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Temperature as an equilibrium relation. Classical thermodynamics begins with the zeroth law of thermodynamics: thermal equilibrium is transitive. A thermometer can therefore assign a common empirical temperature to systems in mutual thermal equilibrium. Different thermometer properties initially give different monotone empirical scales; a resistance reading by itself does not define the absolute scale.

The Second law of thermodynamics provides a substance-independent thermodynamic scale. A reversible engine operating between reservoirs has a heat ratio depending only on their temperatures. One chooses absolute temperature so that $Q_h/Q_c=T_h/T_c$, with $Q_h,Q_c$ the positive heat magnitudes. Its reversible efficiency is $1-T_c/T_h$. Equivalently, reversible heat defines entropy through $dS=\delta Q_{\mathrm{rev}}/T$. For an equilibrium simple compressible system,Classical thermodynamics uses this relation without needing a microscopic account of entropy. A dilute-gas thermometer realizes an empirical scale approaching the thermodynamic scale after corrections for nonideal behavior.

Statistical meaning of the same derivative. In the microcanonical description, an equilibrium energy shell containing $\Omega (U,V,N)$ accessible states has $S=k_B\ln \Omega$. Its statistical temperature is consequentlyTo see why this is an equilibrium temperature, let two weakly coupled systems exchange a small energy while their total energy stays fixed. The combined multiplicity is ${\Omega }_A(U_A){\Omega }_B(U_{\mathrm{tot}}-U_A)$. Maximizing its logarithm giveswhich is equality of the two inverse temperatures. Stability requires the total entropy maximum, not only stationarity. The overwhelmingly dominant macrostate in a large system therefore realizes the thermodynamic equilibrium condition.

A large reservoir gives a small subsystem the canonical probabilities $p_i\propto e^{-\beta E_i}$, with $\beta =1/(k_{B}T)$. In that ensemble, $F=-k_{B}T\ln Z$ and $S=-k_B{\sum }_i{p}_i\ln p_i$ agree with macroscopic thermodynamic relations in their usual equilibrium regime. The equipartition identity for a classical dilute monatomic gas, $⟨p^2/(2m)⟩=3k_{B}T/2$, is a useful operational connection between microscopic motion and temperature, but is not a universal definition: quantum modes need not obey equipartition, and a system away from equilibrium need not have one temperature at all.

For finite systems one must specify the entropy convention and energy resolution; surface and fluctuation corrections can distinguish definitions that agree in the ordinary thermodynamic limit. Bounded-energy population-inverted systems can also have a negative entropy derivative and hence a negative statistical temperature. That possibility lies outside an ordinary positive-temperature ideal-gas thermometer; it does not mean temperature is merely an arbitrary average kinetic energy.

Obtaining the energy–temperature conversion. The Boltzmann constant connects entropy measured in $\mathrm{J}{\mathrm{K}}^{-1}$ with the dimensionless logarithm of a state count. Historically it could be determined from the ideal gas law $pV=N{k}_{B}T$: measure the molar gas constant $R$ and independently determine the Avogadro constant $N_A$, then obtain $k_B=R/N_A$. The independent particle-number determination matters; using $R$ alone would not fix energy per particle.

An electrical method uses Johnson-Nyquist noise. A resistor of resistance $R_e$ at known temperature has, in the classical range $hf\ll k_{B}T$, a one-sided open-circuit voltage-noise spectrum $S_V(f)=4k_{B}T{R}_e$. Over a calibrated flat bandwidth $\Delta f$,A practical measurement uses an equilibrium resistor, a calibrated low-noise amplifier and a known transfer function, subtracts or independently characterizes amplifier noise, and replaces a nominal bandwidth by the actual effective noise bandwidth. Varying temperature and fitting the mean-square voltage is more informative than treating one amplified voltage sample as the thermal variance. Other primary methods relate temperature to sound speed in a dilute gas or to calibrated particle diffusion and mobility.

There is a historically relevant unit-definition qualification: by the date of this exam, the revised SI had already taken effect on 20 May 2019. The kelvin is now defined by fixingThe same experiments now realize or compare thermodynamic temperatures using this fixed constant, whereas historically they measured the conversion constant on an independently defined temperature scale. The physical distinction between equilibrium temperature and microscopic state counting is unchanged.^[ref]

The expression is the canonical partition function for a system in thermal equilibrium with a large heat bath at fixed temperature $T$. The system exchanges energy with the bath, while its particle number and external mechanical parameters, such as volume, are fixed. The bath is large enough that its temperature is not appreciably changed by this exchange.

The system has allowed energy levels ${ϵ}_j$. The index $j$ labels distinct energy values, and the integer degeneracy factor $g_j$ counts distinct quantum states with that same energy. $k_B$ is the Boltzmann constant, and $\exp [-{ϵ}_j/(k_{B}T)]$ is a state's Boltzmann factor. The dimensionless sum $Z$ normalizes the probabilities:Thus the degeneracy multiplies the probability of occupying the level, not the probability assigned to each individual state within it.

The canonical model is not restricted to a classical ideal gas. It applies to a quantum rotor, for example, provided the sum counts all of that system's allowed states with their proper degeneracies. For a many-particle system, ${ϵ}_j$ would be the energy of a many-particle state, with any quantum-statistical restrictions already built into the state count.

For a dilute suspended sphere in Stokes flow, the translational friction coefficient is $f_h=6\pi \eta R$, with dynamic viscosity $\eta$ and hydrodynamic radius $R$. To derive the diffusion-friction relation, put particles in a weak conservative force $F=-U^{\prime }$. Their drift velocity is $F/f_h$, while the diffusive flux is $-Ddc/dx$. At equilibrium the Boltzmann distribution gives $c^{\prime }=-c{U}^{\prime }/(k_{B}T)=cF/(k_{B}T)$. Zero total flux therefore gives $cF/f_h-DcF/(k_{B}T)=0$, hence the Stokes–Einstein relationHere $k_B$ is the Boltzmann constant, $T$ absolute temperature and $D$ the translational diffusion coefficient. The sphere, no-slip continuum-fluid and dilute thermal-equilibrium assumptions enter through the drag law and flux balance.

At $T=293.15\mathrm{K}$ and the supplied viscosity, the predicted product is $DR=2.1472\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The derived predictions and radii inferred from the measured diffusion coefficients are:

The last three agree closely at the precision of the supplied data. The ribonuclease diffusion coefficient is about $16\%$ below the spherical prediction, corresponding to a larger frictional radius than the stated radius. Hydration, shape, a different operational radius and coarse measurement precision can matter; molar mass is not an independent parameter in the spherical diffusion law. A compact constant-density globule would have $R\propto M^{1/3}$, whereas collagen's large frictional radius relative to its mass is consistent with an extended nonspherical molecule. The inferred Stokes radius is a friction-equivalent radius, not automatically a geometric half-length.

In analytical ultracentrifugation, scan the radial protein concentration while a rotor spins at known angular speed $\omega$. Optical absorbance or refractive-index interference records a moving sedimentation boundary and its diffusive broadening.^[ref] With solvent density ${\rho }_s$, partial specific volume $\bar{v}$ and molar mass $M$ in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, the buoyant mass is $m_b=M(1-\bar{v}{\rho }_s)/N_A$. Force balance gives $f_{h}v=m_b{\omega }^{2}r$, so the sedimentation coefficient is $s=v/({\omega }^{2}r)=m_b/f_h$. A boundary trajectory gives $s={\omega }^{-2}d\ln r/dt$; diffusion gives $f_h=k_{B}T/D$.

Combining them gives the molar mass from sedimentation and diffusion:Known mass and buoyancy can instead determine friction from $s$ alone, and comparison with a compact sphere tests shape or hydration. In sedimentation equilibrium, the stationary radial profile obeys $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$, providing buoyant mass without a friction model. This is a sedimentation-diffusion balance in a uniform solvent, rather than density-gradient banding at an isopycnic point.

The constants are the Planck constant $h$, photon frequency $\nu$, speed of light $c$, Boltzmann constant $k_B$ and absolute temperature $T$. The supplied $U_{\nu }$ is a spectral energy density, with units $\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$. It describes equilibrium radiation per volume and frequency interval, rather than emitted power itself. For isotropic cavity radiation the corresponding surface spectral radiant exitance is $M_{\nu }=c{U}_{\nu }/4$.

At low frequency the curve rises as ${\nu }^2$; it has one positive maximum and an exponentially decaying high-frequency tail. The left panel below shades the area $u(T)={\int }_0^{\infty }{U}_{\nu }d\nu$ under one curve. Put $x=h\nu /(k_{B}T)$; thenThe remaining dimensionless integral is independent of temperature, soThus doubling temperature doubles the characteristic frequency scale, raises the spectrum's height by eight and increases its area by sixteen. Total emitted energy also depends on emitting area and elapsed time.

Here $h$ is the Planck constant, $\nu$ is the radiation frequency, $c$ is the speed of light in vacuum, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature in kelvin. The quantity $U_{\nu }$ is spectral energy density per unit frequency, so $U_{\nu }d\nu$ is energy per unit volume in the interval $[\nu ,\nu +d\nu ]$, and $h\nu /(k_{B}T)$ is dimensionless.

Take room temperature as $T=300\mathrm{K}$. With the supplied gyromagnetic ratio, reduced Planck constant and Boltzmann constant,soAt $293\mathrm{K}$ the ratio is $1.05\times 10^{-5}$; the room-temperature convention makes only a small difference. The corresponding fractional nuclear spin polarization has magnitude about five parts per million. The source's negative sign convention assigns the label up to the higher-energy state; it should not be confused with the lower-energy magnetic moment aligned with a positive proton magnetic field.

An amphiphile has a solvent-affine part and a solvent-averse part within one molecule. In water these are commonly called hydrophilic head and hydrophobic tail. Two nonbiological examples are sodium dodecyl sulfate, with a twelve-carbon tail and an anionic sulfate head, and cetyltrimethylammonium bromide, with a sixteen-carbon tail and a cationic quaternary-ammonium head. Their counterions balance the ionic head-group charge; the head and tail do not have interchangeable interactions with water.

Aggregation reduces unfavorable exposure of hydrocarbon tails to water but loses molecular translational entropy. Interfacial free energy, head-group solvation and electrostatic repulsion, chain packing and conformational entropy determine the preferred aggregate. The hydrophobic effect is a solvent-mediated free-energy balance, rather than a literal attractive bond between tails.

The amphiphile packing parametercompares tail volume $v$, effective head-group area $a_0$ and maximum tail length ${\ell }_c$. A spherical core of radius $R$ obeys $v/a_0=R/3$, and a long cylindrical core obeys $v/a_0=R/2$. The condition $R\le {\ell }_c$ gives the conventional spherical window $P\lesssim 1/3$ and cylindrical window $1/3\lesssim P\lesssim 1/2$. Less strongly curved bilayers become compatible for $1/2\lesssim P\lesssim 1$, with planar packing near one; values above one can favor inverse curvature in an appropriate nonpolar surrounding medium. These windows are geometric guides rather than exact phase boundaries.

A larger hydrated or electrostatically repelling head increases $a_0$ and favors positive curvature. Screening of charged heads or a smaller head can lower $a_0$ and favor rods or sheets. Multiple tails increase the tail volume relative to head area and can favor bilayers. Tail length, branching, temperature and solvent selectivity alter both geometry and interaction free energy, so one cannot predict a unique aggregate from hydrophobicity alone.^[ref]

In the supplied chemical-potential expression, $\mu$ is the equilibrium chemical potential per amphiphile, ${\mu }_{\alpha }^0$ the standard free energy per amphiphile within an aggregate of size $\alpha$, and $x_{\alpha }$ the dimensionless molecular fraction belonging to that size class. The aggregate molecular fraction versus aggregate number fraction distinction explains the logarithm: the aggregate count fraction is $x_{\alpha }/\alpha$, and dividing its translational chemical potential by $\alpha$ gives a per-molecule value. ThusHere $k_B$ is the Boltzmann constant, $T$ absolute temperature, and $\alpha$ a positive integer. The logarithm acts on a dimensionless ideal activity; a concentrated nonideal solution would require activity coefficients.

equilibrium exchange of amphiphiles gives the same per-molecule $\mu$ for every size. For monomers, $\mu ={\mu }_1^0+k_{B}T\ln x_1$. Substitution yields the finite-micelle aggregation-number distributionThis formula by itself does not prescribe a preferred micelle size: the size dependence of ${\mu }_{\alpha }^0$ must include the competing packing and interaction terms. If their balance favors a finite size ${\alpha }_{\ast }$, a monomer population and a peak around that size can coexist. Dividing by $\alpha$ would instead plot the distribution of aggregate counts.

At very low $x_{\mathrm{tot}}$, the factors $x_1^{\alpha }$ suppress large aggregates and most molecules remain monomers. As $x_1$ rises, a favorable aggregate class grows extremely rapidly because of this high power. The critical micelle concentration from standard chemical potentials has the large-size estimateMore precisely, the illustrative crossover condition $x_{{\alpha }_{\ast }}=x_1$ givesAdded material above the crossover enters aggregates predominantly. Monomer concentration then increases slowly rather than being exactly pinned: its logarithm enters $\mu$, while aggregate populations respond with a large factor ${\alpha }_{\ast }$. For finite micelles this is a sharp association crossover, not an absolute prohibition on aggregates below one exact concentration.

The drawing uses a declared illustrative standard-free-energy model favoring aggregates near $\alpha =40$. It is not an inferred distribution or critical concentration for either named surfactant. The separate monomer peak is at $\alpha =1$; a distribution of molecular fractions weights each aggregate by its number of molecules.

For the general physics of a nearly planar fluid sheet, let $h(x,y)$ be a small-slope displacement, $\kappa$ membrane bending rigidity and $\sigma$ mechanical tension. At fixed topology and symmetric reference curvature, the quadratic Helfrich energy isThe bending term is the small-slope curvature-elastic energy; additional spontaneous-curvature or boundary terms require extra information.^[ref]

For Fourier amplitudes $h_{\mathbf{q}}=A^{-1}{\int }_{A}h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, equipartition gives the membrane fluctuation spectrum under tension,Higher temperature increases fluctuations, while greater membrane bending rigidity or tension suppresses them. Short-wavelength modes are predominantly bending-controlled; longer wavelengths can be tension-controlled. Finite sheet size, supports, confinement and imposed area set boundary conditions and infrared cutoffs; continuum elasticity also needs a short-distance cutoff. solvent viscosity affects relaxation rates, although it does not enter this equilibrium amplitude at fixed elastic parameters.

TODO: user-deferred biology. The requested importance of these fluctuations for bodily function is deferred. The preceding result covers only the independent continuum and thermal physics.

Entropic elasticity. A rubber contains flexible polymer strands tied into a network by cross-links. Stretching biases the strands toward relatively extended configurations, reducing their configurational entropy. The restoring entropic force follows from the increase in Helmholtz free energy $F=U-TS$. For a Gaussian strand with $N_s$ statistical steps of length $b$, its deformation-dependent free energy is approximately $3k_{B}T{R}^2/(2N_s{b}^2)$. Its stiffness therefore increases with absolute temperature, even if the underlying bond stiffness changes little.

In the ideal affine-network model of rubber elasticity, the shear modulus iswhere ${\nu }_s$ is the number density of elastically active network strands. Thus heating increases the modulus approximately in proportion to $T$, provided the network density and rubbery state remain approximately unchanged. Thermal expansion, changing relaxation times or damage can modify that simple trend.

Examples are lightly crosslinked natural rubber, based on cis-rich polyisoprene, and crosslinked polydimethylsiloxane. The crosslinks prevent the long-time macroscopic flow of an unconnected chain solution.

Density estimate and the unspecified modulus convention. With room temperature $T\simeq 298\mathrm{K}$, the Boltzmann constant gives $k_{B}T\simeq 4.11\times 10^{-21}\mathrm{J}$. If the stated modulus is $G=2\times 10^5\mathrm{Pa}$, thenThe molar value counts moles of active strands. If “modulus” instead means Young's modulus $E$, a nearly incompressible isotropic rubber has $E\simeq 3G$, and the inferred value isThus the robust order-of-magnitude estimate is $10^{25}{\mathrm{m}}^{-3}$. The cross-link density inferred from rubber elasticity is usually an effective strand density; the actual junction density is ${\nu }_j=2{\nu }_s/f$ for junction functionality $f$ in an ideal network without inactive loops or dangling ends. For four-functional junctions this would be half the strand density. The paper supplies neither modulus type nor junction functionality, so a unique chemical junction count cannot be extracted.

Write the membrane height above its mean plane as a sum of Fourier modes, $h(\mathbf{r})={\sum }_{\mathbf{q}}{h}_{\mathbf{q}}{e}^{i\mathbf{q}\cdot \mathbf{r}}$. The supplied average is a thermal mean-squared modal height; for complex Fourier coefficients it is understood as $⟨|h_{\mathbf{q}}{|}^2⟩$, with $h_{-\mathbf{q}}=h_{\mathbf{q}}^{\ast }$ for a real height field. The terms are:

In the small-slope approximation, Helfrich membrane energy containsThe equipartition theorem, with the independent real modes counted consistently, gives the stated membrane fluctuation spectrum under tension. Its units are length squared: $\sigma q^2$ and $\kappa q^4$ both have units $\mathrm{J}{\mathrm{m}}^{-4}$, so $k_{B}T/[A(\sigma q^2+\kappa q^4)]$ has units ${\mathrm{m}}^2$.

At small length scales, $q$ is large and the bending term dominates:The fluctuation amplitude is controlled mainly by the membrane bending modulus and thermal energy, rather than tension. The crossover occurs when $\sigma q_c^2=\kappa q_c^4$, henceThe bending-dominated regime extends up to the membrane elastic length when scale means $1/q$, or up to ${\lambda }_c$ when it means wavelength. At still larger scales tension gives a $q^{-2}$ spectrum. At the opposite end, the continuum formula requires wavelengths larger than the microscopic membrane thickness or molecular cutoff. Without numerical $\kappa$ and $\sigma$, no numerical crossover scale is determined.

Two stabilization mechanisms. Steric stabilization coats each colloid with adsorbed or grafted polymer layers. Bringing two particles close overlaps or compresses those layers, increasing their free energy through loss of chain conformations and, in a suitable solvent, increased local osmotic pressure. The resulting repulsion can keep the cores apart. A well-solvated nonionic coating can remain effective when added salt would screen electrostatic forces. Coatings can fail through desorption, poor solvent conditions or bridging, so simply adding arbitrary free polymer does not guarantee stabilization.

Charge stabilization of colloids instead uses similarly charged surfaces and their diffuse electric double layers. Their approach costs electrostatic and ionic free energy. In DLVO theory, that repulsion competes with attractive Van der Waals forces. It is particularly sensitive to salt concentration, counterion valence and surface ionization.

Bjerrum length. The standard Bjerrum length is the separation at which the Coulomb interaction energy of two elementary charges equals thermal energy:At $T\simeq 298\mathrm{K}$ and ${\epsilon }_r=78$, this isIt compares ionic electrostatic coupling with thermal agitation. For ions of signed valences $z_1,z_2$, $U/(k_{B}T)=z_1{z}_2{\ell }_B/r$. The standard reference length uses $e$, not the whole colloid charge: assigning a colloid charge $Ze$ does not change the solvent's tabulated ${\ell }_B$.

Screening and the printed normalization. In a symmetric $z:z$ electrolyte, $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $n_{\infty }$ the bulk number density of each ionic species, ${\epsilon }_0$ the vacuum permittivity, ${\epsilon }_R$ the solvent's relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. Mobile ions rearrange around a charged surface; the Debye–Hückel screening length ${\kappa }^{-1}$ sets the range of the resulting weak potential.

There is an SI normalization error in the original PDF. Combining the Boltzmann distribution $n_{\pm }=n_{\infty }{e}^{\mp ze\psi /(k_{B}T)}$ with the SI Poisson equation and linearizing at $|ze\psi |\ll k_{B}T$ givesThe extra $4\pi$ in the PDF denominator is inconsistent with those SI charge, permittivity and density conventions. Keeping the printed algebra would make its screening length $\sqrt{4\pi }$ times the SI value at the same ion density. No numerical screening length is required here, and the qualitative dependence ${\kappa }^{-1}\propto n_{\infty }^{-1/2}$ is the same in either normalization. Added salt shortens the range of repulsion, allowing the attractive potential to capture particles more readily.

Interaction-potential sketches. Let $h$ be the surface separation between two spheres of radius $a$. A useful schematic DLVO theory potential has the formaway from the microscopic contact region. Here $A_H>0$ is a Hamaker constant; excluded-core repulsion and molecular details regularize contact. The exponential is the double-layer repulsion and the inverse-gap term the short-gap Van der Waals force contribution. A fixed $B$ illustrates the changing screening range; actual charge regulation can change the repulsive amplitude as well.

At low salt there is a high positive barrier between a deep primary minimum near contact and the dispersed state at $U(\infty )=0$. A shallow secondary minimum can occur outside the barrier and support weak reversible association. Increasing salt narrows the repulsive range and lowers the barrier. Thermally activated primary aggregation becomes appreciable when the barrier is only a few $k_{B}T$; the idealized barrier-free condition defines the fast-aggregation threshold associated with the critical coagulation concentration. Above that threshold, collisions can result in rapid capture in the primary minimum.

Aggregate structures. Well above the critical coagulation concentration, the sticking probability is close to one. In diffusion-limited cluster aggregation, the first encounter attaches particles or clusters at accessible outer branches before they can explore many alternative configurations. The result is an open, tenuous structure. Typical ideal three-dimensional cluster models have mass fractal dimension $d_f\simeq 1.8$.

Near the threshold on its lower-salt, slow-aggregation side, a remaining barrier makes most encounters unsuccessful. Reaction-limited cluster aggregation then permits repeated approaches and orientations before attachment, commonly giving denser clusters with $d_f\simeq 2.1$. Since particle count scales as $N\propto (R_g/a{)}^{d_f}$, the larger exponent means more particles at a given large cluster size. This explanation does not require already formed bonds to rearrange: unsuccessful encounters themselves alter which configurations eventually bind.

The side of the threshold matters. At the ideal barrier-free threshold itself, or just above it, aggregation can already be diffusion limited; proximity to the critical coagulation concentration alone does not universally imply a denser reaction-limited structure. The quoted dimensions are typical of irreversible approximately isotropic cluster models, not exact values for every colloid. Strong bond rearrangement or anisotropic attractions can change this comparison.

Diffusion and mobility. For a dilute isolated sphere in a Newtonian fluid, Stokes's law gives the drag magnitude $F_{\mathrm{drag}}=6\pi \eta Rv=\xi v$, where $\eta$ is the dynamic viscosity and $R$ is the hydrodynamic radius. The mechanical mobility is therefore $1/\xi$; the translational hydrodynamic mobility matrix is ${\xi }^{-1}I$.

To relate mobility to diffusion, put the particles in a weak external potential energy $U(x)$. Their drift speed is $-U^{\prime }(x)/\xi$. For number density $n(x)$, the sum of drift and Fick's first law fluxes isAt thermal equilibrium, the Boltzmann distribution gives $n\propto e^{-U/(k_{B}T)}$, and the net flux is zero. Substituting $n^{\prime }=-n{U}^{\prime }/(k_{B}T)$ into $J=0$ yields $D=k_{B}T/\xi$. Thus the Stokes–Einstein relation isHere $k_B$ is the Boltzmann constant and $T$ is absolute temperature. The assumptions include a no-slip approximately spherical particle, a continuum fluid, a small particle Reynolds number, and negligible hydrodynamic interactions with other particles or boundaries.

Gravitational sedimentation. The buoyant force must be included, since the mass of displaced water is comparable to the protein mass. Using $1\mathrm{Da}=1.66054\times 10^{-27}\mathrm{kg}$ givesso the buoyant mass in the stated spherical model isThe drag coefficient is $\xi =6\pi (10^{-3})(2.16\times 10^{-9})=4.0715\times 10^{-11}\mathrm{kg}{\mathrm{s}}^{-1}$. At terminal speed, $\xi v_g=m_{b}g$, soIgnoring buoyancy would give $1.47\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-1}$ and would not use the supplied water density. This calculation treats the stated radius as both the hydrodynamic radius and the radius determining displaced volume; a real hydrated protein can require a separately measured partial specific volume.

Electrical drift. In the specified simple electrophoresis model, terminal force balance is $Ze\mathbf{E}=\xi \mathbf{v}$. Thereforewhere $\mu$ is the electrophoretic mobility and $e$ the elementary electric charge. For the given positive beta-lactoglobulin charge,in the direction of the electric field. This exceeds the gravitational sedimentation speed by about $8.6\times 10^6$.

Separation time. The two populations have equal drag coefficients because their radii are equal. Their difference in drift speed is therefore set by one elementary charge:Starting with coincident band centres, their separation is $\Delta vt$. HenceAt that time the original band has travelled $20\mathrm{mm}$ and the mutant $21\mathrm{mm}$. Their masses do not enter this ideal electrical force balance: equal-radius particles with different masses but the same charge would have equal electrophoretic mobility in this model.

Practical discrimination. A one-charge difference is enough for the stipulated separation, so the method is sensitive to mutations that change effective charge or friction. It cannot distinguish every mutation: substitutions that preserve charge and hydrodynamic size need not separate. Buffer pH changes protein ionization, and realistic electrophoresis also involves the ionic atmosphere, electro-osmosis, walls or gel interactions, and finite band widths.

For example, the Stokes–Einstein relation at $293.15\mathrm{K}$ gives $D=9.94\times 10^{-11}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. For an initially narrow band, diffusive broadening of electrophoretic bands givesat the calculated separation time. Thus the centres are about $3.15$ diffusion standard deviations apart, rather than forming infinitely sharp bands. The stated $1\mathrm{mm}$ centre-separation criterion is achievable in the model, but actual resolution also depends on the loading width and other dispersion mechanisms.

Equivalent centrifugal acceleration. For the original protein to attain its electrical drift speed by sedimentation, its buoyant centrifugal force must equal $20eE$:ConsequentlyThis is the acceleration at the protein's position in the rotating solution; no rotation rate can be assigned without a centrifuge radius. If the intended comparison is instead the relative separation speed of the two bands, use $\Delta m_b=600\mathrm{Da}$, since their displaced volumes are equal. Matching $|\Delta m_b|a_{\mathrm{eff}}=eE$ then gives $8.04\times 10^7\mathrm{m}{\mathrm{s}}^{-2}$. In that separation experiment the heavier original protein moves faster centrifugally, whereas the more highly charged mutant moves faster electrically.

Stabilization and screening. Charge stabilization of colloids means suppressing aggregation through repulsion between similarly charged particles and their electric double layers. In DLVO theory, this repulsion competes with attractive Van der Waals forces. A substantial repulsive barrier compared with thermal energy $k_{B}T$ can make aggregation slow, even when the dispersed state is only metastable.

The Debye–Hückel screening length ${\lambda }_D={\kappa }^{-1}$ is the characteristic range over which a weak electric potential is attenuated by rearrangement of mobile ions. It measures the thickness of a diffuse screening cloud, not a molecular diameter. For a plane, a weak potential decays as $e^{-\kappa x}$ into the bulk.

Derivation and the source convention. Put a planar interface at $x=0$, with electrolyte in $x>0$, bulk potential zero, and absolute permittivity $\epsilon ={\epsilon }_0{\epsilon }_r$. For a symmetric $z:z$ electrolyte, let $n_{\infty }$ be the bulk number density of each ionic species. The Boltzmann distribution givesso the mobile charge density isSubstitution into the SI Poisson equation, ${\nabla }^2\psi =-{\rho }_e/\epsilon$, gives the Poisson-Boltzmann equationFor $|ze\psi |\ll k_{B}T$, linearize the hyperbolic sine. The Debye–Hückel approximation is thenHere $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $k_B$ the Boltzmann constant, $T$ absolute temperature, ${\epsilon }_0$ the vacuum permittivity, and ${\epsilon }_r$ the solvent's relative permittivity. For several ionic species, the numerator $2z^2{e}^2{n}_{\infty }$ is replaced by $e^2{\sum }_i{z}_i^2{n}_{i,\infty }$.

The original PDF includes an additional $4\pi$ in the denominator. That factor does not follow from SI Poisson equation with number densities and the stated permittivities. The Bjerrum length provides a useful check: ${\ell }_B=e^2/(4\pi \epsilon k_{B}T)$ and the consistent result is ${\kappa }^2=8\pi {\ell }_B{z}^2{n}_{\infty }$. A Gaussian-unit convention moves factors of $4\pi$ together with the charge and permittivity conventions; it does not justify inserting this extra factor while retaining all the given SI quantities. Both numerical interpretations are reported below rather than silently changing the source.

Conditions favoring stability. Increasing the surface-potential or surface-charge magnitude usually strengthens the electrostatic barrier. Lower ionic strength increases ${\lambda }_D$ and generally makes repulsion longer ranged; added salt and especially multivalent counterions shorten it and can promote aggregation. Suitable pH can keep an ionizable surface charged, while conditions near an isoelectric state weaken charge repulsion. The solvent-dependent Van der Waals forces, charge regulation and particle geometry also affect the barrier, so a long screening length is not by itself sufficient. At fixed density and valence, the linear formula gives ${\lambda }_D\propto \sqrt{\epsilon T}$, but actual temperature changes can also alter the permittivity and surface chemistry.

Surface charge at small potential. Take the usual one-sided planar-interface model, with negligible electric displacement on the nonaqueous side. The solution decaying into the solution isThe electric field is $E_x=-{\psi }^{\prime }=\kappa {\psi }_0{e}^{-\kappa x}$. The electrostatic interface boundary conditions then give the surface charge densityOne can also derive it by integrating the diffuse charge density:Thus the diffuse layer neutralizes the surface charge. This is the planar Debye-Hückel double layer. If instead the object is an isolated sheet wetted by identical electrolyte on both sides, each side screens half its total charge and ${\sigma }_{\mathrm{sheet}}=2\epsilon \kappa {\psi }_0$. The one-sided assumption is the conventional local model for the exterior of a large charged colloid; it must be stated because the question does not specify the material on the other side.

Numerical ion density. For $T=298.15\mathrm{K}$, $z=1$, $\kappa =10^9{\mathrm{m}}^{-1}$ and $\epsilon =80{\epsilon }_0=7.0834\times 10^{-10}\mathrm{F}{\mathrm{m}}^{-1}$, the consistent SI result isThis is $\boxed{0.0943\mathrm{mol}{\mathrm{L}}^{-1}}$ for each ion species, and hence also the salt concentration for a monovalent $1:1$ salt. The combined number density of both signs is $2n_{\infty }$.

Literal substitution into the formula printed in the PDF instead givesThese values are distinct consequences of the source's extra factor; they cannot both describe the same physical screening length in SI with the same definition of $n_{\infty }$.

Surface potential and validity. The requested linear estimate in the one-sided model isHowever, the thermal voltage is $k_{B}T/e=25.7\mathrm{mV}$, so $e{\psi }_{0,\mathrm{linear}}/(k_{B}T)\simeq 13.7$. Thus the small-potential approximation is not self-consistent for the supplied surface charge. The boxed value is its formal extrapolation, not a reliable prediction of the actual surface potential.

For a valid nonlinear comparison within the ideal one-sided planar Poisson-Boltzmann equation, multiply the differential equation by ${\psi }^{\prime }$ and use zero potential and field in the bulk. This givesUsing the surface boundary condition yields the Grahame equation and its inverse:With the SI screening length, the latter gives $\boxed{{\psi }_{0,\mathrm{PB}}\simeq 0.135\mathrm{V}}$. This is a nonlinear mean-field estimate under explicit assumptions, not a unique experimentally determined potential: compact interfacial layers, finite ion size and surface chemistry were not supplied. The source's normalization problem and its use of a linear formula far outside the small-potential regime are separate issues.

Here $T$ is the absolute temperature and $k_B$ the Boltzmann constant. An adsorption blob contains $g\sim (\xi /b{)}^2$ monomers, so there are $N/g\sim N(b/\xi {)}^2$ such blobs. Confining a blob removes an amount of conformational entropy of order $k_B$; thus the positive confinement contribution to the Helmholtz free energy isThe statement about one blob's entropy is used as an entropy loss relative to the free chain, rather than as an absolute entropy.

In the assumed slab model, contacts occur within a monomer-scale distance $b$ of the substrate. The contact fraction is approximately $b/\xi$, giving $N_c\simeq Nb/\xi$. Multiplying this by the negative contact energy yieldsThe two contributions express the competition between loss of conformational entropy and gain of adsorption energy. The numerical coefficients are the ones stipulated by this blob approximation.

Availability and total entropy. The reservoir changes by $d{U}_R=-dU$ and $d{V}_R=-dV$. Its particle number and the system's particle number are fixed. The reservoir's first law of thermodynamics therefore givesAdd the system's entropy change. Because $T_R,p_R$ are constant,Thus define the availabilityAn additive constant in $A$ is immaterial; its value above the equilibrium reference is the exergy. The Second law of thermodynamics gives $\Delta S_{\mathrm{tot}}\ge 0$, hence $\Delta A\le 0$. A stable equilibrium maximizes total entropy over the allowed variations and therefore minimizes $A$. The environmental $T_R,p_R$ enter this definition even when the system has not yet reached their values. Because both thermal and mechanical exchanges are allowed, equilibrium here includes mechanical balance; equality of temperatures alone does not ensure the minimum.

Probability and curvature. For a fixed value of a macroscopic coordinate $x$, let $S(x)$ count the compatible system microstates. The product of the system and reservoir multiplicities isAll factors independent of $x$ enter a normalization constant, givingHere $A(x)$ is the constrained or effective availability for this coordinate, including the multiplicity of any eliminated variables. A nonuniform coordinate measure must likewise be retained or incorporated in that effective availability.

At a stable interior minimum, $A^{\prime }(x_0)=0$ and $A^{\prime \prime }(x_0)>0$. Its Taylor expansion givesKeeping the quadratic term yields a normal distribution,This is availability curvature and equilibrium fluctuations. The quoted fluctuation formula holds at quadratic order in the Gaussian fluctuation approximation; it is not an exact identity for an arbitrary nonlinear $A(x)$. Higher terms can shift the mean away from its most probable value $x_0$.

Bubble equilibrium. Keep the number of air molecules fixed and include the surface energy in the system's total internal energy. For constant surface tension $\Gamma$, this contribution is $\Gamma \Omega =\Gamma \alpha V^{2/3}$, with no surface entropy under the stated approximation. Its differential isHenceIndependent thermal and volume variations give $T=T_R$ andThe air pressure is evaluated at $V_0$; since the bubble is closed it generally depends on volume, so this equation is implicit until an air equation of state is supplied. For a sphere, $\alpha =(36\pi {)}^{1/3}$, $\Omega =4\pi R^2$, and the pressure jump becomes $2\Gamma /R$, the Young–Laplace equation for a single gas–liquid interface. There is no second surface as in a thin soap film.

Isothermal volume fluctuations. With the air maintained at $T_R$, eliminate its entropy in favor of its Helmholtz free energy:The identity $(\partial F_{\mathrm{air}}/\partial V{)}_{T,N}=-p_{\mathrm{air}}$ givesUsing the curvature result,These are the volume fluctuations of a closed gas bubble. Stability requires the denominator to be positive. The negative surface contribution reflects the concavity of $V^{2/3}$; gas compression supplies the positive restoring curvature. For ideal air, ${\kappa }_T^{-1}=p_{\mathrm{air}}$, and mechanical equilibrium reduces the denominator toThe units are consistent: $D$ is pressure, so $k_B{T}_R{V}_0/D$ has units ${\mathrm{m}}^6$, the units of a volume variance. Holding $p_{\mathrm{air}}$ externally fixed while varying $V$ would remove the gas-compression term and change this stability problem; that is not the fixed-particle-number situation here.

Numerical size of the fluctuations. The volume conversion is important:Use the specified $\alpha =4.84$, $\Gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$ and $p_R=10^5\mathrm{Pa}$. ThenWith the Boltzmann constant $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$,Thus the fractional root mean square fluctuation and fractional variance are respectivelyThe absolute root mean square fluctuation is $2.01\times 10^{-19}{\mathrm{m}}^3$. The very small fractional value supports the local Gaussian fluctuation approximation. This estimate neglects exchange of dissolved air, variations of surface tension, shape fluctuations and temperature fluctuations, consistently with the prescribed fixed-number, isothermal bubble model.