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In a laterally uniform dense polymer brush with approximately fixed polymer volume fraction $\varphi$, conservation of occupied volume gives $\varphi h=N{v}_0\sigma$, with repeat-unit volume $v_0$. At fixed solvent quality this gives $h\propto N\sigma$, unlike the good-solvent brush height. Isolated collapsed polymer globules and laterally segregated grafted layers need not obey this uniform-layer model.

A small insulating object of excluded volume $\Omega$ perturbs an Ohmic pore most strongly where the pore cross-sectional area $A$ is small. In a local weak-blockage model, replacing a small conducting volume gives the displayed resistance scaling, with a shape-dependent coefficient. At fixed voltage, $\Delta I\simeq -I_0\Delta \mathcal{R}/{\mathcal{R}}_0$. A symmetric pore with narrow ends and a wide middle can therefore resolve two end features in a passage pulse. This is a conditional volume-exclusion model: charged-particle ion transport, surface interactions and bandwidth can change the observed sign, amplitude and shape.

A power law is a relation $f(x)=C{x}^p$, with fixed constants $C$ and $p$. For positive $x$, any real number $p$ is allowed. Its useful feature is a simple scaling rule:The same proportional change of input produces the same proportional change of output, independently of the starting value. The exponent controls that response; the coefficient fixes the magnitude.

For example, the area of a circle is $A(r)=\pi r^2$. Doubling the radius multiplies its area by four, and tripling it multiplies the area by nine. The volume of a sphere is $V(r)=4\pi r^3/3$: the same changes multiply its volume by eight and twenty-seven. The distinction explains why similarly shaped objects do not preserve their ratio of surface area to volume when their size changes.

When the output is positive, a logarithm turns the power law into a straight-line relation. Using positive reference values $x_0$ and $f_0=f(x_0)$,These ratios are dimensionless quantities, so the formula also applies to physical measurements with units. Plotting both ratios on a logarithmic scale gives a line of slope $p$. Changing a unit changes the numerical coefficient, while the exponent stays the same. Positive $p$ describes increasing values; negative $p$ describes decreasing values; $p=0$ gives a constant. The zero coefficient is a degenerate case with no useful logarithmic slope.

Exact scaling is stronger than regular variation. The function $x^2\log x$ has a doubling ratio tending to four, but its logarithmic correction keeps growing. The diagram contrasts it with the exact power law $x^2$.

A power law for a probability density and a power law for a tail probability have different exponents. For a Pareto distribution with $\alpha >0$ and lower endpoint one,The tail probability has exponent $-\alpha$; differentiating it and changing sign gives the probability density exponent $-\alpha -1$. Confusing them changes the interpretation of a fitted exponent. The normalizing coefficient and lower endpoint also matter: a decreasing power law is not automatically a valid probability density over the entire positive real line.

The appearance of a line over a finite logarithmic range is evidence to investigate, rather than a proof of an exact power law or its mechanism. In their study of power-law distributions, Clauset, Shalizi and Newman combine maximum likelihood estimation, goodness-of-fit tests and comparisons with alternative models; their abstract explains why ordinary least-squares fitting can give misleading results for tail data. See Aaron Clauset, Cosma Rohilla Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Review 51 (2009), 661–703. The scaling rule is mathematically exact when it defines a model; whether a dataset supports that model is a question of statistical inference.

A power-law propensity uses the same reactant monomial as a deterministic law of mass action, scaled by reactor volume. For example, a bimolecular channel may be assigned $a(x)=\alpha x^2/V$. This convention differs at small copy number from the combinatorial propensity $\alpha x(x-1)/V$ and must therefore be stated explicitly.

Consider an original toy AB salt model with $K_{\mathrm{sp},c}=10^{-8}{\mathrm{M}}^2$ stipulated at fixed temperature and fixed activity coefficient product. This is a conditional concentration constant, not measured data for calcium carbonate or any identified material. A stock contains A cations and X anions; another contains Y cations and B anions. All four ions have unit electric charge magnitude. Only AB can form a solid; X and Y remain spectator ions. Assume full dissociation, no other chemical reactions, and a mixed volume equal to the sum of the initial volumes.

Mix two stocks, each of volume $V$,, each at molar concentration $C$ of its indicated salt. Before any precipitation, the mixed volume is $2V$, soMultiplying the two stock concentrations would overestimate the mixed ion product by a factor of four. The saturation threshold is $C=2\sqrt{K_{\mathrm{sp},c}}=2\times 10^{-4}\mathrm{M}$.

If $C=10^{-4}\mathrm{M}$, mixing gives $Q_{c,0}=2.5\times 10^{-9}{\mathrm{M}}^2<K_{\mathrm{sp},c}$: no AB precipitate is favored. If $C=4\times 10^{-4}\mathrm{M}$, mixing gives $Q_{c,0}=4\times 10^{-8}{\mathrm{M}}^2>K_{\mathrm{sp},c}$. Suppose nucleation occurs and bulk AB solid remains at chemical equilibrium. Equal starting inventories and one-to-one stoichiometry require equal remaining concentrations, henceFor two $10\mathrm{mL}$ portions in the second case, each reacting ion starts with $4\mu \mathrm{mol}$; the final $20\mathrm{mL}$ chemical solution retains $2\mu \mathrm{mol}$ of each, and $2\mu \mathrm{mol}$ of AB enters the solid. The respective X and Y inventories stay at $4\mu \mathrm{mol}$. Exactly half of each reacting inventory precipitates in this toy model.

The chemical equilibrium calculation adds a phase constraint to the inventory balance. Write $x=n_{\mathrm{AB},s}/(2V)$ and $c=C/2-x$. In the equilibrated bulk model,When $x>0$, the last condition requires saturation; when $x=0$, the inventory can leave an undersaturated chemical solution. Together these give $x=\max \{0,C/2-\sqrt{K_{\mathrm{sp},c}}\}$. This excludes a persistent supersaturated solution by its explicit chemical equilibrium assumption; it does not model a nucleation delay.

At fixed temperature, $dG=VdP$ gives the pressure correction for the participating solids. Constant reaction volume reduces the integral to $\Delta V_{\mathrm{solids}}(P-P_0)$. A gas product treated by chemical fugacity instead contributes $\nu RT\ln (f/P_0)$ relative to its reference state. Adding a separate gas-volume correction to that term would double count the same pressure dependence.

The resultant of distributed surface pressure; in a fluid its force per unit volume is $-\nabla P$. A spherical shell in hydrostatic equilibrium balances this against its gravitational weight.

The pressure gradient is the vector $\nabla p$ pointing in the direction of fastest pressure increase. The force density exerted by pressure on a continuum is $-\nabla p$.

A spatially varying pressure exerts net force per unit volume $-\nabla p$ on a small fluid element. Its contribution to acceleration is $-\nabla p/\rho$. Pressure increases in the direction opposite to this force.

Suppose a small volume change obeys $dV=-V_1{\kappa }_{T}dP+V_1\alpha dT$, with its displayed coefficients held constant. Work on the body is $\delta W=V_1{\kappa }_{T}PdP-V_1\alpha PdT$. Two rectangular pressure-temperature paths between the same endpoints differ by $W_a-W_b=V_1\alpha \Delta P\Delta T$, where path a changes temperature first. The straight interpolation samples the mean pressure during its temperature change, giving $W_c=(W_a+W_b)/2$. Variable material coefficients need not preserve that average relation.

In the local rest frame of a gas, the pressure tensor is the directional momentum flux $P_{ij}=\int p_i{v}_{j}f(\mathbf{p})d^{3}p$, where $f$ is the phase-space distribution function per unit momentum volume and $\mathbf{v}$ is particle velocity. For an isotropic distribution, $P_{ij}=P{\delta }_{ij}$, recovering the kinetic pressure of an isotropic gas. An anisotropic distribution can exert different normal pressures along different directions.

Let $m$ be the transferred sample mass, $p$ the known mass fraction of the intended chemical compound, and $M$ its molar mass. In a model with complete dissolution and chemically inactive impurities,The purity fraction is dimensionless: $99.95\%$ would mean $p=0.9995$, not $99.95$. These equations assume that the impurity contributes no titrant demand; an impurity that also reacts needs its own inventory equation. The value of $p$ must refer to the same preparation state as $m$, for example the specified dry material. A mass fraction assigned before unaccounted uptake of water cannot simply be used after it.

For an original worked example, stipulate pure potassium hydrogen phthalate, abbreviated KHP, so $p=1$, a selected mass $m=0.51055\mathrm{g}$, and rounded molar mass $M=204.22\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The NIST Chemistry WebBook lists formula ${\mathrm{C}}_8{\mathrm{H}}_5\mathrm{K}{\mathrm{O}}_4$ and molecular weight $204.2212$, consistent with the rounded molar mass used here.^[ref] The selected mass, purity assumption and volumes in this example are chosen inputs, not measured results or a claim about a real batch's purity.

The acid-bearing anion of potassium hydrogen phthalate loses one proton to hydroxide. The net chemical equation isThus sodium hydroxide has one-to-one stoichiometry with potassium hydrogen phthalate in this neutralization, givingChoose a delivered endpoint titre of $24.85\mathrm{mL}$ and matched titration blank of $0.05\mathrm{mL}$. ThenThe extra displayed digits show the arithmetic of the stipulated model; they do not establish experimental accuracy. A compact rounded result is $0.1008\mathrm{mol}{\mathrm{L}}^{-1}$.

When the entire weighed portion is titrated, added water changes concentration and the pH path but preserves the initial acid inventory. The dissolution volume therefore does not appear in $n_s=mp/M$. If instead a volumetric flask prepares a stock and a volumetric pipette selects an aliquot, the selected inventory is $n_s{V}_a/V_f$ for homogeneous stock, with delivered aliquot volume $V_a$ and final stock volume $V_f$. The same bookkeeping explains why complete transfer matters even when the amount of dissolution water is approximate.

A fundamental cell of a Bravais lattice, of volume equal to the absolute determinant of any primitive lattice basis. Direct and reciprocal lattice primitive cell volumes multiply to $(2\pi {)}^d$ in dimension $d$ with the angular-wavenumber convention.

A uniform-depth particle-laden gravity current in fixed triangular cross section has conserved volume proportional to $h^{2}X$. A gravity-current front condition and absorbing-boundary particle deposition flux give $\dot{X}=\mathrm{Fr}\sqrt{G\varphi h}$ and $\dot{\varphi }=-2w(\varphi )\varphi /h$. The box describes late-time bulk evolution; it does not resolve the nose or immediate dam-break flow.

A $3\times 1.5\times 1.5\mathrm{m}{\mathrm{m}}^3$ single crystal of gadolinium antimonide carried a $2\mathrm{mm}$ fibre Bragg grating cyanoacrylate-bonded along its $a$ axis. The pulsed magnetic field paralleled that axis. Broadband light, a circulator and a spectrometer resolved reflected-wavelength shifts at $5.4\mathrm{K}$. Pre-pulse readings measured noise; low fibre thermal expansion reduced thermal background. A residual post-pulse shift indicated possible heating. Spectral distortion checked bond uniformity, while the fibre's smaller cross-section limited loading; bonding was not assumed force-free. Longitudinal strain showed a kink near $32.6\mathrm{T}$.^[ref]

The distinction between readout and inference is essential. The fibre Bragg grating senses a change of reflected wavelength; interpreting this as the specimen's strain requires good strain transfer and a controlled temperature. A useful independent control would compare a bonded sensor with a nearby unloaded sensor through the same pulse. Another would vary bond stiffness or specimen cross-section to test loading. These are proposed checks, not additional procedures claimed for the experiment above. The pulsed magnet's winding details were not specified in the inspected account.

This experiment measures one longitudinal component of strain. Establishing the first-order volume change would require three orthogonal normal components under the same conditions. GdSb also has its own antiferromagnetism and field-dependent magnetic structure: this measured curve does not establish the simple saturated isotropic law. Its role here is to make the measurement chain and its limitations concrete.

Water is pumped to an upper reservoir and later discharged through a turbine. For approximately constant head, useful discharge energy is $\eta \rho Vgh$ while power is $\eta \rho Qgh$; reservoir volume and flow rating set different properties.

A pycnometer makes the distinction between mass and mass density experimentally tangible. NIST's 1994 report describes a glass vessel with filling capillaries, wire plugs and a covered overflow bowl in a controlled water bath. The operators soaked a pure-water filling to establish the bath temperature, dried the outside, then left it overnight in the balance enclosure to reach thermal equilibrium before weighing. Empty and full force-restoration balance readings, together with known water mass density, calibrated its capacity at the bath temperature. A tin-oxide coating suppressed stray electric charge; an electrometer detected none during use. The authors distinguish bath and weighing temperatures and correct for buoyancy and changes of displacement volume. Capacity and water-reference uncertainties carry into later liquid mass density determinations. Their pycnometer data were not yet analyzed, so this account provides no measured density or achieved precision.^[ref]

The ideal accounting behind this measurement calibration can be derived without numerical observations. Let $m_0,m_w,m_x$ denote corrected masses of the empty vessel, water-filled vessel and sample-filled vessel. At the same filling temperature, with known reference water mass density ${\rho }_w$ and reproducibly filled volume $V_c$,The empty-vessel subtraction removes vessel mass; the water filling sets volume. Doubling $V_c$ for the same uniform sample doubles the sample mass increment but preserves ${\rho }_x$. Uncorrected balance indications, trapped bubbles, incomplete filling or unequal filling temperatures break the simple identification with corrected mass and a common $V_c$. This explains why replication is a statement about comparable states rather than arbitrary instrument readings.

A pyramid is a polyhedron with a polygon as its base and triangular faces joining the base edges to one apex outside the base plane. If its base area is $A$ and perpendicular height is $h$, its volume is $Ah/3$. At height $z$ above the base, similarity gives cross-sectional area $A(1-z/h{)}^2$. Integrating this area from $0$ to $h$ proves the formula.

One isotropic three-dimensional bosonic branch with dispersion $ϵ=\alpha p^2$ and zero chemical potential has density of states per volume $\sqrt{ϵ}/(4{\pi }^2{\hslash }^3{\alpha }^{3/2})$. Integrating its Bose-Einstein energy and differentiating temperature gives the title coefficient. Identical independent branches multiply it by their number. Thus a measured $T^{3/2}$ coefficient fixes branch multiplicity divided by ${\alpha }^{3/2}$, not a dispersion coefficient independent of that counting convention. A gap or low-dimensional spectrum changes the asymptotic law.

The quasiparticle vacuum contains physical particles at nonzero momentum because the Bogoliubov transformation mixes creation and annihilation operators. Their occupation is $v_k^2=(A_k/E_k-1)/2$, and the depletion density is its momentum sum divided by the volume. At nonzero temperature an additional $(A_k/E_k)n_B(E_k)$ contribution is required; quantum depletion alone is a zero-temperature expression.