# Van der Waals equation

The Van der Waals equation is an equation of state for gases with which the behavior of real gases can be described in a better approximation than with the general gas equation for the ideal gas . The Van der Waals equation contains, going beyond the general gas equation, two parameters for the repulsive and the attractive forces between the gas particles. These are characteristic of the respective gas. It thus leads to a simple and approximately quantitative understanding of liquefaction and many other properties in which the real gases deviate from the ideal gas. The equation was established in 1873 by Johannes Diderik van der Waals , for which he received the Nobel Prize in Physics in 1910 .

## The equation and its interpretation

### equation

Van der Waals constants in the gas phase
gas Cohesion
pressure a in (kPa dm 6 ) / mol 2
= 10 −3 (Pa m 6 ) / mol 2
= 10 −3 (J m 3 ) / mol 2
Co-volume b
in cm 3 / mol
= 10 −6 m 3 / mol
Helium (He) 3.45 23.7
Neon (ne) 21.3 17.1
Argon (Ar) 136.3 32.2
Hydrogen (H 2 ) 24.7 26.6
Nitrogen (N 2 ) 140.8 39.1
Oxygen (O 2 ) 137.8 31.8
Air (80% N 2 , 20% O 2 ) 135.8 36.4
Carbon dioxide (CO 2 ) 363.7 42.7
Water (H 2 O) 557.29 31
Chlorine (Cl 2 ) 657.4 56.2
Ammonia (NH 3 ) 422.4 37.1
Methane (CH 4 ) 225 42.8
Benzene (C 6 H 6 ) 52.74 304.3
Decane (C 10 H 22 ) 37.88 237.4
Octane (C 8 H 18 ) 18.82 119.3

The Van der Waals equation is

${\ displaystyle p = {\ frac {nRT} {V-nb}} - {\ frac {n ^ {2} a} {V ^ {2}}} \ quad}$ or.
${\ displaystyle p = {\ frac {RT} {V_ {m} -b}} - {\ frac {a} {{V_ {m}} ^ {2}}}}$

or equivalent

${\ displaystyle \ Leftrightarrow \ left (p + {\ frac {a} {{V_ {m}} ^ {2}}} \ right) \ left (V_ {m} -b \ right) = RT \.}$

The symbols represent the following quantities :

• ${\ displaystyle p}$- pressure of the gas
• ${\ displaystyle T}$- temperature
• ${\ displaystyle V_ {m}}$- molar volume ( with volume and amount of substance of the gas)${\ displaystyle V_ {m} = V / n}$ ${\ displaystyle V}$ ${\ displaystyle n}$
• ${\ displaystyle R}$- Universal gas constant (as with ideal gas)
• ${\ displaystyle a}$- cohesion pressure (parameter)
• ${\ displaystyle b}$- covolume

Cohesion pressure (parameters) and covolume are also referred to as van der Waals constants of the gas concerned (see examples in the table). For the ideal gas, and . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a = 0}$${\ displaystyle b = 0}$

By introducing the amount of substance , the Van der Waals equation can also be represented as ${\ displaystyle n}$

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

The Van der Waals equation describes both the gas and the liquid phase qualitatively correctly, but is often too imprecise for quantitative applications. As a result, the van der Waals parameters of a gas can depend on the range of states in which they were obtained. More precisely are z. B. the Redlich-Kwong equation or the equation of state of Soave-Redlich-Kwong , both semi- empirical modified Van der Waals equations.

There are also empirical equations of state, such as B. the Benedict-Webb-Rubin equation .

### Interpretation within the framework of the kinetic gas theory

The equation of state of the ideal gas

${\ displaystyle p = {\ frac {RT} {V_ {m}}}}$

can be justified in the kinetic gas theory assuming point-like particles without mutual forces ( derivation ). In comparison, with van der Waals gas is the pressure

• reduced by subtracting the cohesive pressure (or internal pressure )${\ displaystyle {\ frac {a} {V_ {m} ^ {2}}}}$
• increased by a factor by effectively reducing the molar volume by the covolume .${\ displaystyle {\ frac {V_ {m}} {V_ {m} -b}} = {\ frac {1} {1-b / V_ {m}}}}$${\ displaystyle b}$

These modifications can also be clearly interpreted using the kinetic gas theory.

#### Cohesive pressure

The Van der Waals parameter takes into account the fact that gas molecules attract each other. This is ignored in the model of the ideal gas, but is, among other things, decisive for the possibility of liquefaction (hence the term cohesion pressure). These attractive forces, which are summarized under the name of Van der Waals forces, are ultimately of an electromagnetic nature, but have a much smaller range. They are based on the fact that molecules, even those without a permanent dipole moment, can polarize one another electrically, particularly markedly when two molecules are the same. These forces reduce the pressure on the container wall because they are mainly directed inwards for the molecules near the wall (hence the name internal pressure). For each particle, its strength is proportional to the density of the other particles, for all particles together it is proportional to the square of the particle density . Therefore, the total negative contribution to pressure is inversely proportional to . ${\ displaystyle a}$${\ displaystyle V_ {m} ^ {2}}$

#### Co-volume

The Van der Waals parameter takes into account that the gas molecules are not mass points, as assumed in the model of the ideal gas. Due to their finite size, the mean free path is shortened and the number of collisions against the wall that generate the pressure, as well as the collisions with each other, is greater than for point-like particles. The increase in the number of impacts can be parameterized by an apparent reduction in the volume. To put it clearly, the particle centers can only approach each other up to twice the radius, which effectively blocks not eight times but only four times their own volume (because only the possible particle pairs are to be counted here). This means that the covolume is roughly 4 times the size of the molecules' own volume. The co-volume also roughly corresponds to the volume of the liquefied gas. ${\ displaystyle b}$${\ displaystyle b}$

### Justification in statistical mechanics

A systematic reasons for the van der Waals equation and its parameters , and is used in the statistical mechanics given. There the equation of state is calculated as part of a series expansion according to the particle density (see virial expansion ). For a classic gas, the equation of state of the ideal gas results if this series expansion is broken off after the first term, and the van der Waals equation (in a corresponding approximation) if the following second term is taken into account. ${\ displaystyle a}$${\ displaystyle b}$

## Isotherms in the pV diagram

The isotherm according to the solved Van der Waals equation ${\ displaystyle p}$

${\ displaystyle p (V_ {m}) = {\ frac {RT} {V_ {m} -b}} - a {\ frac {1} {V_ {m} ^ {2}}}}$

is represented in the pV diagram by a curve which is the difference between a simple hyperbola and a squared hyperbola, the simple hyperbola being shifted by. ${\ displaystyle b}$

Van der Waals loops (dark blue)
and Maxwell construction

For high temperatures and low density (i.e. large ) it approaches the simple hyperbola according to the equation for ideal gases. ${\ displaystyle T}$${\ displaystyle V_ {m}}$

Below a critical temperature (isotherms marked in red in the picture), however, a maximum and a minimum occur (points C and A), so that with an isothermal compression from C to A, both the volume and the pressure would decrease. This area is therefore unstable and does not occur in nature in a metastable form. Instead, the isotherm runs isobarically in this volume range, i.e. H. in the - diagram as a horizontal line (black). Assuming a large initial volume through isothermal compression to smaller volumes, the pressure initially increases according to the van der Waals isotherm, but remains constant from a certain value (point G in the figure), even if the volume continues to decrease becomes. From this point - if the temperature is kept constant by heat dissipation - the condensation of the gas to liquid takes place at constant pressure . Both states of aggregation are in coexistence along the black line, with only their quantitative ratio changing. The gaseous part retains the density it had at point G and the liquid part the density of point F. The area of ​​coexistence on this isotherm ends when the isobaric part meets the van der Waals curve again (point F in the figure). From this point on, the pressure rises steeply with a further decrease in volume, which is characteristic of the low compressibility of a pure liquid. Together these isobaric areas of the isotherms of all temperatures up to the critical temperature form the gas-liquid coexistence area of ​​the gas under consideration. ${\ displaystyle T_ {c}}$${\ displaystyle p}$${\ displaystyle V}$

The position of the horizontal piece, i.e. the saturation vapor pressure of the liquefied gas at the temperature of the isotherm under consideration, can be determined from the fact that the two surface pieces towards the van der Waals curve (lime green in the figure) must be the same size. James Clerk Maxwell , who introduced this Maxwell construction , justified it with the conservation of energy in an isothermal reversible cycle along the Van der Waals loop that borders the two surfaces. However, this justification suffers from the fact that the cycle cannot actually take place because of the instability between A and C. A better justification is based on the fact that the states on the relevant part of the Van der Waals isotherm are unstable and change automatically to states on the Maxwell line because these have the lowest possible free enthalpy at the same temperature and pressure . The homogeneous matter described by the Van der Waals equation spontaneously breaks down into two parts with different densities. The vapor pressure determined in this way depends on the temperature, as required by the Clausius-Clapeyron equation .

Condensation and evaporation are phase transitions that may only start with a delay if the state changes sufficiently quickly. Therefore, the states on the curve pieces of the van der Waals isotherm to the left of the minimum (points FA) and to the right of the maximum (points CG) can sometimes be reached briefly if the volume of the pure gas is rapidly reduced or that of the pure liquid is increased rapidly , or by other suitable processes that are not isothermal such as e.g. B. Boiling delay . These states represent metastable states with a homogeneous density, which disintegrate into a suitable mixed state when the vapor formation or condensation occurs. For example, water droplets could be heated briefly to 279 ° C under atmospheric pressure until they evaporated explosively. The course according to the van Waals equation was approximately confirmed.

## Critical point

At a temperature above the critical temperature, the van der Waals isotherm in the diagram shows a negative slope everywhere. There can therefore be no area where the volume (or density) varies, but the temperature remains the same. A coexistence of two phases with different densities is therefore excluded. This agrees with all observations made on real gases. The critical temperature is given by the fact that its isotherm goes through the critical point, where it has a turning point with a horizontal tangent in the diagram : At the critical point, the first and second derivatives of the pressure with respect to volume vanish (at constant temperature): ${\ displaystyle pV}$${\ displaystyle T_ {crit}}$${\ displaystyle pV}$ ${\ displaystyle K}$

${\ displaystyle \ left ({\ frac {\ mathrm {d} p} {\ mathrm {d} V}} \ right) _ {T} = 0 \; \ quad \ left ({\ frac {\ mathrm {d } ^ {2} p} {\ mathrm {d} V ^ {2}}} \ right) _ {T} = 0}$

It follows from this:

${\ displaystyle T_ {crit} = {\ frac {8a} {27bR}}}$
${\ displaystyle p_ {crit} = {\ frac {a} {27b ^ {2}}}}$
${\ displaystyle V_ {m, crit} = 3b}$

Some examples of critical temperatures: water = 374.12 ° C, nitrogen = -146 ° C, carbon dioxide = 31 ° C, helium-4 = 5.2 K. The gas in question can only be liquefied below these temperatures. ${\ displaystyle T_ {crit}}$${\ displaystyle T_ {crit}}$${\ displaystyle T_ {crit}}$${\ displaystyle T_ {crit}}$

Conversely, the van der Waals constants and can be calculated from the experimentally found data for the critical point . Since there are three critical data, but only two Van der Waals constants, the system of equations is overdetermined and would only have an exact solution if the gas behaved exactly according to the Van der Waals equation. In tables, the values ​​obtained from the van der Waals parameters are often given: ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle T_ {crit} \, p_ {crit}}$

${\ displaystyle b = {\ frac {RT_ {crit}} {8p_ {crit}}}}$
${\ displaystyle a = {\ frac {27 (RT_ {crit}) ^ {2}} {64p_ {crit}}}}$

The value for is usually only less precisely known, but would lead to a significantly different value. This is also expressed when calculating the compression factor. For every ideal gas is in all states . According to the Van der Waals equation, the following should apply regardless of the parameters and , i.e. for every real gas: ${\ displaystyle V_ {crit}}$${\ displaystyle b = V_ {m, crit} / 3}$ ${\ displaystyle Z = {\ tfrac {pV} {R \, T}}}$${\ displaystyle Z = 1}$${\ displaystyle a}$${\ displaystyle b}$

${\ displaystyle Z = {\ frac {p_ {crit} V_ {m, crit}} {R \, T_ {crit}}} = {\ frac {3} {8}} = 0 {,} 375}$

In fact, the values ​​for real gases are relatively close together, but are even further away from the value 1 of the ideal gas than in the van der Waals model. Examples:

${\ displaystyle Z (H_ {2}) = 0 {,} 304; \ quad Z (C_ {2} H_ {6}) = 0 {,} 267; \ quad Z (NH_ {3}) = 0 {, } 238; \ quad Z (H_ {2} O) = 0 {,} 226}$

The effectiveness of the van der Waals model is shown in the relatively good agreement with one another, and its approximate character in the deviation from the model value.

## Reduced form of the van der Waals equation

If you press pressure, temperature and molar volume according to

${\ displaystyle p = p_ {red} \, p_ {crit}}$, ,${\ displaystyle T = T_ {red} \, T_ {crit}}$${\ displaystyle V_ {M} = V_ {red} \, V_ {m, crit}}$

from the dimensionless reduced state variables , then the Van der Waals equation is called: ${\ displaystyle p_ {red}, \ T_ {red}, \ V_ {red}}$

${\ displaystyle \ left (p_ {red} + {\ frac {3} {V_ {red} ^ {2}}} \ right) (V_ {red} - {\ frac {1} {3}}) = { \ frac {8} {3}} T_ {red}}$

This equation does not contain any material-specific parameter and therefore applies in an identical form to all substances that follow a Van der Waals equation. This is an example of the coincident state theorem . If two substances assume the same values , one speaks of matching / corresponding states. ${\ displaystyle p_ {red}, V_ {red}, T_ {red}}$

Sorted according to the powers of the volume results in a 3rd degree polynomial:

${\ displaystyle \ Leftrightarrow V_ {red} ^ {3} - \ left ({\ frac {1} {3}} + {\ frac {8T_ {red}} {3p_ {red}}} \ right) V_ {red } ^ {2} + {\ frac {3} {p_ {red}}} V_ {red} - {\ frac {1} {p_ {red}}} = 0}$

The dimensionless compression factor relates to the molar volume of an ideal gas: ${\ displaystyle Z}$

${\ displaystyle Z = {\ frac {p \, V_ {m}} {R \, T}} = {\ frac {p_ {c} \, V_ {m, c}} {R \, T_ {c} }} ~ {\ frac {p_ {red} \, V_ {red}} {T_ {red}}} = {\ frac {3} {8}} ~ {\ frac {p_ {red} \, V_ {red }} {T_ {red}}}}$

The Van der Waals equation thus reads:

${\ displaystyle \ left (p_ {red} + {\ frac {27} {64}} ~ {\ frac {p_ {red} ^ {2}} {T_ {red} ^ {2} Z ^ {2}} } \ right) \ left (Z - {\ frac {p_ {red}} {8}} \ right) = 1}$

This is a 3rd degree equation for the compression factor:

${\ displaystyle Z ^ {3} - \ left (1 + {\ frac {p_ {red}} {8T_ {red}}} \ right) Z ^ {2} + {\ frac {27} {64}} ~ {\ frac {Zp_ {red}} {T_ {red} ^ {2}}} - \ left ({\ frac {3} {8}} \ right) ^ {3} {\ frac {p_ {red} ^ {2}} {T_ {red} ^ {3}}} = 0}$

An approximation for small pressures and / or high temperatures is:

${\ displaystyle Z = 1 - {\ frac {p_ {red}} {8T_ {red}}} + {\ frac {27} {64}} ~ {\ frac {p_ {red}} {T_ {red} ^ {2}}} - \ left ({\ frac {3} {8}} \ right) ^ {3} {\ frac {p_ {red} ^ {2}} {T_ {red} ^ {3}}} }$

## Coefficient of thermal expansion

The isobaric expansion coefficient results from the equation of state if you differentiate it in the form given above:

${\ displaystyle \ mathrm {d} p = {\ frac {1} {V_ {m} -b}} \ mathrm {d} T- \ left ({\ frac {T} {(V_ {m} -b) ^ {2}}} - {\ frac {2a} {RV_ {m} ^ {3}}} \ right) \ mathrm {d} V_ {m}}$

and then continues. It follows ${\ displaystyle \ mathrm {d} p = 0}$

${\ displaystyle \ beta = {\ frac {1} {V_ {m}}} \ left ({\ frac {\ partial V_ {m}} {\ partial T}} \ right) _ {p} = {\ frac {V_ {m} -b} {V_ {m} T - {\ frac {2a} {R}} \ left ({\ frac {V_ {m} -b} {V_ {m}}} \ right) ^ {2}}}}$

The difference to the expansion coefficient of the ideal gas , which results for , is calculated ${\ displaystyle \ beta _ {idGas} = {\ tfrac {1} {T}}}$${\ displaystyle a = b = 0}$

${\ displaystyle \ beta - \ beta _ {idGas} = {\ frac {{\ frac {2a} {RT}} \ left ({\ frac {V_ {m} -b} {V_ {m}}} \ right ) ^ {2} -b} {V_ {m} T - {\ frac {2a} {RT}} \ left ({\ frac {V_ {m} -b} {V_ {m}}} \ right) ^ {2}}}}$

In the normal temperature range, the difference for gases such as oxygen, nitrogen and air is positive; these gases expand a little more than an ideal gas. With hydrogen and the noble gases it is the other way round. With them, the forces of attraction between the molecules or atoms, and thus the van der Waals parameter a , are so small that the numerator becomes negative (if the temperature is not too low). The zero point of the counter also marks the states in which the Joule-Thomson effect changes between cooling and heating (inversion point).

## Inner energy

For a van der Waals gas without internal degrees of freedom, the caloric equation of state applies :

${\ displaystyle U = {\ frac {3} {2}} nRT - {\ frac {na} {V_ {m}}}.}$

or more generally: where is the volume-independent portion of the internal energy per particle. ${\ displaystyle U = n \ cdot N_ {A} \ cdot u (T) - {\ frac {n ^ {2} a} {V}} \ quad}$${\ displaystyle u (T)}$

It depends not only on the kinetic energy of the molecules, but also on the potential energy of the cohesive forces given by the parameter . ${\ displaystyle a}$

## literature

• Johannes Diderik van der Waals: Over de Continuiteit van den gas en vloeistoftoestand . Sijthoff, Leiden 1873, German: The continuity of the gaseous and liquid state. Barth, Leipzig 1881.
• James Clerk Maxwell: On The Dynamical Evidence of the Molecular Constitution of Bodies. In: Nature . Volume 11, 1875, pp. 357-359 and 374-377.
• Peter W. Atkins: Physical Chemistry. Wiley-VCH, Weinheim 2001, pp. 43-46.
• D. Lüdecke, C. Lüdecke: Thermodynamics - Physico-chemical basics of thermal process engineering. Springer Verlag, 2000, ( books.google.de ).
• Torsten Fließbach: Statistical Physics. Spectrum Academic Publishing House, 2006.

## Individual evidence

1. C. San Marchi et al., WSRC-STI-2007-00579
2. Richard Becker: Theory of heat . Springer-Verlag, Heidelberg 1985, p. 32 .
3. Frederick Reif: Statistical Physics and Theory of Heat . 3. Edition. Walter de Gruyter, Berlin 1987, p. 354 ff .
4. Pablo Debenedetti: Metastable Liquids . Princeton University Press, Princeton 1996, pp. 5 ff .
5. ^ Arnold Sommerfeld: Thermodynamics and Statistics , Akademische Verlagsgesellschaft Leipzig, 1965
6. see Van-der-Waals Gas (tu-freiberg.de, from textbook by T. Fliessbach, Statistische Physik, Chapter 37).

## Web links

Commons : Van der Waals equation  - collection of images, videos and audio files