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STATES OF MATTER - properties of gases and liquids (fluids) and solids

24. Deviations from ideal gas behaviour, intermolecular forces, Van der Waals equation, compressibility factors and the critical pressure and critical temperature of a gas

Doc Brown's chemistry revision notes:

Helpful for UK advanced level chemistry students aged ~16-18, IB courses and US grades 11-12 K12 honors.

24. Non–ideal real gas behaviour and Van der Waals Equation

deviation from ideal gas behaviour PV versus P graph 224a. The deviations of a gases from ideal behaviour and their causes

  • Certain postulates in the kinetic theory of gases (see section 17) are far from true in real gases, particularly at higher pressures and a lower temperatures.

  • This can be clearly seen in the diagram on the right.

  • If the gases conformed to the ideal gas law equation PV=nRT, the product PV should be constant with increasing pressure at constant temperature, clearly this is not the case.

  • It can also be seen that the greatest deviation from ideal behaviour always tend to occur at higher pressures (right diagram) and often at lower temperatures (see the compressibility factor diagram further down the page, and both positive and negative deviation occur).

  • Several points in the theoretical kinetic particle model cannot be ignored in 'real gases'.

    • The actual volume of the molecules (Vmolecules) is significant at high pressures i.e. the free space for random particle movement (Videal) is less than it appears from volume measurements.

      • Vreal = Videal + Vmolecules

      • At very high pressures therefore, the value of PV becomes greater than the ideal value and presumably outweighs the intermolecular force of attraction factor which would tend to increase the closer the molecules are and decrease P (see forces arguments next).

      • The deviation from ideal gas behaviour due to the molecular volume factor will generally increase with (i) the greater the pressure and (ii) the larger the volume of the molecule (~increasing Mr).

    • Intermolecular forces always exist i.e. instantaneous dipole – induced dipole forces (Van der Waals forces) between ANY molecules and at high pressures the molecules are closer together and so attraction is more likely to occur. As a particle hits the container side there is an imbalance of the intermolecular forces which act in all directions in the bulk of the gas. Just as the particle is about to hit the surface there will be a net greater attraction towards the bulk of the gas as the molecule, so reducing its impact force i.e. reduces its 'ideal' pressure (pideal) by an amount (preduction).

    • preal = pideal – preduction

    • At lower temperatures when the KE of the molecules are at their lower values, the intermolecular forces can have more of an effect in reducing P, so the PV value is less than the ideal value. The effect becomes less as the temperature increases (graph above-right) and also as the pressure becomes much higher when the molecule volume factor outweighs the intermolecular force factor.

    • These intermolecular forces will increase the bigger the molecule (~increasing number of electrons) and the more polar the molecule where permanent dipole – permanent dipole forces can operate in addition to the instantaneous dipole – induced dipole forces.

    • Also, the lower the temperature, the kinetic energies are lower so its more likely that neighbouring molecules can affect each other. The reduction in pideal also increases with increasing pressure too, since the molecules will be on average closer together.

    • There is direct experimental evidence for the effects of intermolecular forces in gases from adiabatic expansion or compression situations. Adiabatic means to effect a change in a system fast enough to avoid heat transfer to or from the surroundings. e.g.

    • (i) If a gas at high pressure is suddenly released through a small nozzle it rapidly cools on expansion into the lower pressure zone. The reason for the cooling is that in order to expand the intermolecular forces must be overcome by energy absorption, an endothermic process. The change is so rapid that the source of heat energy can only come from the kinetic energy of the gas molecules themselves, so the gas rapidly cools. This is observed when a carbon dioxide fire extinguisher is used, just for a second bits of solid CO2 can be seen, which rapidly vaporise. However, it proves that the gas was rapidly cooled from room temperature to –78oC!

    • (ii) When you rapidly pump air into a bicycle tyre the gas warms up because the molecules are forced closer together so the intermolecular forces can operate more strongly, this, just like bond formation, is always an exothermic process.

  • Therefore generally speaking for any gas the lower its pressure and the higher its temperature, the more closely it will be 'ideal', i.e. closely obey the ideal gas equation PV=nRT etc. Also the smaller the molecular mass or the weaker the intermolecular forces, the gas will be closer to ideal behaviour.

    • However, for any gas at a particular P and T, its all a question of what factor outweighs the others.

    • Note that both positive and negative deviation from ideal gas behaviour can occur and there will be situations where the different causes of non–ideal behaviour cancel each other out.

    • Check out the graphs at the start of this age

  • The measurement and predictions of gas behaviour is very important in industrial processes and so many mathematical developments have been devised to accurately describe the real behaviour of gases.

  • The Van der Waals equation is one of the earliest and simplest equations to model real gas behaviour.

24b. The Van der Waals equation of state

  • Equations such as the Van der Waals equation for real non–ideal gases attempt to take into account the volume occupied by the molecules and the intermolecular forces between them. The idea is to incorporate 'corrective' terms to reproduce or model real gas P–V–T behaviour with a modified equation of state.

  • The Van der Waals equation for one mole of gas can be most simply stated in as

  • (i)  (p + a') (V – b') = RT

  • This is called an 'equation of state' and is the simplest version for a non-ideal gas.

  • The term a' represents the extra pressure the gas would exert if it behaved ideally.

  • In real gases the intermolecular forces are imbalanced at the point of impact on the container wall, with a net attraction in the direction of the bulk of the gas.

  • In the bulk of the gas, each molecules is subjected to the same 'time averaged' attractions in all directions, but heading for the container wall it is considered to be 'dragged back a bit' by attraction with the bulk of the gas surrounding it on all sides bar the surface of impact, which is therefore reduced in force. (see also intermolecular forces discussion)

  • The term b' represents the volume that the molecules occupy, so V–b' represents the actual volume of free space the molecules can move in. (see also molecule volume discussion)

  • For n moles of gas the Van der Waals equation is ...

  • (ii)  [p + (an2/V2)] (V – nb) = nRT

    • a and b are the Van der Waal equation constants.

    • The factor n2/V2 is related to the gas density, the more dense the gas (i.e. moles/volume), at higher pressures, the more intense will be the intermolecular attractive force field effects.

    • Dividing through by n, using the (V – nb) term, gives the alternative version ...

  • (iii)  [p + (an2/V2)] [(V/n) – b)] = RT

    • and from these equations an expression for predicting pressure can be derived i.e. from (ii) we get ...

  • (iv)  p = [nRT/(V – nb)] – (an2/V2)

    • and from (iii) we get ...

  • (v)  p = [RT/(V/n – b)] – (an2/V2)

  • For 1 mole of gas the equation simplifies to

    • (p + a/V2)] (V – b) = RT

  • A selection of a and b Van der Waal's constants are given below.


Van der Waals constants

critical values of the gas


a (Pa m6 mol–2)

b (m3 mol–1)

pressure pc (Pa)

temp. Tc (K)
air, av Mr(mix) ~ 29 0.1358 3.64 x 10–5 3.77 x 106 133 K
ammonia, Mr(NH3) = 17 0.4233 3.73 x 10–5 11.3 x 106 406 K
butane, Mr(C4H10) = 59 1.466 12.2 x 10–5 3.78 x 106 425 K
carbon dioxide, Mr(CO2) = 44 0.3643 4.27 x 10–5 7.39 x 106 304 K
dichlorodifluoromethane, Freon CFC–11, Mr(CCl2F2) = 121 1.078 9.98 x 10–5 4.12 x 106 385 K
helium, Mr(He) = 4 0.00341 2.34 x 10–5 0.23 x 106 5 K
hydrogen, Mr(H2) = 2 0.0247 2.65 x 10–5 1.29 x 106 33 K
nitrogen, Mr(N2) = 28 0.1361 3.85 x 10–5 3.39 x 106 126 K
water, Mr(H2O) = 18 0.5507 3.04 x 10–5 22.1 x 106 647 K
  • The constant a varies considerably from gas to gas because of the wide variety of intermolecular forces e.g. very low for helium and non–polar hydrogen (2 e's each, just instantaneous dipole–induced dipole forces), to much higher a values for larger polar molecules like water or dichlorodifluoromethane (more electrons and extra permanent dipole–permanent dipole intermolecular forces).

  • The constant b varies less, and not unexpectedly, just tends to rise with increase in molecule size.

  • Critical values of gas behaviour.

    • Critical temperature Tc

      • This is the maximum temperature at which a substance can exist as a liquid. Above Tc, only the gaseous state can exist, however great the density or pressure! It might be truer to say that above Tc, the gaseous and liquid state become indistinguishable as the meniscus just disappears!

    • Critical pressure pc

      • This is the pressure the gas exerts at the critical temperature.

    • Generally speaking the critical values for a gas/liquid increase with increase in intermolecular forces e.g. due to increase in molecular mass or increasing polarity of molecule.

compressibility factors z for methane gas24c. Compressibility factors

The compressibility factor z, is defined as the ratio PV/nRT.

Since PV = nRT for an ideal gas, then z = 1 for an ideal gas.

z varies with pressure or temperature for any gas, see the PV versus P graph in start of this page, which gives an indication of how z might vary with pressure at a given temperature).

Clearly from the graph on the right for methane, z can be at least as high as 2, and, at least as low as 0.6, showing considerable deviation from ideal gas behaviour, particularly at low temperatures (influence of intermolecular forces stronger) and high pressures (where the effect of both actual molecule volume and intermolecular forces are important). See more detailed discussion at start of this page.

As the pressure becomes lower and/or temperature higher, the gas becomes more ideal in terms of its physical behaviour and particularly 'ideal' as the pressure tends towards zero.

Known values of z can be used to calculate the real P–V values for a non–ideal gas.

z = pV/nRT, pV = znRT, p = znRT/V and V = znRT/p

24d. The Critical Point – The Critical Temperature and Critical Pressure for a gas

Question! If you increase the pressure of a gas it can change into a liquid. But, increasing the pressure, also increases the temperature, so shouldn't the gas remain a gas?

  • Gases can be converted to liquids by compressing the gas at a suitable temperature and this is done commercially at as lower temperature as possible e.g. liquefaction of air to fractionally distil off nitrogen and oxygen or liquefying petroleum gas.

  • Gases become more difficult to liquefy as the temperature increases because the kinetic energies of the particles that make up the gas also increase and the intermolecular forces have less influence i.e. more easily overcome.

  • When you increase the pressure of a gas you force the molecules closer together and if the extra intermolecular force is strong enough liquefaction occurs. Remember the force of electrical attraction is proportional to the numerical +ve charge multiplied by the  –ve charge divided by the distance squared.

    • F = constant x C+ x C / d2

    • where d = distance between the centres of the charged particles

  • However when you compress a gas it can heat up. This is because heat is generated by the increased intermolecular interaction (remember bond formation is also exothermic) but here its just weak molecule association due to the intermolecular attractive forces.

  • BUT liquefaction = condensation and is an exothermic process, so heat must be removed to effect the state change of gas ==> liquid. If the temperature is low enough and the heat is dispersed liquefaction can still happen.

  • If it is too hot it would stay as a gas. So liquefaction conditions are all about temperature, pressure and heat transfer i.e. the ambient conditions.

  • However, above a certain temperature called the critical temperature (Tc) you cannot get a liquid with a 'surface', what you get is an extremely dense gas that is close to being a liquid but not quite!

  • The critical temperature of a substance is the temperature at and above which vapour of the substance cannot be liquefied, no matter how much pressure is applied.

  • The critical pressure (Pc) of a substance is the minimum pressure required to liquefy a gas at its critical temperature i.e. the critical pressure is the vapour pressure at the critical temperature.

  • The vapour-liquid critical point denotes the conditions above which distinct liquid and gas phases do not exist and a meniscus no longer exists!

  • The point at the critical temperature and critical pressure is called the critical point of the substance.

Learning objectives

Know that the principal reasons for the deviation of gaseous behaviour from the ideal gas laws, at most temperatures and pressures is due to the influence of intermolecular attractive forces between the particles in the gas.

Know that significant deviations also occur at high pressure (again due to above) AND the particles occupy an actual volume - they are not points of zero volume!

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