Concentration

Concentration quantifies how much of a named constituent is present relative to a specified sample measure. In chemical terminology, volumetric concentrations use the final mixture volume: mass concentration is mass per volume and molar concentration is amount of substance per volume. Mass fraction and amount fraction instead compare a constituent with the corresponding total quantity in the mixture.^[ref]

The distinction matters in transport models. If an isolated sample expands without exchanging constituents, its volumetric concentration decreases even though its composition fractions stay fixed. A scalar called concentration in an advection model may represent a fraction instead, so its definition must be stated.

Conservation applies to the transported constituent amount. Reactions, diffusion and exchange then determine how the chosen concentration changes relative to its carrier and denominator.

Molality is the amount of substance of a solute divided by the mass of the solvent: $b=n_{\mathrm{solute}}/m_{\mathrm{solvent}}$. Its SI unit is $\mathrm{mol}\mathrm{k}{\mathrm{g}}^{-1}$. Dissolving $0.10\mathrm{mol}$ of solute in $0.50\mathrm{kg}$ of solvent produces $b=0.20\mathrm{mol}\mathrm{k}{\mathrm{g}}^{-1}$.

The denominator contains solvent mass, rather than the volume of the resulting chemical solution. Thus molality avoids changes caused solely by thermal expansion of the chemical solution, making it useful in colligative properties and chemical thermodynamics. It must be distinguished from molar concentration, whose denominator is solution volume.

The amount fraction of component $i$ is $x_i=n_i/{\sum }_j{n}_j$, where $n_i$ is its amount of substance. It is dimensionless and the fractions sum to one for a complete composition specification.

For an ideal gas mixture at common temperature and pressure, amount fraction equals partial-pressure fraction. It generally differs from mass fraction because different species have different molar masses. For one mole each of hydrogen and oxygen molecules, both amount fractions are one half, although their masses are very different. The composition basis must therefore accompany a quoted concentration.

Parts per million, written ppm, expresses a dimensionless ratio multiplied by $10^6$. A mass fraction of $3\times 10^{-6}$ is 3 ppm by mass; an amount fraction of the same numerical value is 3 ppm by amount of substance. The two compare different quantities and need not describe the same mixture composition.

A concentration written in milligrams per litre is a mass concentration, not automatically a parts-per-million ratio. Its approximate numerical equality to ppm by mass for a dilute aqueous sample uses an additional mass density approximation of $1kg/L$. Stating the basis prevents a unit conversion from silently becoming an assumption about the sample.

Mass concentration is the mass of a specified constituent divided by the final sample volume, ${\rho }_i=m_i/V$. Its SI unit is $\mathrm{kg}{\mathrm{m}}^{-3}$; laboratory data also commonly use mass per litre.^[ref]

For example, $10\mathrm{g}$ of a solute in a final solution volume of $1\mathrm{L}$ gives $10\mathrm{g}{\mathrm{L}}^{-1}$. The denominator is the solution volume after mixing, not necessarily the volume of solvent initially added.

If $w_i$ is mass fraction and $\rho$ the mixture mass density, then ${\rho }_i=w_i\rho$. Heating or compressing the sample can change the mass concentration through a change of volume while leaving the mass fraction unchanged.

An equilibrium concentration is a constant concentration satisfying the fixed point equations of a specified dynamical model. Balance between production and loss can maintain such a value without implying thermodynamic chemical equilibrium.

Protein concentration specifies protein amount per sample volume, commonly as mass concentration or molar concentration. A molar value must state the counted entity, such as chains or oligomeric particles, and quantification must account for contaminants and optical properties.

A concentration gradient is the spatial gradient $\nabla c$ of a scalar concentration field. It points toward the greatest local increase of $c$; its components are spatial partial derivatives, so its units are those of concentration divided by length.

For a one-dimensional profile $c(x)=c_0+ax$, the concentration gradient is the constant $a\hat{\mathbf{x}}$. Fick's first law gives a diffusive flux proportional to $-\nabla c$ when that constitutive model applies: the minus sign describes transport toward lower concentration. A fluid velocity can simultaneously advect the constituent in another direction, so a concentration gradient alone does not determine its total flux.

Salinity measures dissolved salt content. A transport model must specify its units; salt mass fraction is one convenient convention. A dilute solution's ice liquidus is often approximated by $T_L=T_m-mC$. Salt-free ice has zero salinity, and the moving interface's salt balance distinguishes salt rejection during freezing from dilution during melting.

The conservation law for salt in a reservoir: the rate of change of its salt content equals inflowing minus outflowing salt transport and any explicit salt sources. Freshwater addition changes salinity without directly adding salt. For a well-mixed fixed-volume reservoir, salt content is proportional to its volume times salinity.

The salt forcing used to represent freshwater exchange while retaining a fixed reference volume. Removing freshwater at volume flux $E$ is represented by a salt input $E{S}_0$, and adding freshwater by the opposite salt forcing, with reference salinity $S_0$. Replacing the actual salinity in a compensating-flow salt budget by $S_0$ is an approximation when salinity contrasts are small.

Molar concentration is amount of substance of a named chemical species per final chemical solution volume: $c=n/V$. State whether the count describes dissolved species or a conserved analytical concentration.