Partial molar volume

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Partial molar properties are thermodynamic quantities which indicate how any extensive property of a solution or mixture varies with changes in the molar composition of the mixture at constant temperature and pressure. Every extensive property of a mixture has a corresponding partial molar property.

1 Definition
2 Applications
3 Properties
- 3.1 Relations of the partial molar properties of the thermodynamic potentials
- 3.2 Differential form of the thermodynamic potentials
4 Calculating partial molar properties
5 External links

Definition

If, by $Z$ , one denotes a generic extensive property of a mixture, it will always be true that it depends on the pressure ( $P$ ), temperature ( $T$ ), and the amount of every component of the mixture ( $N i$ ). For a n-component mixture, this is expressed as

$Z=Z(T,P,N_1,N_2,\cdots,N_n).$

Now if T and P are held constant, $Z=Z(N_1,N_2,\cdots,N_n)$ is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double $Z$ . More generally, for any $λ$ :

$Z(\lambda N_1,\lambda N_2, \cdots, \lambda N_n)=\lambda Z(N_1,N_2,\cdots,N_n).$

By Euler's first theorem for homogeneous functions, this implies

$Z=\sum _{i=1}^n N_i \bar{Z_i},$

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

$\bar{Z_i}=\left( \frac{\partial Z}{\partial N_i} \right)_{T,P,N_{j\neq i}}.$

By Euler's second theorem for homogeneous functions, $\bar{Z_i}$ is a homogeneous function of degree 0 which means that for any $λ$ :

$\bar{Z_i}(\lambda N_1,\lambda N_2,\cdots , \lambda N_n)=\bar{Z_i}(N_1,N_2,\cdots ,N_n).$

In particular, taking $λ = 1 / N T$ where $N_T=N_1+N_2+ \cdots +N_n$ , one has

$\bar{Z_i}(x_1,x_2, \cdots , x_n)=\bar{Z_i}(N_1,N_2,\cdots,N_n),$

where $x_i=\frac{N_i}{N_T}$ is the concentration, or mole fraction, of component $i$ . Since the molar fractions satisfy the relation

$\sum _{i=1}^n x_i = 1,$

the x_i are not independent, and the partial molar property is a function of only $n - 1$ mole fractions:

$\bar{Z_i}=\bar{Z_i}(x_1,x_2, \cdots , x_{n-1}).$

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

Applications

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. By definition, properties of mixing are related to those of the pure substance by:

$z^M=z-\sum_i x_iz^*_i.$

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

$z=\sum_i x_i \bar{Z_i},$

substitution yields:

$Z^M=\sum_i x_i(\bar{Z_i}-z_i^*).$

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

Properties

Relations of the partial molar properties of the thermodynamic potentials

The internal energy U, enthalpy H, Helmholtz free energy A, and Gibbs free energy G, are the four thermodynamic potentials. Partial molar properties satisfy relations analogous to those of the extensive properties:

$\bar{H_i}=\bar{U_i}+P\bar{V_i},$

$\bar{A_i}=\bar{U_i}-T\bar{S_i},$

$\bar{G_i}=\bar{H_i}-T\bar{S_i},$

where $P$ is the pressure, $V$ the volume, $T$ the temperature, and $S$ the enthropy.

Differential form of the thermodynamic potentials

The thermodynamic potentials also satisfy

$dU= TdS-PdV+\sum_i \mu_i dN_i,\,$

$dH= TdS+VdP+\sum_i \mu_i dN_i,\,$

$dA=-SdT-PdV+\sum_i \mu_i dN_i,\,$

$dG=-SdT+VdP+\sum_i \mu_i dN_i,\,$

where $μ i$ is the chemical potential defined as (for constant N_i≠j $N_{j\neq i}$ ):

$\mu_i=\left( \frac{\partial U}{\partial N_i}\right)_{S,V}=\left( \frac{\partial H}{\partial N_i}\right)_{S,P}=\left( \frac{\partial A}{\partial N_i}\right)_{T,V}=\left( \frac{\partial G}{\partial N_i}\right)_{T,P}=\bar{G_i}.$

This is another reason why partial molar properties are important: the chemical potential, one of the most important properties in thermodynamics and chemistry, is actually a partial molar property. Under isobaric (constant P) and isothermal (constant T ) conditions, knowledge of the chemical potentials, $\mu_i(x_1,x_2,\cdots , x_n)$ , yields every property of the mixture as they completely determines the Gibbs free energy.

Calculating partial molar properties

To calculate the partial molar property $\bar{Z_1}$ of a binary solution, one begins with the pure component denoted as $2$ and, keeping the temperature and pressure constant during the entire process, add small quantities of component $1$ ; measuring $Z$ after each addition. After sampling the compositions of interest one can fit a curve to the experimental data. This function will be $Z (N 1)$ . Differentiating with respect to $N 1$ will give $\bar{Z_1}$ . $\bar{Z_2}$ is then obtained from the relation:

$Z=\bar{Z_1}N_1+\bar{Z_2}N_2.$

External links

Lecture notes from the University of Arizona detailing mixtures, partial molar quantities, and ideal solutions

Wikipedia content modification information:

This page was last modified on 20 October 2008, at 12:58.

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