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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 volume 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, who received the Nobel prize in 1910 for "his work on the equation of state for gases and liquids". The equation is based on a modification of the ideal gas law and approximates the behavior of real fluids, taking into account the nonzero size of molecules and the attraction between them.


The first form of this equation is

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


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 the Avogadro constantNA, the number of molesn, and the total number of particles nNA, the equation can be cast into the second (better known) form

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


p is the pressure of the fluid
V is the total volume of the container containing the fluid
a is a measure of the attraction between the particles \scriptstyle a=N_\mathrm{A}^2 a'
b is the volume excluded by a mole of particles \scriptstyle \, b=N_\mathrm{A} b'
n is the number of moles
R is the universal gas constant, \scriptstyle \,R= N_\mathrm{A} k
T is the absolute temperature

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 \scriptstyle \,v refers to the empty space available per particle. That is, \scriptstyle \,v is the volume V of the container divided by the total number nNA 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 \scriptstyle \,v because of the space taken up by one particle. In van der Waals' original derivation, given below, \scriptstyle 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.


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 is not appropriate for rigorous quantitative calculations, remaining useful only for teaching and qualitative purposes.

In the first-order phase transition range of (P,V,T), where the liquidphase 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: although this behavior can be easily inserted into the van der Waals model (see Maxwell's correction below), the result is no longer a simple analytical model, and others (such as those based on the principle of corresponding states) achieve a better fit with roughly the same work.


Most textbooks give 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 the major advantage that it makes explicit the intermolecular potential, which is neglected in the first derivation.

The excluded volume per particle is \scriptstyle 4\pi d^3/3 = 8\times (4\pi r^3/3), which we must divide by two to avoid overcounting, so that the excluded volume b' is \scriptstyle 4\times (4\pi 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, which is not true as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). 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 C=N_\mathrm{A}/V_\mathrm{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,

Van der Waals radius

The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms had a finite size (i.e., that atoms were not simply points) and to demonstrate the physical consequences of their size through the van der Waals equation of state.

Van der Waals volume

The van der Waals volume, V, also called the atomic volume or molecular volume, is the atomic property most directly related to the van der Waals radius. It is the volume "occupied" by an individual atom (or molecule). The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances and angles) are known. For a spherical single atom, it is the volume of a sphere whose radius is the van der Waals radius of the atom:

V_{\rm w} = {4\over 3}\pi r_{\rm w}^3 .

For a molecule, it is the volume enclosed by the van der Waals surface. The van der Waals volume of a molecule is always smaller than the sum of the van der Waals volumes of the constituent atoms: the atoms can be said to "overlap" when they form chemical bonds.

The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constantb, the polarizabilityα or the molar refractivityA. In all three cases, measurements are made on macroscopic samples and it is normal to express the results as molar quantities. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constantN.

The molar van der Waals volume should not be confused with the molar volume of the substance. In general, at normal laboratory temperatures and pressures, the atoms or molecules of a gas only occupy about 1/1000 of the volume of the gas, the rest being empty space. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about 1000-times smaller than the molar volume for a gas at standard temperature and pressure.

Methods of determination

Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). These various methods give values for the van der Waals radius which are similar (1–2 Ã…, 100–200 pm) but not identical. Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case.

Van der Waals equation of state

The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases:

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

where n is the amount of substance of the gas in question and a and b are adjustable parameters; a is a correction for intermolecular forces and b corrects for finite atomic or molecular sizes; the value of b equals the volume of one mole of the atoms or molecules. Their values vary from gas to gas.

The van der Waals equation also has a microscopic interpretation: molecules interact with one another. The interaction is strongly repulsive at very short distance, becomes mildly attractive at intermediate range, and vanishes at long distance. The ideal gas law must be corrected when attractive and repulsive forces are considered. For example, the mutual repulsion between molecules has the effect of excluding neighbors from a certain amount of space around each molecule. Thus, a fraction of the total space becomes unavailable to each molecule as it executes random motion. In the equation of state, this volume of exclusion (nb) should be subtracted from the volume of the container (V), thus: (V - nb). The other term that is introduced in the van der Waals equation, a\left (\frac{n}{\tilde{V}}\right )^2, describes a weak attractive force among molecules (known as the van der Waals force), which increases when n increases or V decreases and molecules become more crowded together.

The van der Waals constantb volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases.

For helium, b = 23.7 cm/mol. Helium is a monoatomic gas, and each mole of helium contains 6.022×10 atoms (the Avogadro constant, N):

V_{\rm w} = {b\over{N_{\rm A}}}

Therefore the van der Waals volume of a single atom V = 39.36 Ã…, which corresponds to r = 2.11 Ã…. This method may be extended to diatomic gases by appro

Johannes Diderik van der Waals

Johannes Diderik van der Waals (November 23, 1837 – March 8, 1923) was a Dutchtheoretical physicist and thermodynamicist famous for his work on an equation of state for gases and liquids.

His name is primarily associated with the van der Waals equation of state that describes the behavior of gases and their condensation to the liquidphase. His name is also associated with van der Waals forces (forces between stable molecules), with van der Waals molecules (small molecular clusters bound by van der Waals forces), and with van der Waals radii (sizes of molecules).

He became the first physics professor of the University of Amsterdam when it opened in 1877 and won the 1910 Nobel Prize in physics.


Early years and education

Johannes Diderik was the oldest of ten children born to Jacobus van der Waals and Elisabeth van den Berg. His father was a carpenter in the Dutch city of Leiden. As was usual for working class children in the 19th century, he did not go to the kind of secondary school that would have given him the right to enter university. Instead he went to a school of "advanced primary education", which he finished at the age of fifteen. He then became a teacher's apprentice in an elementary school. Between 1856 and 1861 he followed courses and gained the necessary qualifications to become a primary school teacher and head teacher.

In 1862, he began to attend lectures in mathematics, physics and astronomy at the University in his city of birth, although he was not qualified to be enrolled as a regular student in part because of his lack of education in classical languages. However, the University of Leiden had a provision that enabled outside students to take up to four courses a year. In 1863 the Dutch government started a new kind of secondary school (HBS, a school aiming at the children of the higher middle classes). Van der Waals—at that time head of an elementary school—wanted to become a HBS teacher in mathematics and physics and spent two years studying in his spare time for the required examinations.

In 1865, he was appointed as a physics teacher at the HBS in Deventer and in 1866, he received such a position in The Hague, which is close enough to Leiden to allow van der Waals to resume his courses at the University there. Just before moving to Deventer Johannes Diderik married the eighteen-year-old Anna Magdalena Smit, in September 1865.


Van der Waals still lacked the knowledge of the classical languages that would have given him the right to enter university as a regular student and to take examinations. However, it so happened that the law regulating the university entrance was changed and dispensation from the study of classical languages could be given by the minister of education. Van der Waals was given this dispensation and passed the qualification exams in physics and mathematics for doctoral studies.

At Leiden University, on June 14, 1873, he defended his doctoral thesis Over de Continuiteit van den Gas en Vloeistoftoestand (on the continuity of the gaseous and liquid state) under Pieter Rijke. In the thesis, he introduced the concepts of molecular volume and molecular attraction .

In September 1877 van der Waals was appointed the first professor of physics at the newly founded Municipal University of Amsterdam. Two of his notable colleagues were the physical chemist Jacobus Henricus van 't Hoff and the biologist Hugo de Vries. As could be expected from a former elementary school teacher, van der Waals was an excellent and impressive lecturer . Until his retirement at the age of 70 van der Waals remained at the Amsterdam University. He was succeeded by his son Johannes Diderik van der Waals, Jr., who also was a theoretical physicist. At the age of 72, (in 1910) van der Waals was awarded the Nobel Prize in physics. He died at the age of 85 on March 8, 1923.

Scientific work

The main interest of van der Waals was in the field of thermodynamics. He was much influenced by Rudolf Clausius' 1857 treatise entitled Über die Art der Bewegung, welche wir Wärme nennen (On the Kind of Motion which we Call Heat). Van der Waals was later greatly influenced by the writings of James Clerk Maxwell, Ludwig Boltzmann, and Willard Gibbs. Clausius' work led him to look for an explanation of Thomas Andrews' experiments that had revealed, in 1869, the existence of critical temperatures in fluids. He managed to give a semi-quantitative description of the phenomena of condensation and critical temperatures in his 1873 thesis, entitled Over de Continuïteit van den Gas- en Vloeistoftoestand (On the continuity of the gas and liquid state). This dissertation represented a hallmark in physics and was immediately recognized as such, e.g. by James Clerk Maxwell who reviewed it in Nature in a laudatory manner.


From Yahoo Answers

Question:I am givev: Calculate the pressure exerted by .5000 mol N2 in a 10.000-+ container at 25.0 celsius Use va der Waals equation to calculate.

Answers:bebeh!, You need to use the Van der Waals equation: [P + (an^2/V^2)] * (V - nb) = nRT As well as the Van der Waals constants for nitrogen: a (L2-atm/mol2) = 1.390 b (L/mol) = 0.03913 Solve for P: [P + (an^2/V^2)] * (V - nb) = nRT P = nRT/(V-nb) - (an^2/V^2) P = ( (0.500 mol*0.0821*298.15K) / (10 L - (0.5 mol * 0.03913)) - ( 1.390* (0.5*0.5) / (10.000*10.000) ) = 1.22 atmospheres Hope that helped!

Question:How can you do this question only using the equation, (P+a/V^2)(V-B)=RT Please help, its for a chemistry exam review!

Answers:Boyle's temperature is the temperature at which the second virial coefficient, B (T), is zero. For a v-d-W gas that is where T = a/Rb, where a and b are the van der Waals equation of state parameters. See the attached image for a derivation.

Question:What is the pressure of 2.5mol CO2 in a 1L flask at 298K when calculated using the van der waals equation? For CO2, the values of the van der waals constants are a = 3.59L^2 atm/mol^2 and b = 0.0427L/mol please show work! thanks!

Answers:van der waals equation is... (P + a n / V ) x (V/n - b) = RT P = RT/(V/n - b) - (a n / V ) R = 0.0821 Latm/moleK T = 298K V = 1 L n = 2.5 moles b = 0.0427 L/mole a = 3.59 L atm / mole P = [(0.0821 Latm/moleK) x (298K) / ((1 L / 2.5 mole) - 0.0427 L/mole)] - [(3.59 L atm / mole ) x (2.5 moles) / (1 L) ] P = 46.0 atm ******* via the ideal gas PV = nRT P = nRT/V = 2.5 moles x (0.0821 Latm/moleK) x (298K) / 1 L = 61.2 atm there's a pretty significant difference there. and it's because gases deviate from ideality when the pressures are high and when intermolecular forces of attraction are large. The "a" in the Van Der Waals equation is term that adjusts the "ideal gas law" (which assumes no intermolecular attractions) for real attractions between molecules. The "b" adjusts for the real volume of the gas molecules. Ideal gas equation assumes it is zero. now.. you notice what happens to the Van Der Waals equation if a=b=0 (as in the ideal gas law?) P = RT/(V/n - 0) - (0 x n / V ) = RT/(V/n) = nRT/V ----> PV = nRT...

Question:Use the van der Waals equation for real gases to calculate the pressure exerted by 1.00 mole of ammonia at 27.0 C in a 750.0 mL container. (a = 4.17 atmL2/mol2, b = 0.0371 L/mol)

Answers:(P + 4.17/0.75x0.75)(0.75 - 0.0371) = 0.082057x300.15 (P+7.4133)x0.7129 = 24.6294 0.7129P +5.2847 = 24.6294 P = 27.14 atm (vab der Waals) P = nRT/V = 32.84 atm (ideal)

From Youtube

Van der Waals Isotherms The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The Van der Waals equation of state for one mole of an imperfect fluid reads (P + a / V^2)(V - b) = R T. The critical constants are predicted to be Pc = a / 27b^2, Vc = 3b, Tc = 8a / 27R b. The Van der Waals equation can be recast in the form of a unive... Contributed by: SM Blinder

Van der Waals Isotherms for Real and Ideal Gases The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The van der Waals equation of state, P = RT/(Vm - b) - a/Vm^2, represents the behavior of real gases in response to pressure, molar volume, and temperature at finite concentrations. The constant a stands for the reduction in pressure due to the attract... Contributed by: Kristen Aramthanapon, Wiktor Macura, and Chris French