Partial molar property

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In thermodynamics, a partial molar property is a quantity which describes the variation of an extensive property of a solution or mixture with changes in the molar composition of the mixture at constant temperature and pressure. It is the partial derivative of the extensive property with respect to the amount (number of moles) of the component of interest. Every extensive property of a mixture has a corresponding partial molar property.

The partial molar volume is broadly understood as the contribution that a component of a mixture makes to the overall volume of the solution. However, there is more to it than this:

When one mole of water is added to a large volume of water at 25 °C, the volume increases by 18 cm³. The molar volume of pure water would thus be reported as 18 cm³ mol⁻¹. However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14 cm³. The reason that the increase is different is that the volume occupied by a given number of water molecules depends upon the identity of the surrounding molecules. The value 14 cm³ is said to be the partial molar volume of water in ethanol.

In general, the partial molar volume of a substance X in a mixture is the change in volume per mole of X added to the mixture.

The partial molar volumes of the components of a mixture vary with the composition of the mixture, because the environment of the molecules in the mixture changes with the composition. It is the changing molecular environment (and the consequent alteration of the interactions between molecules) that results in the thermodynamic properties of a mixture changing as its composition is altered.

If, by ${\displaystyle Z}$, one denotes a generic extensive property of a mixture, it will always be true that it depends on the pressure (${\displaystyle P}$), temperature (${\displaystyle T}$), and the amount of each component of the mixture (measured in moles, n). For a mixture with q components, this is expressed as

${\displaystyle Z=Z(T,P,n_{1},n_{2},\cdots ,n_{q}).}$

Now if temperature T and pressure P are held constant, ${\displaystyle Z=Z(n_{1},n_{2},\cdots )}$ is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double ${\displaystyle Z}$. More generally, for any ${\displaystyle \lambda }$:

${\displaystyle Z(\lambda n_{1},\lambda n_{2},\cdots ,\lambda n_{q})=\lambda Z(n_{1},n_{2},\cdots ,n_{q}).}$

By Euler's first theorem for homogeneous functions, this implies^[1]

${\displaystyle Z=\sum _{i=1}^{q}n_{i}{\bar {Z_{i}}},}$

where ${\displaystyle {\bar {Z_{i}}}}$ is the partial molar ${\displaystyle Z}$ of component ${\displaystyle i}$ defined as:

${\displaystyle {\bar {Z_{i}}}=\left({\frac {\partial Z}{\partial n_{i}}}\right)_{T,P,n_{j\neq i}}.}$

By Euler's second theorem for homogeneous functions, ${\displaystyle {\bar {Z_{i}}}}$ is a homogeneous function of degree 0 (i.e., ${\displaystyle {\bar {Z_{i}}}}$ is an intensive property) which means that for any ${\displaystyle \lambda }$:

${\displaystyle {\bar {Z_{i}}}(\lambda n_{1},\lambda n_{2},\cdots ,\lambda n_{q})={\bar {Z_{i}}}(n_{1},n_{2},\cdots ,n_{q}).}$

In particular, taking ${\displaystyle \lambda =1/n_{T}}$ where ${\displaystyle n_{T}=n_{1}+n_{2}+\cdots }$, one has

${\displaystyle {\bar {Z_{i}}}(x_{1},x_{2},\cdots )={\bar {Z_{i}}}(n_{1},n_{2},\cdots ),}$

where ${\displaystyle x_{i}={\frac {n_{i}}{n_{T}}}}$ is the concentration expressed as the mole fraction of component ${\displaystyle i}$. Since the molar fractions satisfy the relation

${\displaystyle \sum _{i=1}^{q}x_{i}=1,}$

the x_i are not independent, and the partial molar property is a function of only ${\displaystyle q-1}$ mole fractions:

${\displaystyle {\bar {Z_{i}}}={\bar {Z_{i}}}(x_{1},x_{2},\cdots ,x_{q-1}).}$

The partial molar property is thus an intensive property - it does not depend on the size of the system.

The partial volume is not the partial molar volume.

Partial molar properties are useful because chemical mixtures are often maintained at constant temperature and pressure and under these conditions, the value of any extensive property can be obtained from its partial molar property. They are especially useful when considering specific properties of pure substances (that is, properties of one mole of pure substance) and properties of mixing (such as the heat of mixing or entropy of mixing). By definition, properties of mixing are related to those of the pure substances by:

${\displaystyle \Delta z^{M}=z-\sum _{i}x_{i}z_{i}^{*}.}$

Here ${\displaystyle *}$ denotes a pure substance, ${\displaystyle M}$ the mixing property, and ${\displaystyle z}$ corresponds to the specific property under consideration. From the definition of partial molar properties,

${\displaystyle z=\sum _{i}x_{i}{\bar {Z_{i}}},}$

substitution yields:

${\displaystyle \Delta z^{M}=\sum _{i}x_{i}({\bar {Z_{i}}}-z_{i}^{*}).}$

So from knowledge of the partial molar properties, deviation of properties of mixing from single components can be calculated.

Partial molar properties satisfy relations analogous to those of the extensive properties. For the internal energy U, enthalpy H, Helmholtz free energy A, and Gibbs free energy G, the following hold:

${\displaystyle {\bar {H_{i}}}={\bar {U_{i}}}+P{\bar {V_{i}}},}$

${\displaystyle {\bar {A_{i}}}={\bar {U_{i}}}-T{\bar {S_{i}}},}$

${\displaystyle {\bar {G_{i}}}={\bar {H_{i}}}-T{\bar {S_{i}}},}$

where ${\displaystyle P}$ is the pressure, ${\displaystyle V}$ the volume, ${\displaystyle T}$ the temperature, and ${\displaystyle S}$ the entropy.

The thermodynamic potentials also satisfy

${\displaystyle dU=TdS-PdV+\sum _{i}\mu _{i}dn_{i},\,}$

${\displaystyle dH=TdS+VdP+\sum _{i}\mu _{i}dn_{i},\,}$

${\displaystyle dA=-SdT-PdV+\sum _{i}\mu _{i}dn_{i},\,}$

${\displaystyle dG=-SdT+VdP+\sum _{i}\mu _{i}dn_{i},\,}$

where ${\displaystyle \mu _{i}}$ is the chemical potential defined as (for constant n_j with j≠i):

${\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial n_{i}}}\right)_{S,V}=\left({\frac {\partial H}{\partial n_{i}}}\right)_{S,P}=\left({\frac {\partial A}{\partial n_{i}}}\right)_{T,V}=\left({\frac {\partial G}{\partial n_{i}}}\right)_{T,P}.}$

This last partial derivative is the same as ${\displaystyle {\bar {G_{i}}}}$, the partial molar Gibbs free energy. This means that the partial molar Gibbs free energy and the chemical potential, one of the most important properties in thermodynamics and chemistry, are the same quantity. Under isobaric (constant P) and isothermal (constant T ) conditions, knowledge of the chemical potentials, ${\displaystyle \mu _{i}(x_{1},x_{2},\cdots ,x_{m})}$, yields every property of the mixture as they completely determine the Gibbs free energy.

To measure the partial molar property ${\displaystyle {\bar {Z_{1}}}}$ of a binary solution, one begins with the pure component denoted as ${\displaystyle 2}$ and, keeping the temperature and pressure constant during the entire process, add small quantities of component ${\displaystyle 1}$; measuring ${\displaystyle Z}$ after each addition. After sampling the compositions of interest one can fit a curve to the experimental data. This function will be ${\displaystyle Z(n_{1})}$. Differentiating with respect to ${\displaystyle n_{1}}$ will give ${\displaystyle {\bar {Z_{1}}}}$. ${\displaystyle {\bar {Z_{2}}}}$ is then obtained from the relation:

${\displaystyle Z={\bar {Z_{1}}}n_{1}+{\bar {Z_{2}}}n_{2}.}$

The relation between partial molar properties and the apparent ones can be derived from the definition of the apparent quantities and of the molality.

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

The relation holds also for multicomponent mixtures, just that in this case subscript i is required.

• Apparent molar property
• Ideal solution
• Excess molar quantity
• Partial specific volume
• Thermodynamic activity

• P. Atkins and J. de Paula, "Atkins' Physical Chemistry" (8th edition, Freeman 2006), chap.5
• T. Engel and P. Reid, "Physical Chemistry" (Pearson Benjamin-Cummings 2006), p. 210
• K.J. Laidler and J.H. Meiser, "Physical Chemistry" (Benjamin-Cummings 1982), p. 184-189
• P. Rock, "Chemical Thermodynamics" (MacMillan 1969), chap.9
• Ira Levine, "Physical Chemistry" (6th edition, McGraw Hill 2009), p. 125-128

• Lecture notes from the University of Arizona detailing mixtures, partial molar quantities, and ideal solutions^[archive]
• On-line calculator for densities and partial molar volumes of aqueous solutions of some common electrolytes and their mixtures, at temperatures up to 323.15 K.