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The chemical element with atomic number eighteen. Argon is a colourless noble gas. It can provide a protective atmosphere around hot, reactive materials, for example in welding, where excluding oxygen limits oxidation. Commercial production separates it by distillation of liquid air.^[ref]

Its discovery shows why a small disagreement can matter more than a reassuringly repeatable result. Two preparations can each give consistent measurements while producing different mixtures. The Rayleigh-Ramsay argon density experiment investigated just such a discrepancy. Gas density links the weighing to a mean molar mass; a separate separation process tests which component accounts for it.

Consider an original simplified model containing only nitrogen and argon, with argon mole fraction $x$. Let their molar masses be $M_N$ and $M_A$. Adding the component masses gives the mixture's mean molar massFor an ideal gas mixture at the same pressure and temperature as a pure-nitrogen reference, the gas density ratio isThe inversion requires known identities, the binary-mixture assumption and $M_A\ne M_N$. A density ratio cannot uniquely recover an arbitrary multicomponent mixture. Nor should $x$ be interpreted as the argon fraction of untreated air: the model sample has only the two specified components.

Choose round model values $M_N=28\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ and $M_A=40\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. For $x=0.010$, the calculated relative density excess isThus a one-percent mole fraction does not mean a one-percent gas density change. In a chosen $2.00\mathrm{L}$ globe at $100\mathrm{kPa}$ and $300\mathrm{K}$, the ideal gas law gives approximately $2.24508\mathrm{g}$ for pure nitrogen and $2.25471\mathrm{g}$ for this mixture: only $9.62\mathrm{mg}$ separates the weighings. These are model calculations with rounded molar masses, not historical measurements.

A direct control calculation explains why the filling conditions matter. At fixed pressure, pure nitrogen athas the same gas density as that mixture at $300\mathrm{K}$. A roughly $1.28\mathrm{K}$ mismatch can therefore imitate the entire stipulated composition signal. For small changes, the ideal gas law givesAn independently measured pressure and temperature, matched weighing procedure, and checks for residual vapour distinguish composition from these alternative causes. The model explains the need for controls; it does not supply the complete corrections for a real gas-weighing instrument.

The Clapeyron equation connects the slope of a phase coexistence curve to the latent heat and the change in volume during a first-order phase transition. A steep boundary can indicate a large latent heat, a small change in volume, or both. This lets a measured boundary constrain caloric properties and predicts how changing pressure shifts a melting temperature.

For a pure substance in bulk phase coexistence, label the initial and final thermodynamic phases $\alpha$ and $\beta$. Both have the same absolute temperature $T$, pressure $p$, and chemical potential. With $s$, $h$, and $v$ denoting entropy, enthalpy, and volume on one common basis, define $\Delta s=s_{\beta }-s_{\alpha }$, $\Delta h=h_{\beta }-h_{\alpha }$, and $\Delta v=v_{\beta }-v_{\alpha }$. ThenThe derivative follows the phase coexistence curve; it is not a partial derivative within one thermodynamic phase. Use molar volume $v_m$ with molar latent heat $L_m$ in $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}$, or specific volume $v$ with latent heat $L$ in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. Both give $dp/dT$ in $\mathrm{Pa}{\mathrm{K}}^{-1}$. For molar mass $M$, $L_m=ML$ and $\Delta v_m=M\Delta v$, so the predicted slope is independent of the chosen basis. This formulation assumes a simple compressible system, fixed composition, and negligible interface contributions; a curved interface can instead require the Young–Laplace equation.

For positive latent heat in the stated conversion direction, the sign of the slope is the sign of $\Delta v$. Converting liquid to gas normally increases volume and gives a positive slope. The negative melting-point slope of water illustrates the opposite sign. At a thermodynamic critical point, the coexisting gas and liquid merge: both $L$ and $\Delta v$ vanish, so their ratio needs a limiting analysis rather than substitution of zero into the Clapeyron equation.^[ref]

A short derivation uses the molar Gibbs free energy $g_m$, equal to the chemical potential for a pure substance. Equality persists along the boundary, so $d{g}_{m,\alpha }=d{g}_{m,\beta }$ with $d{g}_m=-s_{m}dT+v_{m}dp$. Therefore $-\Delta s_{m}dT+\Delta v_{m}dp=0$. Equality of $g_m=h_m-T{s}_m$ also gives $\Delta h_m=T\Delta s_m$. Eliminating $\Delta s_m$ yields the Clapeyron equation.

In a one-carrier Drude model, the Hall coefficient is $R_H=1/(nq)$. Combining its inferred electron density with atomic density ${\rho }_m{N}_A/M$ gives the displayed number of mobile electrons per atom. The molar mass and mass density must be supplied independently. Multiple bands, anisotropic carrier motion and Hall factors limit this simple interpretation; field reversal helps distinguish the transverse response from voltage offsets.

Gas density is the mass density of a gas. For an ideal gas sample of amount of substance $n$, mean molar mass $\overline{M}$ and volume $V$, combine $m=n\overline{M}$ with $pV=nRT$ to obtainHere $R$ is the molar gas constant. For a homogeneous mixture of gases, $\overline{M}={\sum }_i{x}_i{M}_i$, where $x_i$ is the mole fraction of component $i$. Equal pressure and temperature make a gas density ratio equal to a mean-molar mass ratio within this model. Argon detection in a binary gas mixture gives a worked example and a temperature confounder.

A gel band is a localized population of stained or otherwise detected molecules with sufficiently similar migration in one sample lane. One visible gel band need not contain only one nucleotide sequence. Two different linear DNA fragments of 375 base pairs can occupy the same unresolved gel band; equal migration supplies size evidence under suitable conditions, not a sequence determination.

Fluorescence from a bound stain can support an estimate of DNA quantity, but suitable staining and detector response matter.^[ref]

For a quantitative model with equal staining response per unit mass, background-subtracted, integrated fluorescence $I$ satisfies $I=km$. This requires an unsaturated detector and a common proportionality constant $k$; visual peak brightness alone does not supply $I$.

The amount of substance is $n=m/M$, where $M$ is molar mass. For double-stranded DNA with approximately equal average mass per base pair, $M$ is proportional to length $L$, so $n$ is proportional to $I/L$. In a constructed example, equal integrated signals from 250-base pair and 1000-base pair gel bands imply approximately equal masses but four times as many short DNA molecules. Conversely, equal numbers of those DNA molecules would give about four times the signal for the longer fragments in this model. Neither conclusion follows from darkness in a schematic drawing.

An endpoint experiment uses a polarimeter with a $1.0\mathrm{dm}$ sample cell and supplied $4\%$, $6\%$ and $8\%$ D-glucose chemical solutions. Collect about $15\mathrm{mL}$ of each. Fill the cell first with distilled water, dry its windows, move bubbles to the air trap, and retain its orientation. Scan light intensity against polarimeter analyzer angle through $360^{\circ }$. The protocol fits a Gaussian function to the upper third of a symmetric peak to locate its maximum. Rinse with the next sample, repeat for each glucose chemical solution, and subtract the water maximum's angle from each sample maximum. This is an operational peak-location method; the Gaussian function is not the physical transmission law.^[ref]

The existing optical activity and polarimetry articles explain the apparatus, Malus's law and instrument limitations. For an extension that observes mutarotation in time, freshly dissolve a specified crystalline anomer, record the mixing time, and repeat the optical rotation readout at fixed concentration, path length, wavelength and recorded temperature. Keep pH consistent and record the delay before the first reading. These are proposed additions to the endpoint protocol, not its published steps or a claim of measured results. Recheck the water blank, repeat the preparation, and compare with a separately equilibrated portion under the same conditions. A stable endpoint alone does not show the time evolution.

Interpret each concentration separately. If $\theta$ is blank-corrected optical rotation, $l$ is path length and $c$ is total glucose mass concentration, first calculate specific rotation $r=\theta /(lc)$. In an additive two-anomer approximation, let $r_{\alpha },r_{\beta }$ be reference specific rotations under the same conditions and $x_{\alpha }$ the mole fraction of alpha within that pair. Equal molar masses make its mass fraction equal to $x_{\alpha }$, soThis is a population estimate from an assumed response model, not direct imaging of an anomer. For a simple consistency check, doubling $c$ at fixed $l$ and unchanged composition doubles $\theta$ in this model while leaving $r$ unchanged. Averaging raw optical rotations at different concentrations would fail this test; normalize first.

Changing temperature, pH, contamination, incomplete dissolution, window stress, bubbles or instrumental drift can imitate or obscure a changing signal. Reference specific rotations must match the sample conditions. Minor open-chain and other ring forms limit the two-anomer approximation; a single optical rotation cannot separately determine every population. Also, these anomers are diastereomers, not an enantiomer pair: the equal-and-opposite cancellation formula for enantiomeric excess does not apply.

The radius of gyration of a dilute linear polystyrene coil in a good solvent depends on its molar mass as $R_g=K{M}^{\nu }$, with a material- and solvent-dependent amplitude. A manufacturer-supplied light-scattering calibration for polystyrene in benzene or toluene near room temperature is the displayed relation, quoted over $M\simeq 2\times 10^5$ to $5\times 10^7\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The exponent is close to the Flory approximation $3/5$. Values outside the quoted mass range are extrapolations, not extra measured data. Using $c^{\ast }=M/[N_A(4\pi /3)R_g^3]$ then gives the polymer overlap concentration scaling $c^{\ast }\propto M^{1-3\nu }$. ^[ref]

The incremental radiative forcing per increment of a specified atmospheric gas burden. State whether the abundance unit is emitted mass, mole fraction or ppb, and specify the radiation-adjustment convention. For equal emitted masses in a fixed atmospheric air inventory, a per-ppb coefficient must be divided by the gas molar mass before comparing efficiencies. Spectral overlap with existing absorption makes efficiency depend on the background state.

The number of water units included in a specified crystalline hydrate formula. A weighed hydrate and its anhydrous compound have different molar masses.

Hydrodynamic size describes the effective dimensions of a dissolved molecule moving with its surrounding solvent. It depends on molecular geometry, internal shape, and solvent conditions, rather than just molar mass. SEC pore access probes size-dependent accessibility of the packing.

The Mixture measurements and the Gibbs-Duhem constraint article describes the inspected acetone–chloroform preparation, apparatus, paired readouts, controls and sampling limits.^[ref]

That existing apparatus account supplies the experimental controls and limitations. Its refractive index calibration yields mass fractions, which must be converted to mole fractions. For a measured component-A mass fraction $w_A$ and component molar masses $M_A,M_B$,Apply this conversion separately to the flask sample and to completely condensed vapor to obtain $x_A$ and $y_A$. Comparing $y_{i}P$ with $x_i{p}_i^{\ast }(T)$ tests Raoult's law at the measured temperature; using the changing temperature at each composition is essential. The chosen A/B values above are unrelated to the protocol's substances.

The more general diagnostic, retaining an ideal gas vapor and negligible pure-liquid pressure correction, isThis follows by replacing $\ln x_i$ with $\ln (x_i{\gamma }_i)$ in the chemical potential subtraction used for Raoult's law. A reproducible discrepancy from ${\gamma }_i=1$ can expose failure of the ideal solution approximation, but interpreting it requires checking measurement uncertainty, representative paired sampling and the ideal gas approximation. Apparent agreement within uncertainty supports an approximation over the tested conditions; it cannot establish exact ideality at every composition.

Positive or negative deviations of component partial pressures require activity coefficients different from one under these assumptions. An azeotrope is also absent from the chosen unequal-vapor pressure model: the explicit inequality in ideal binary vapor composition gives $y_A>x_A$ throughout $0<x_A<1$. An interior equality would require $p_A^{\ast }=p_B^{\ast }$; in that degenerate case every composition has $y_A=x_A$, rather than an isolated enrichment reversal. Neither an azeotrope nor liquid phase separation should be inferred from the ideal curves.

Gerald M. Edelman reduced IgG disulfide bonds and used dissociating chemical solutions, including $6\mathrm{mol}{\mathrm{L}}^{-1}$ urea. Ultracentrifugal analysis showed a decrease in molar mass, supporting a multichain protein. In the comparison reproduced in his lecture, starch gel electrophoresis in urea separated serum IgG and a urinary protein from the same patient, alongside their reduced and alkylation-treated preparations. The untreated samples provided the treatment comparison; the urinary material provided a chain-identity comparison. The treated serum sample revealed heavy- and light-chain bands. Two-dimensional peptide mapping compared the urinary protein with the isolated immunoglobulin light chain. Together these readouts supported a shared chain identity.^[ref]

The controls address distinct alternatives. Treatment-dependent separation supports chains joined in the starting protein; a matching gel electrophoresis position alone would not uniquely identify a chain. Agreement of peptide mapping adds an independent chemical comparison. Neither readout gives atomic coordinates, and this patient's preparation does not establish a common protein primary structure for all antibodies.

Reduction targets protein disulfide bonds, while protein denaturation weakens the folded assembly. Neither operation is the proteolysis that cuts peptide bonds. This explains why chain separation and Papain digestion of IgG give different inventories: four intact chain backbones in the first case, three assembled fragments in the second.

If the molar latent heat $L_m$ is approximately constant on the chosen temperature interval, the ideal-vapor Clausius-Clapeyron approximation integrates toFor $L_m>0$, higher absolute temperature gives higher saturation pressure. A plot of $\ln (p/p_1)$ against inverse temperature $1/T$ is a straight line of slope $-L_m/R$. One reference pair $(T_1,p_1)$ fixes its intercept. On a per-unit-mass basis replace $L_m/R$ by $L/R_s$, with $R_s=R/M$ and $M$ the molar mass; using $L$ in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ with the molar $R$ would give incompatible units.

If only the constant-latent heat approximation is relaxed, the same ideal-vapor model instead givesConsequently an apparent straight-line fit across a broad temperature range need not represent a single material constant. The simple exponential also approaches a finite value as $T\to \infty$; this is a property of the stipulated formula, not a prediction that liquid–gas phase coexistence extends beyond a thermodynamic critical point.

A global warming potential compares equal-mass pulses. If ${ϵ}_i$ is the greenhouse gas radiative efficiency per ppb, an emitted mass produces a mixing-ratio increment inversely proportional to molar mass $M_i$. For two gases with exponential pulse removal, their GWP ratio at horizon $H$ is $({ϵ}_i/{ϵ}_j)(M_j/M_i){\tau }_i(1-e^{-H/{\tau }_i})/[{\tau }_j(1-e^{-H/{\tau }_j})]$. The common carbon-dioxide reference integral cancels. Comparing only the per-ppb coefficients would instead compare unequal emitted masses.

The mass mixing ratio used in a dilute radiative transfer model is absorber mass density divided by total atmospheric mass density. A hydrostatic pressure interval contributes absorber mass path $qdp/g$. Some meteorological conventions divide by dry-air mass instead; the convention must be stated. A volume or mole mixing ratio instead contributes absorber moles $xdp/(gM)$, with background molar mass $M$.

For a dry background inventory of column weight $P_d$, a mixed layer of fixed pressure thickness $\Delta p$, and saturated surface vapor partial pressure $E$, put $x=E/P$ and let $ϵ$ be the vapor-to-background molar mass ratio. The vapor mass fraction is $ϵx/[1+(ϵ-1)x]$, so hydrostatic balance and mass conservation give the displayed quadratic equation. Physical solutions require $x\le 1$ and a nonnegative upper-layer pressure. At $x=1$, $P=E=P_d+\Delta p$. A real positive algebraic root beyond that value need not be physically admissible.

Molar mass turns a weighed sample into a count of specified particles. For a pure chemical substance with specified chemical formula and isotopic abundances, its molar mass iswhere $m$ is the sample mass and $n$ its amount of substance. The SI unit is $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$; the common laboratory unit is $\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, with $1\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}=10^{-3}\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. Equal values of amount of substance correspond to equal counts of specified entities, but heavier entities give greater masses.

Combining this definition with molecular count and amount of substance givesThe Avogadro constant is exact under the modern definition of the mole. The mass $M/N_{\mathrm{A}}$ is the mean mass per specified entity; if isotopes differ within the sample, individual molecules need not all have that mean mass.^[ref]

Specify what $n$ counts: water molecules, hydrogen chemical atoms, and oxygen chemical atoms give different values of amount of substance for the same sample. For a mixture, dividing total mass by total amount of substance of its chosen entities gives a composition-dependent mean. A polymer sample can contain many chain molar masses; its number-average molar mass and weight-average molar mass answer different averaging questions. Relating either to degree of polymerization also requires a unit-mass convention and end-group accounting.

The molar mass constant converts relative atomic mass or relative molecular mass into molar mass. It is approximately $1\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. Its SI numerical value is determined experimentally: the exact Avogadro constant multiplies an experimentally determined unified atomic mass unit in kilograms. Treating $M_{\mathrm{u}}$ as exactly $1\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ is therefore an excellent elementary-calculation approximation, rather than a defining equality of the modern SI.^[ref]

The weight-average molar mass divided by the number-average molar mass. Its conventional written symbol is Đ. With number weighting it equals $1+\mathrm{Var}(M)/⟨M{⟩}^2$, so it is at least one and equals one only for a uniform molar mass. It measures one aspect of distribution breadth, not the full shape of a molar mass distribution.