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A primary standard supplies a reliably known reacting inventory from which a titrant concentration can be determined.

The defining advantage is known high purity: the weighed portion can be assigned a known amount of substance. A titrant whose concentration is established against that inventory is a secondary standard solution in IUPAC terminology.^[ref]

An accurate balance determines total mass, including any unwanted water or other material. To infer reacting inventory, one also needs known composition, stability during storage and weighing, complete transfer and dissolution, and a sufficiently complete chemical reaction of known stoichiometry. A finely resolved balance cannot establish these chemical facts. Potassium hydrogen phthalate is useful because a suitably prepared portion provides a known acid inventory for sodium hydroxide titrant standardization; the KHP standardization protocol below shows how that claim is put into practice.

Preparation belongs to the particular material and method. For example, the MIT handout specifies drying potassium hydrogen phthalate at $110{}^{\circ }\mathrm{C}$ for $1.5$ hours or overnight, cooling in a desiccator, and weighing by difference; its footnote identifies removal of superficial moisture as the purpose. This is a condition of that potassium hydrogen phthalate procedure, not a universal instruction to heat every primary standard in the same way.^[ref]

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 black hole formed in the early universe rather than as the endpoint of a star. It is not subject to the ordinary stellar-remnant baryon budget in the same way as a black hole formed from a baryonic progenitor. Formation scenarios and observational constraints depend strongly on its mass.

At fixed cosmological weak freeze-out temperature $T_f$ and fixed neutron survival factor, neutron-proton chemical equilibrium gives $r_f=e^{-Q/T_f}$ and neutron-limited helium synthesis gives $Y_p\propto 2r_f/(1+r_f)$. Differentiating yields the displayed sensitivity. A finite change $Q\mapsto (1+\alpha )Q$ gives $Y_p^{\prime }/Y_p=(1+e^{Q/T_f})/(1+e^{(1+\alpha )Q/T_f})$. Increasing the positive mass difference reduces helium production. Changes to the weak conversion rates, neutron lifetime or nuclear history must be supplied separately for a full abundance prediction.

The principal moments of inertia about a specified point $O$ are the eigenvalues $I_1,I_2,I_3$ of the inertia tensor ${\mathbf{I}}_O$. Their associated unit vectors that are eigenvectors specify principal axes through $O$. The spectral theorem for real symmetric matrices guarantees an orthonormal basis of these eigenvectors, even for an irregular rigid body. Choose their signs to make a right-handed basis and collect them as the columns of an orthogonal matrix $Q$:Thus diagonalization removes the off-diagonal coupling in the rotational kinetic energy. The eigenvalues belong to the mass distribution and reference point, independently of which orthonormal basis initially describes them. ^[ref]

In this principal body frame, write angular velocity as $\mathbf{\omega }={\sum }_i{\omega }_i{\mathbf{e}}_i$. ThenUse these as total angular momentum and kinetic energy only under the reference-point conditions stated under inertia tensor. Along a single principal axis with $I_i>0$, angular momentum is parallel to angular velocity. For a mixture of directions with different principal moments of inertia, it generally is not. A principal axis need not be a geometric symmetry axis: it is selected by the complete mass distribution. A principal axis also need not give stable free rotation; for three distinct principal moments of inertia, the intermediate axis theorem supplies the stability distinction.

For any unit vector $\mathbf{n}={\sum }_i{n}_i{\mathbf{e}}_i$,The axial moment of inertia is a weighted average of the principal moments of inertia. At fixed angular speed, the least and greatest rotational kinetic energies occur in the corresponding eigenspaces. At fixed nonzero angular momentum magnitude $L$, instead, $K=\frac{1}{2}{\sum }_i{L}_i^2/I_i$, so the largest principal moment of inertia gives the smallest rotational kinetic energy. These comparisons hold the reference point fixed.

There is an additional physical restriction beyond positive semidefiniteness. In principal axes coordinates,with two analogous cyclic inequalities. Therefore three arbitrary positive numbers do not always represent a possible inertia tensor. For a distribution in the $xy$ plane through $O$, $I_3=I_1+I_2$, as in the planar inertia tensor of paired point masses. A uniform solid ellipsoid of mass $M$ and semiaxes $a,b,c$ supplies a continuous three-dimensional example: using its mass second-moment tensor, its central principal moments of inertia are $M(b^2+c^2)/5$, $M(a^2+c^2)/5$, $M(a^2+b^2)/5$. Setting $a=b=c=R$ recovers the moment of inertia of a uniform solid sphere.

With Minkowski metric signature $(+---)$ and source coupling $-{\mu }_0{J}_{\mu }{A}^{\mu }$ added to the Proca action, a massive vector field coupled to a conserved current obeysIn vacuum this gives $(\square +m^2)A^{\nu }=0$ after imposing the dynamically derived Lorenz constraint in Proca theory. The explicit mass term breaks the gauge invariance of the massless Maxwell theory.

The semimajor axis $a_1$ of the pulsar's orbit about the binary barycenter, projected along the line of sight and divided by the speed of light. The orbital inclination $i$ is zero for a face-on orbit. Thus $x$ is a time, often quoted as the numerical number of light-seconds in $a_1\sin i$, not the true binary separation. The relative-orbit semimajor axis is instead $a=a_1+a_2$; conservation of momentum gives $a_1=a{m}_2/(m_1+m_2)$.

Combining Kepler's third law with that barycenter relation gives the timing form of the binary mass function, also valid for eccentric Kepler orbits:Choose $x=2\mathrm{s}$ and $P_b=8\mathrm{h}$ for an original example: $a_1\sin i=599,585\mathrm{km}$ and $f\simeq 0.0773M_{\odot }$, where $M_{\odot }$ is a solar mass. For a specified pulsar mass $m_1=1.4M_{\odot }$, the smallest allowed companion mass occurs at $\sin i=1$ and is approximately $0.698M_{\odot }$. A different assumed $m_1$ changes this bound. One measured $f$ does not determine both masses and orbital inclination.

Projected surface mass density is mass per unit projected area, obtained by integrating mass density along the line of sight. It determines the enclosed cylindrical mass used by a thin gravitational lens equation. It need not equal a physical disk's density, and its aperture mass differs from a spherical three-dimensional enclosed mass.

With well-mixed gas-phase metallicity $Z$, zero mass return, star formation rate $\psi$, pristine galactic gas inflow $\nu \psi$, and constant stellar yield $y$, mass conservation and metal accounting give $\dot{M}=-(1-\nu )\psi$ and $M\dot{Z}=(y-\nu Z)\psi$. Thus, for $0<\nu <1$,The initial gas-phase metallicity $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.

Exchanging a down-type squark of mass $M$ between $LQD$ and $UDD$ vertices produces a dimension-six effective operator with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/M^2$. Dimensional analysis then gives a proton decay width ${\Gamma }_p=c_h|{\lambda }^{\prime }{\lambda }^{\prime \prime }{|}^2{m}_p^5/M^4$, with a dimensionless factor $c_h$ containing hadronic matrix elements and phase space. A lifetime bound ${\tau }_p>{\tau }_{\min }$ implies $|{\lambda }^{\prime }{\lambda }^{\prime \prime }|<M^2/\sqrt{c_h{\tau }_{\min }{m}_p^5}$. The bound is on the relevant flavour product and depends on the exchanged mass; dimensional analysis alone does not determine $c_h$.

With bare nuclear masses, the proton separation energy is the energy needed to remove a proton. A negative value means the daughter plus proton has lower rest energy. Proton emission can still have a finite lifetime because of a Coulomb or centrifugal barrier.

Growth of solid aggregates through collisions and sticking in a protoplanetary disk. Relative motion can arise from Brownian motion, settling, turbulence and differential radial drift. Bouncing and fragmentation can interrupt growth; collision frequency alone does not determine whether an encounter increases aggregate mass.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

A pseudo-Goldstone boson is a light mode produced when a spontaneously broken continuous symmetry is also weakly broken explicitly. Its mass vanishes with the explicit symmetry-breaking parameters.

A hanging mass $m$ pulls the centre of a uniform disc of mass $M$ up an incline of angle $\theta$, using a light string and light frictionless pulley. If the disc rolls without slipping and the string is parallel to the slope, its effective translational mass is $M+I/a^2=3M/2$. With downward hanging acceleration $q$, $q=g(m-M\sin \theta )/(m+3M/2)$. Static friction has signed downslope component $Mq/2$ and supplies the rolling torque. The acceleration threshold is unchanged from a sliding block, although rotational inertia reduces the magnitude.

A calibrated vessel used to infer a liquid's mass density from its mass and filled volume; see pycnometer density measurement for an inspected experiment and the accounting formula.

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.

The QCD scale is the renormalization-group invariant energy at which the perturbative running coupling becomes strong. It sets the characteristic scale of confinement and most hadron masses.

It is a characteristic energy scale of Quantum chromodynamics.

With renormalized Dirac fields, the coefficient of $-\bar{\psi }\psi$ in the counterterm Lagrangian is $\delta m_L=Z_2{m}_0-m$, whereas the bare-parameter shift is $m_0-m=\delta m_L-m\delta Z_2$ at first order. In Feynman gauge and $d=4-ϵ$, QED has $\delta Z_2=-e^2/(8{\pi }^2ϵ)$ and $\delta m_L=-4m{e}^2/(8{\pi }^2ϵ)$. Their difference gives $m_0-m=-3m{e}^2/(8{\pi }^2ϵ)$. Defining the insertion and the mass coefficient before quoting its sign avoids confusing these two counterterms.

Let $S=s/s_{\infty }$, let the interstellar medium have mass $g=s_{\infty }S(1-S)$, and retain fraction $f=S$ of returned material. For $0\le \beta <1$ put $q=(2-\beta )/(1-\beta )$. The integrating factor for the metal mass is $S^{1/(1-\beta )}/(1-S)$. A finite initial gas-phase metallicity removes the homogeneous solution and gives the displayed expression, equivalently $Z=y{\sum }_{n\ge 1}{S}^n/(q+n)$ for $0\le S<1$. The increasing initial gas mass requires pristine galactic gas inflow: gas accounting gives $d{M}_{\mathrm{in}}/ds=q(1-S)$. A fixed initial mass function makes the birth metallicity distribution function proportional to $ds/dZ$ when enrichment is monotone.