NationMaster - Encyclopedia: Van der Waals equation

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. In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... In physics, force is an influence that may cause a body to accelerate. ... In chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer to intermolecular forces that deal with forces due to the polarization of molecules. ... van der Waals Johannes Diderik van der Waals (November 23, 1837 â€“ March 8, 1923) was a Dutch scientist famous for his work on the equation of state for gases and liquids, for which he won the Nobel Prize in physics in 1910. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by BenoÃ®t Paul Ã‰mile Clapeyron in 1834. ...

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

Equation

The first form of 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

v is the volume of the container holding the particles divided by the total number of particles

k is Boltzmann's constant

T is the absolute temperature

a' is a measure for the attraction between the particles

b' is the average volume excluded from v by a particle

Upon introduction of Avogadro's constant N_A, the number of moles n, and the total number of particles nN_A, the equation can be cast into the second (better known) form The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Fig. ... Avogadros number, also called Avogadros constant (NA) is a large constant used in chemistry and physics. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ...

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

where

V is the total volume of the container containing the fluid

a is a measure of the attraction between the particles $a=N_mathrm{A}^2 a'$

b is the volume excluded by a mole of particles $, b=N_mathrm{A} b'$

n is the number of moles

R is the gas constant, $,R= N_mathrm{A} k$

A careful distinction must be drawn between the volume available to a particle and the volume of a particle. In particular, in the first equation $,v$ refers to the empty space available per particle. That is, $,v$ is the volume V of the container divided by the total number nN_A of particles. The parameter $b'$ , on the other hand, is proportional to the proper volume of a single particle—the volume bounded by the atomic radius. This is the volume to be subtracted from $,v$ because of the space taken up by one particle. In van der Waals' original derivation, given below, $b'$ is four times the proper volume of the particle. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move. This occurs when V = n b. The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... Atomic radius: Ionic radius Covalent radius Metallic radius van der Waals radius edit Atomic radius, and more generally the size of an atom, is not a precisely defined physical quantity, nor is it constant in all circumstances. ...

Validity

Above the critical temperature the van der Waals equation is an improvement of the ideal gas law, and for lower temperatures the equation is also qualitatively reasonable for the liquid state and the low-pressure gaseous state. However, the van der Waals model cannot be taken seriously in a quantitative sense, it is only useful for qualitative purposes.^[1] The critical temperature, Tc, of a material is the temperature above which distinct liquid and gas phases do not exist. ...

In the first-order phase transition range of (p,V,T) (where the liquid phase and the gas phase are in equilibrium) it does not exhibit the empirical fact that p is constant as a function of V for a given temperature. In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ... A liquid will usually assume the shape of its container A liquid is one of the main states of matter. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... This article or section does not cite its references or sources. ...

Reference

^ T. L. Hill, Statistical Thermodynamics, Addison-Wesley, Reading (1960), p. 280

Derivation

In the usual textbooks one find two different derivations. One is the conventional derivation that goes back to van der Waals and the other is a statistical mechanics derivation. The latter has its major advantage that it makes explicit the intermolecular potential, which is out of sight in the first derivation.

Conventional derivation

Consider first one mole of gas which is composed of non-interacting point particles that satisfy the ideal gas law Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by BenoÃ®t Paul Ã‰mile Clapeyron in 1834. ...

$p = frac{RT}{V_mathrm{m}}.$

Next assume that all particles are hard spheres of the same finite radius r (the van der Waals radius). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace V by V-b, where b is called the excluded volume. The corrected equation becomes The van der Waals radius of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. ...

$p = frac{RT}{V_mathrm{m}-b}.$

The excluded formula is not just equal to the volume occupied by the solid, finite size, particles, but actually four times that volume. To see this we must realize that a particle is surrounded by a sphere of radius d = 2r that is forbidden for the centers of the other particles. If the distance between two particle centers would be smaller than 2r, it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do. The excluded volume per particle is $4π d 3 / 3$ , which we must divide by two to avoid overcounting, so that the excluded volume $b'$ is $4times (4pi r^3/3)$ , which is four times the proper volume of the particle. It was a point of concern to van der Waals that the factor four yields actually an upper bound, empirical values for $b'$ are usually lower. Of course molecules are not infinitely hard, as van der Waals thought, but are often fairly soft.

Next, we introduce a pairwise attractive force between the particles. Van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous. Further he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size. That is, the bulk of the particles do not notice that they have more attracting particles to their right than to their left when they are relatively close to the left-hand wall of the container. The same statement holds with left and right interchanged. Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density $ρ = N A / V m$ . The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density and the pressure (force per unit surface) is decreased by

$a'rho^2= a' left(frac{N_mathrm{A}}{V_mathrm{m}}right)^2 = frac{a}{V_mathrm{m}^2}$ ,

so that

$p = frac{RT}{V_mathrm{m}-b}-frac{a}{V_mathrm{m}^2} Longrightarrow (p + frac{a}{V_mathrm{m}^2})(V_mathrm{m}-b) = RT.$

Upon writing n for the number of moles and nV_m = V, the equation obtains the second form given above,

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

It is of some historical interest to point out that van der Waals in his Nobel price lecture gives credit to Laplace for the argument that pressure is reduced proportional to the square of the density. Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ...

Other thermodynamic parameters

In the following, we will use the extensive volume V instead of volume per particle v=V/N where N = nN_A 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: It has been suggested that this article or section be merged into Helmholtz energy. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

$A(T,V,N)=-NkTleft(1+lnleft(frac{(V-Nb)T^{hat{c}_V}}{NPhi}right)right) -frac{aN^2}{V}$

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 It has been suggested that this article or section be merged into Helmholtz energy. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, p = pressure, V = volume Differential definitions The following differential relations hold for the four potentials: If we write the above four equations generally as Then it is seen that yielding expressions for...

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

The entropy equation of state yields the entropy (S ) Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

$S = -left(frac{partial A}{partial T}right)_V =Nkleft[ lnleft(frac{(V-Nb)T^{hat{c}_V}}{NPhi}right)+hat{c}_V+1 right]$

from which we can calculate the internal energy In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

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

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. This article needs to be cleaned up to conform to a higher standard of quality. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... In thermodynamics, the Gibbs energy or Gibbs energy function is the energy portion of a thermodynamic system available to do work. ...

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), Critical variables are defined, for example in thermodynamics, in terms of the values of variables at the critical point. ...

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

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

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

where

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

$displaystyle{v_C=3nb}$

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

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. The principle (law) of corresponding states can, most generally, be summarized as Material constants that vary for each type of material are eliminated, in a recast reduced form of a constitutive equation. ...

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: PVT in the physics sense is an Equation of state that connects pressure, specific volume and temperature for a fluid. ... A polymer is a long, repeating chain of atoms, formed through the linkage of many molecules called monomers. ... Specific volume is the volume of a unit of mass of a material. ...

$(p+A)(V-B)=CT,$ ,

where

p is the pressure

V is specific volume

T is the temperature

A, B and C are parameters.

The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... Specific volume is the volume of a unit of mass of a material. ... Fig. ...

External links

Some values of a and b in the 2nd equation

Results from FactBites:

Van der Waals equation - Wikipedia, the free encyclopedia (901 words)
	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 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:
	The equation is also usable as a PVT equation for compressible fluids (e.g.

NationMaster.com - Encyclopedia: Johannes Diderik van der Waals (1034 words)
	Van der Waals was the first to realize the necessity of taking into account the volumes of molecules and the intermolecular forces (now generally called "van der Waals forces") in establishing the relationship between the pressure, volume and temperature of gases and liquids.
	Van der Waals was born in Leiden, the Netherlands, as the son of Jacobus van der Waals and Elisabeth van den Burg.
	Van der Waals is one of the most distinguished men in thermodynamics and author of an equation of state that has become the basis of all modern equations of state and theories of mixtures.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Equation GA_googleFillSlot("encyclopedia_square");

Validity

Reference

Derivation

Conventional derivation

Other thermodynamic parameters

Reduced form

Application to compressible fluids

See also

External links

Equation