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| A '''partial molar property''' is a [[Thermodynamics|thermodynamic]] quantity which indicates how an [[Intensive and extensive properties|extensive property]] of a [[solution]] or [[mixture]] varies with changes in the [[Mole (unit)|molar]] composition of the mixture at constant [[temperature]] and [[pressure]]. Essentially it is the partial derivative with respect to the quantity (number of moles) of the component of interest. Every extensive property of a mixture has a corresponding partial molar property.
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| ==Definition==
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| [[File:Excess Volume Mixture of Ethanol and Water.png|thumb|right|Water and ethanol always have negative excess volumes when mixed, indicating the partial molar volume of each component is less when mixed than its molar volume when pure.]]
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| 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 rather more to it than this:
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| When one mole of water is added to a large volume of water at 25 ºC, the volume increases by 18 cm<sup>3</sup>. The molar volume of pure water would thus be reported as 18 cm<sup>3</sup> mol<sup>-1</sup>. However, addition of one mole of water to a large volume of pure [[ethanol]] results in an increase in volume of only 14 cm<sup>3</sup>. 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 cm<sup>3</sup> is said to be the partial molar volume of water in ethanol.
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| '''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.''' | |
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| 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
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| If, by <math>Z</math>, one denotes a generic extensive property of a mixture, it will always be true that it depends on the [[pressure]] (<math>P</math>), [[temperature]] (<math>T</math>), and the amount of each component of the mixture (measured in [[Mole (unit)|moles]], ''n''). For a mixture with ''m'' components, this is expressed as
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| :<math>Z=Z(T,P,n_1,n_2,\cdots).</math>
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| Now if temperature ''T'' and pressure ''P'' are held constant, <math>Z=Z(n_1,n_2,\cdots)</math> is a [[homogeneous function]] of degree 1, since doubling the quantities of each component in the mixture will double <math>Z</math>. More generally, for any <math>\lambda</math>:
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| :<math>Z(\lambda n_1,\lambda n_2, \cdots)=\lambda Z(n_1,n_2,\cdots).</math>
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| By [[homogeneous function#Elementary theorems|Euler's first theorem for homogeneous functions]], this implies<ref>[http://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html Wolfram Mathworld: Euler's homogeneous function theorem]</ref>
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| :<math>Z=\sum _{i=1}^m n_i \bar{Z_i},</math>
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| where <math>\bar{Z_i}</math> is the partial molar <math>Z</math> of component <math>i</math> defined as:
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| :<math>\bar{Z_i}=\left( \frac{\partial Z}{\partial n_i} \right)_{T,P,n_{j\neq i}}.</math>
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| By [[homogeneous function#Elementary theorems|Euler's second theorem for homogeneous functions]], <math>\bar{Z_i}</math> is a homogeneous function of degree 0 which means that for any <math>\lambda </math>:
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| :<math>\bar{Z_i}(\lambda n_1,\lambda n_2,\cdots )=\bar{Z_i}(n_1,n_2,\cdots ).</math>
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| In particular, taking <math>\lambda = 1/n_T</math> where <math>n_T=n_1+n_2+ \cdots </math>, one has
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| :<math>\bar{Z_i}(x_1,x_2, \cdots )=\bar{Z_i}(n_1,n_2,\cdots),</math>
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| where <math>x_i=\frac{n_i}{n_T}</math> is the [[concentration]], or [[mole fraction]] of component <math>i</math>.
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| Since the molar fractions satisfy the relation
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| :<math>\sum _{i=1}^m x_i = 1,</math>
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| the ''x<sub>i</sub>'' are not independent, and the partial molar property is a function of only <math>m-1</math> mole fractions:
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| :<math>\bar{Z_i}=\bar{Z_i}(x_1,x_2, \cdots , x_{m-1}).</math>
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| The partial molar property is thus an [[Intensive and extensive properties|intensive property]] - it does not depend on the size of the system. | |
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| ==Applications==
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| Partial molar properties are useful because chemical [[mixture]]s are often maintained at constant temperature and pressure and under these conditions, the value of any [[Intensive and extensive properties|extensive property]] can be obtained from its partial molar property. They are especially useful when considering [[specific property|specific properties]] of [[pure substance]]s (that is, properties of one mole of pure substance) and [[property of mixing|properties of mixing]]. By definition, properties of mixing are related to those of the pure substance by:
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| :<math>\Delta z^M=z-\sum_i x_iz^*_i.</math>
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| Here <math>*</math> denotes the pure substance, <math>M</math> the mixing property, and <math>z</math> corresponds to the specific property under consideration. From the definition of partial molar properties,
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| :<math>z=\sum_i x_i \bar{Z_i},</math>
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| substitution yields:
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| :<math>\Delta z^M=\sum_i x_i(\bar{Z_i}-z_i^*).</math>
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| So from knowledge of the partial molar properties, deviation of properties of mixing from single components can be calculated.
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| ==Relationship to thermodynamic potentials==
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| 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:
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| :<math>\bar{H_i}=\bar{U_i}+P\bar{V_i},</math>
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| :<math>\bar{A_i}=\bar{U_i}-T\bar{S_i},</math>
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| :<math>\bar{G_i}=\bar{H_i}-T\bar{S_i},</math>
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| where <math>P</math> is the pressure, <math>V</math> the [[volume]], <math>T</math> the temperature, and <math>S</math> the [[entropy]].
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| ==Differential form of the thermodynamic potentials==
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| The thermodynamic potentials also satisfy
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| :<math>dU= TdS-PdV+\sum_i \mu_i dn_i,\,</math>
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| :<math>dH= TdS+VdP+\sum_i \mu_i dn_i,\,</math>
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| :<math>dA=-SdT-PdV+\sum_i \mu_i dn_i,\,</math>
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| :<math>dG=-SdT+VdP+\sum_i \mu_i dn_i,\,</math>
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| where <math>\mu_i</math> is the [[chemical potential]] defined as (for constant n<sub><big>j</big></sub> with j≠i):
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| :<math>\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}.</math>
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| This last [[partial derivative]] is the same as <math>\bar{G_i}</math>, 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 process|isobaric]] (constant ''P'') and [[Isothermal process|isothermal]] (constant ''T '') conditions, knowledge of the chemical potentials, <math>\mu_i(x_1,x_2,\cdots , x_m)</math>, yields every property of the mixture as they completely determine the Gibbs free energy.
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| ==Calculating partial molar properties==
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| To calculate the partial molar property <math>\bar{Z_1}</math> of a binary solution, one begins with the pure component denoted as <math>2</math> and, keeping the temperature and pressure constant during the entire process, add small quantities of component <math>1</math>; measuring <math>Z</math> after each addition. After sampling the compositions of interest one can fit a curve to the experimental data. This function will be <math>Z(n_1)</math>.
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| Differentiating with respect to <math>n_1</math> will give <math>\bar{Z_1}</math>.
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| <math>\bar{Z_2}</math> is then obtained from the relation:
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| :<math>Z=\bar{Z_1}n_1+\bar{Z_2}n_2.</math> | |
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| ==See also==
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| *[[Ideal solution]]
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| *[[Excess molar quantity]]
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| *[[Partial specific volume]]
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| *[[Thermodynamic activity]]
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| == Further reading ==
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| * P. Atkins and J. de Paula, "Atkins' Physical Chemistry" (8th edition, Freeman 2006), chap.5
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| * T. Engel and P. Reid, "Physical Chemistry" (Pearson Benjamin-Cummings 2006), p.210
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| * K.J. Laidler and J.H. Meiser, "Physical Chemistry" (Benjamin-Cummings 1982), p.184-189
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| * P. Rock, "Chemical Thermodynamics" (MacMillan 1969), chap.9
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{Commons category|Excess volume diagrams}}
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| *Lecture notes from the University of Arizona detailing [http://www.chem.arizona.edu/~salzmanr/480a/480ants/mixpmqis/mixpmqis.html mixtures, partial molar quantities, and ideal solutions]
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| *[http://www.aim.env.uea.ac.uk/aim/density/density_electrolyte.php 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.]
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| [[Category:Physical chemistry]]
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| [[Category:Thermodynamics]]
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| [[Category:Chemical thermodynamics]]
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| [[de:Exzessgröße]]
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| [[es:Magnitud molar parcial]]
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| [[it:Grandezza parziale molare]]
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| [[nl:Partieel molair volume]]
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| [[ro:Mărimi molare parțiale]]
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