polymerase radius $a=15\mathrm{nm}$ at an illustrative temperature $T=300\mathrm{K}$, givesNeither cell volume nor the effective $r_0$ is specified, so a unique numerical time is unavailable. A representative Escherichia coli volume is $V=1\mu {\mathrm{m}}^3=10^{-18}{\mathrm{m}}^3$. The $40$-base pair target has a DNA contour extent of about $40(0.34)=13.6\mathrm{nm}$; using this as an order-of-magnitude capture radius gives $k_a\simeq 2.50\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. For $M=3000$ independent searchers, the target's approximately constant encounter hazard is $M{k}_a/V$, henceUsing half the target's contour extent for $r_0$ instead gives about $2.7\mathrm{ms}$; the spherical capture model is not exact target geometry. One searcher's mean would be about $4\mathrm{s}$ under the first convention, whereas the first arrival among $3000$ is much faster.

If only a fraction $f_3=0.1$ of each searcher's time is available for three-dimensional diffusion, and nonspecific DNA binding neither scans nor delivers the target, the effective searching population is $f_{3}M$. Thus $\boxed{\tau \simeq {\tau }_3/f_3=10{\tau }_3}$, about $13\mathrm{ms}$ under the stated estimate. This assumes stationarity, well-mixed searchers and no additional target-access route; nonspecific binding alone is sequestration, not facilitated diffusion.

Each Cartesian component of one link has zero mean and variance $b^2/3$, with zero cross-component covariance. By the multivariate central limit theorem, for large $N$ on the coil scale the vector probability density isIt is normalized with respect to $d^{3}r$; the radial probability density is $4\pi r^{2}p(\mathbf{r})$. The Gaussian limit of the end-to-end distribution of a freely jointed chain is an approximation, not the exact finite-$N$ law near $r=Nb$.

At prescribed vector separation the configurational entropy is $S(\mathbf{r})=k_B\ln p(\mathbf{r})+\text{constant}$. The ideal chain has no extension-dependent internal energy, so its Helmholtz free energy isThe externally required force is ${\nabla }_{\mathbf{r}}A$, while the chain's restoring force has the opposite sign. ThusThe last equation also follows by multiplying $p(\mathbf{r})$ by the Boltzmann factor $e^{fz/(k_{B}T)}$ and completing the square. This entropic spring resists extension because fewer link configurations realize a stretched chain. At the same small applied force, increasing temperature reduces extension; at the same extension it increases the required force. Do not differentiate $-k_{B}T\ln [4\pi r^{2}p(\mathbf{r})]$ to obtain the fixed-vector Hooke law, since that radial measure includes an additional geometric multiplicity. The low-force condition is $fb/(k_{B}T)\ll 1$, equivalently $⟨z⟩\ll Nb$.

An ionic atmosphere. A negatively charged molecule attracts positive counterions and repels negative co-ions; signs reverse for a positive molecule. The resulting electric double layer partially screens the bare electric field. Far from an isolated molecule, the electrolyte recovers its neutral bulk composition. At thermodynamic equilibrium, spatial uniformity of each species' electrochemical potential requiresThe logarithm implicitly uses a common reference density. This result for Boltzmann ion densities in a symmetric electrolyte neglects ion correlations, finite size and nonideal activities, but shows immediately why equal concentrations of the two signs do not persist close to a charged surface.

Self-consistent screening. Insert these densities into Poisson's equation, with solvent permittivity $ϵ$ and fixed charge density ${\rho }_f$:This is the Poisson-Boltzmann equation. For a symmetric monovalent salt with bulk density $n_s$ of each sign, outside the fixed charge it becomesFor $|z_{i}e\psi |\ll k_{B}T$, linearizing and using bulk neutrality gives the Debye–Hückel approximationThus added salt shortens the Debye length: electrostatic screening becomes stronger over a shorter distance. The factor of two in a $1:1$ salt comes from two ionic species, not from double-counting the density of salt pairs.

For example, a charged sphere of radius $a$ and total charge $Q$ has the exterior linearized solutionassuming fixed uniform surface charge and no ions inside. It follows from the decaying spherical solution and $-ϵ{\psi }^{\prime }(a)=Q/(4\pi a^2)$. At zero salt it reduces to the Coulomb potential; at finite salt it decays exponentially as well as geometrically. The excess mobile charge integrates to $-Q$ in an infinite neutralizing reservoir, although a finite sphere surrounding the molecule generally contains only part of that ionic atmosphere.

Biophysical scales and limitations. In water at room temperature, the monovalent formula is approximately ${\lambda }_D=0.304\mathrm{nm}/\sqrt{I/(\mathrm{mol}{\mathrm{L}}^{-1})}$, with $I$ the ionic strength. It gives about $0.79\mathrm{nm}$ at $0.15\mathrm{mol}{\mathrm{L}}^{-1}$ and $9.6\mathrm{nm}$ at $1\mathrm{mmol}{\mathrm{L}}^{-1}$. Consequently the same charged protein can exert very different long-range interactions mediated by electrostatics at different salt concentrations. Close to strongly charged DNA, the linear approximation fails; nonlinear screening, counterion condensation, ion-specific adsorption and changes of surface ionization through charge regulation may matter. Mean-field theory also misses strong multivalent-ion correlations and finite-size saturation: the exponential law cannot be extrapolated to arbitrarily large near-surface concentrations.

The calculated rotational constant gives ${\theta }_{\mathrm{rot}}=1.18721\mathrm{K}$. At room temperature, taken as $300\mathrm{K}$, the continuous maximum is $J_{\ast }=10.7404$; comparing the adjacent integer populations gives $J_{\max }=11$. Taking $298.15\mathrm{K}$ instead gives $J_{\ast }=10.7057$ and the same integer answer.

At $8.7\mathrm{K}$, $J_{\ast }=1.41417$. In particular,and $N_1>N_0$, so $J_{\max }=1$. The integer comparison avoids treating a noninteger angular momentum quantum number as a physical state.

Substitutional arsenic has one more valence electron than germanium, so it acts as a donor impurity. Thermal ionization of its shallow bound state supplies mobile conduction-band electrons and leaves fixed positive donor charge. The resulting n-type semiconductor can have $n_e\gg n_i$ even when only a small fraction of host atoms are replaced. Its electrical conductivity then rises because the mobile carrier count increases; the opposite donor charge preserves macroscopic charge neutrality.

This argument assumes that the donors are substantially ionized at the measurement temperature and that the added impurity scattering does not overwhelm the carrier increase. The screened bound-donor picture and its departures from a simple hydrogenic model are described in institutional semiconductor spectroscopy notes.^[ref]

The equal supplied masses give equal effective density of states. At the specified temperature,Using the same energy units in the exponential,For equal electron and hole mobilities, both carriers contribute to the intrinsic electrical conductivity:The factor two is essential. A dimensional check givesusing $\mathrm{A}=\mathrm{C}{\mathrm{s}}^{-1}$.

At low temperature, the upper orientation has negligible Boltzmann weight, whereas at high temperature all three orientations become equally probable. ConsequentlyThe high-temperature limit is a finite arithmetic mean of three available one-site energy levels. It is not an unbounded translational or rotational equipartition contribution: the prescribed site model has only three states per molecule. Their common binding offset can keep this limiting internal energy negative without any inconsistency.