Van der Waals equation

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The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the van der Waals force.) It was derived by Johannes Diderik van der Waals in 1873, based on a modification of the ideal gas law. The equation approximates the behavior of real fluids, taking into account the nonzero size of molecules and the attraction between them.

1 Equation
2 Validity
3 Derivation
4 Other thermodynamic parameters
5 Reduced form
6 Application to compressible fluids
7 See also
8 External links

[edit] Equation

The van der Waals equation is

$\left(p + \frac{a'}{v^2}\right)\left(v-b'\right) = kT$

where

p is the pressure of the fluid

a' is a measure of the attraction between the particles

b' is the volume enclosed within a particle

v is the volume per particle of the fluid in the 1st equation;

k is Boltzmann's constant

T is the absolute temperature

This equation is more commonly seen as

$\left(p + a \frac{n^2}{V^2}\right)\left(V-nb\right) = nRT$

where

V is the total volume

a is a measure of the attraction between the particles $a=n_A^2 a'$

b is the volume enclosed within a mole of the particles

b = n A b'

n is the number of moles,

R is the gas constant

A careful distinction must be drawn between the properties of the bulk fluid and the properties of the particles. In particular, in the first equation v refers to the volume of the bulk fluid (i.e. the volume of the container) divided by the number of particles, whereas b is the volume enclosed by a single particle (i.e. the volume bounded by the atomic radius).

[edit] Validity

Above the critical temperature it is an improvement of the ideal gas law, and for lower temperatures the equation is also reasonable for the liquid state and the low-pressure gaseous state.

However, in the first-order phase transition range of (P,V,T) (between a liquid phase and a gaseous phase) it does not exhibit that for a given temperature the vapor pressure is constant for varying values of V, i.e. for various amounts of the material being in the vaporous state.

[edit] Derivation

The derivation of the van der Waals equation begins with the equation of state of an ideal gas, which is composed of non-interacting point particles:

$p = \frac{kT}{v}$

We now stop treating the fluid's constituent particles as point particles, instead modelling them as hard spheres with a small radius (the van der Waals radius.) Denoting the volume of each sphere by b, we modify the equation of state to

$p = \frac{kT}{v - b}$

The volume occupied per particle, v, has been replaced by the "excluded volume" v - b, reflecting the fact that the particles cannot overlap. If the fluid is compressed, its pressure goes to infinity as the total volume approaches the volume enclosed within the particles.

Next, we introduce a pairwise attractive force between atoms. This causes the average Helmholtz energy per particle to be reduced by an amount proportional to the fluid density. However, the pressure obeys the thermodynamic relation

$p = - \left(\frac{\partial A^*}{\partial v}\right)_T$

where A* is the Helmholtz energy per particle. The attraction therefore reduces the pressure by an amount proportional to 1/v². Denoting the constant of proportionality by a, we obtain

$p = \frac{kT}{v-b}-\frac{a}{v^2}$

which is the van der Waals equation.

[edit] Other thermodynamic parameters

In the following, we will use the extensive volume V instead of volume per particle v=V/N where N is the number of particles in the system.

The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the Helmholtz energy for an ideal gas and modify it in full agreement with the above development to yield:

$A(T,V,N)=-NkT\left(1+\ln\left(\frac{(V-Nb)T^{\hat{c}_V}}{N\Phi}\right)\right) -\frac{aN^2}{V}$

where A is the Helmholtz energy, $\hat{c}_v$ is the dimensionless heat capacity at constant volume (assumed constant) and $Φ$ is an undetermined entropy constant. The above equation expresses A in terms of its natural variables V and T , and therefore gives us all thermodynamic information about the system. The mechanical equation of state is identical to the one derived above

$P = -\left(\frac{\partial A}{\partial V}\right)_T = \frac{NkT}{V-Nb}-\frac{aN^2}{V^2}$

The entropy equation of state yields the entropy (S )

$S = -\left(\frac{\partial A}{\partial T}\right)_V =Nk\left[ \ln\left(\frac{(V-Nb)T^{\hat{c}_V}}{N\Phi}\right)+\hat{c}_V+1 \right]$

from which we can calculate the internal energy

$U = A+TS = \hat{c}_V\,NkT-\frac{aN^2}{V}$

Similar equations can be written for the other thermodynamic potentials and the chemical potential, but expressing any potential as a function of pressure P will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated.

[edit] Reduced form

Although the material constants a and b in the usual form of the van der Waals equation differs for every single gas/fluid considered, the equation can be recast into an invariant form applicable to all gases/fluids.

Defining the following reduced variables ( $f R$ , $f c$ is the reduced and critical variables version of $f$ , respectively),

$p_R=\frac{p}{p_C}$ ,

$v_R=\frac{v}{v_C}$ ,

$T_R=\frac{T}{T_C}$ ,

where

$p_C=\frac{a}{27b^2}$

$\displaystyle{v_C=3b}$

$kT_C=\frac{8a}{27b}$ .

The van der Waals equation of state can be recast in the following reduced form:

$\left(p_R + \frac{3}{v_R^2}\right)(v_R - 1/3) = \frac{8}{3} T_R$

This equation is invariant (i.e., the same equation of state, viz., above, applies) for all gases.

Thus, when measured in intervals of the critical values of various quantities, all gases obey the same equation of state -- the reduced van der Waals equation of state. This is also known as the Principle of corresponding states. In the sense that we have eliminated the appearance of the individual material constants a and b in the equation, this can be considered unity in diversity.

[edit] Application to compressible fluids

The equation is also usable as a PVT equation for compressible fluids (e.g. polymers). In this case specific volume changes are small and it can be written in a simplified form:

$(p+A)(V-b)=CT\,$ ,