Apparent molar property

The apparent (molal) volume of a solute can be expressed as a function of the molality b of that solute (and of the densities of the solution and solvent). The volume of solution per mole of solute is

${\displaystyle {\frac {1}{\rho }}\left({\frac {1}{b}}+M_{1}\right).}$

Subtracting the volume of pure solvent per mole of solute gives the apparent molal volume:

${\displaystyle {}^{\phi }{\tilde {V}}_{1}={\frac {V-V_{0}}{n_{1}}}=\left({\frac {m}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}=\left({\frac {m_{1}+m_{0}}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}=\left({\frac {m_{0}}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}+{\frac {m_{1}}{\rho n_{1}}}}$

${\displaystyle {}^{\phi }{\tilde {V}}_{1}={\frac {1}{b}}\left({\frac {1}{\rho }}-{\frac {1}{\rho _{0}^{0}}}\right)+{\frac {M_{1}}{\rho }}}$

For more solutes the above equality is modified with the mean molar mass of the solutes as if they were a single solute with molality b_T:

${\displaystyle {}^{\phi }{\tilde {V}}_{12..}={\frac {1}{b_{T}}}\left({\frac {1}{\rho }}-{\frac {1}{\rho _{0}^{0}}}\right)+{\frac {M}{\rho }}}$, ${\displaystyle M=\sum y_{i}M_{i}}$

The sum of products molalities – apparent molar volumes of solutes in their binary solutions equals the product between the sum of molalities of solutes and apparent molar volume in ternary of multicomponent solution mentioned above.

${\displaystyle {}^{\phi }{\tilde {V}}_{123..}(b_{1}+b_{2}+b_{3}+...)=b_{1}{}^{\phi }{\tilde {V}}_{1}+b_{2}{}^{\phi }{\tilde {V}}_{2}+b_{3}{}^{\phi }{\tilde {V}}_{3}+...}$,

A relation between the apparent molar of a component of a mixture and molar mixing ratio can be obtained by dividing the definition relation

${\displaystyle V=V_{0}+{}^{\phi }{V}_{1}\ ={\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}\,}$

to the number of moles of one component. This gives the following relation:

${\displaystyle {}^{\phi }{\tilde {V}}_{1}={\frac {V}{n_{1}}}-{\tilde {V}}_{0}{\frac {n_{0}}{n_{1}}}={\frac {V}{n_{1}}}-{\tilde {V}}_{0}r_{01}}$

Note the contrasting definitions between partial molar quantity and apparent molar quantity: in the case of partial molar volumes ${\displaystyle {\bar {V_{0}}},{\bar {V_{1}}}}$, defined by partial derivatives

${\displaystyle {\bar {V_{0}}}={\Big (}{\frac {\partial V}{\partial n_{0}}}{\Big )}_{T,p,n_{1}},{\bar {V_{1}}}={\Big (}{\frac {\partial V}{\partial n_{1}}}{\Big )}_{T,p,n_{0}}}$,

one can write ${\displaystyle dV={\bar {V_{0}}}dn_{0}+{\bar {V_{1}}}dn_{1}}$, and so ${\displaystyle V={\bar {V_{0}}}n_{0}+{\bar {V_{1}}}n_{1}}$ always holds. In contrast, in the definition of apparent molar volume, the molar volume of the pure solvent, ${\displaystyle {\tilde {V}}_{0}}$, is used instead, which can be written as

${\displaystyle {\tilde {V_{0}}}={\Big (}{\frac {\partial V}{\partial n_{0}}}{\Big )}_{T,p,n_{1}=0}}$,

for comparison. In other words, we assume that the volume of the solvent does not change, and we use the partial molar volume where the number of moles of the solute is exactly zero ("the molar volume"). Thus, in the defining expression for apparent molar volume ${\displaystyle {}^{\phi }{\tilde {V}}_{1}}$,

${\displaystyle V=V_{0}+{}^{\phi }{V}_{1}\ ={\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}\,}$,

the term ${\displaystyle V_{0}}$ is attributed to the pure solvent, while the "leftover" excess volume, ${\displaystyle {}^{\phi }V_{1}}$, is considered to originate from the solute. At high dilution with ${\displaystyle n_{0}\gg n_{1}\approx 0}$, we have ${\displaystyle {\tilde {V_{0}}}\approx {\bar {V_{0}}}}$, and so the apparent molar volume and partial molar volume of the solute also converge: ${\displaystyle {}^{\phi }{\tilde {V}}_{1}\approx {\bar {V}}_{1}}$.

Quantitatively, the relation between partial molar properties and the apparent ones can be derived from the definition of the apparent quantities and of the molality. For volume,

${\displaystyle {\bar {V_{1}}}={}^{\phi }{\tilde {V}}_{1}+b{\frac {\partial {}^{\phi }{\tilde {V}}_{1}}{\partial b}}.}$

The ratio r_a between the apparent molar volume of a dissolved electrolyte in a concentrated solution and the molar volume of the solvent (water) can be linked to the statistical component of the activity coefficient ${\displaystyle \gamma _{s}}$ of the electrolyte and its solvation shell number h:[3]

${\displaystyle \ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln(1+{\frac {br_{a}}{55.5}})-{\frac {h}{\nu }}\ln(1-{\frac {br_{a}}{55.5}})+{\frac {br_{a}(r_{a}+h-\nu )}{55.5(1+{\frac {br_{a}}{55.5}})}}}$,

where ν is the number of ions due to dissociation of the electrolyte, and b is the molality as above.

For multicomponent solutions, apparent molar properties can be defined in several ways. For the volume of a ternary (3-component) solution with one solvent and two solutes as an example, there would still be only one equation ${\displaystyle (V={\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}+{}^{\phi }{\tilde {V}}_{2}n_{2})}$, which is insufficient to determine the two apparent volumes. (This is in contrast to partial molar properties, which are well-defined intensive properties of the materials and therefore unambiguously defined in multicomponent systems. For example, partial molar volume is defined for each component i as ${\displaystyle {\bar {V_{i}}}=(\partial V/\partial n_{i})_{T,p,n_{j\neq i}}}$.)

One description of ternary aqueous solutions considers only the weighted mean apparent molar volume of the solutes,[6] defined as

${\displaystyle {}^{\phi }{\tilde {V}}(n_{1},n_{2})={}^{\phi }{\tilde {V}}_{12}={\frac {V-V_{0}}{n_{1}+n_{2}}}}$,

where ${\displaystyle V}$ is the solution volume and ${\displaystyle V_{0}}$ the volume of pure water. This method can be extended for mixtures with more than 3 components.[7]

${\displaystyle {}^{\phi }{\tilde {V}}(n_{1},n_{2},n_{3},..)={}^{\phi }{\tilde {V}}_{123..}={\frac {V-V_{0}}{n_{1}+n_{2}+n_{3}+...}}}$,

The sum of products molalities – apparent molar volumes of solutes in their binary solutions equals the product between the sum of molalities of solutes and apparent molar volume in ternary of multicomponent solution mentioned above.

${\displaystyle {}^{\phi }{\tilde {V}}_{123..}(b_{1}+b_{2}+b_{3}+...)=b_{1}{}^{\phi }{\tilde {V}}_{1}+b_{2}{}^{\phi }{\tilde {V}}_{2}+b_{3}{}^{\phi }{\tilde {V}}_{3}+...}$,

Another method is to treat the ternary system as pseudobinary and define the apparent molar volume of each solute with reference to a binary system containing both other components: water and the other solute.[8] The apparent molar volumes of each of the two solutes are then

${\displaystyle {}^{\phi }{\tilde {V}}_{1}={\frac {V-V(solvent+solute\ 2)}{n_{1}}}}$ and ${\displaystyle {}^{\phi }{\tilde {V}}_{2}={\frac {V-V(solvent+solute\ 1)}{n_{2}}}}$

The apparent molar volume of the solvent is:

${\displaystyle {}^{\phi }{\tilde {V}}_{0}={\frac {V-V(solute\ 1+solute\ 2)}{n_{0}}}}$

However, this is an unsatisfactory description of volumetric properties.[9]

The apparent molar volume of two components or solutes considered as one pseudocomponent ${\displaystyle {}^{\phi }{\tilde {V}}_{12}}$ or ${\displaystyle {}^{\phi }{\tilde {V}}_{ij}}$ is not to be confused with volumes of partial binary mixtures with one common component V_ij, V_jk which mixed in a certain mixing ratio form a certain ternary mixture V or V_ijk.

Of course the complement volume of a component in respect to other components of the mixture can be defined as a difference between the volume of the mixture and the volume of a binary submixture of a given composition like:

${\displaystyle {}^{c}{\tilde {V}}_{2}={\frac {V-V_{01}}{n_{2}}}}$

There are situations when there is no rigorous way to define which is solvent and which is solute like in the case of liquid mixtures (say water and ethanol) that can dissolve or not a solid like sugar or salt. In these cases apparent molar properties can and must be ascribed to all components of the mixture.

• Volume fraction
• Ideal solution
• Regular solution
• Enthalpy change of solution
• Enthalpy of mixing
• Block design
• Heat of dilution
• Hydration energy
• Ion transport number
• Solvation shell
• Partial molar property
• Excess molar quantity
• Salting in
• Ternary plot
• Thermodynamic activity

• Apparent Molar Properties: Solutions: Background
• The (p,ρ,T) Properties and Apparent Molar Volumes of ethanol solutions of LiI or ZnCl2 Archived 2016-03-04 at the Wayback Machine
• Apparent molar volumes and apparent molar heat capacities of Pr(NO3)3(aq), Gd(NO3)3(aq), Ho(NO3)3(aq), and Y(NO3)3(aq) at T = (288.15, 298.15, 313.15, and 328.15) K and p = 0.1 MPa
• Isotopic effects for electrolytes apparent properties