Doc Brown's Chemistry  States of Matter Revision Notes continued

(including more advanced ideas)

Part 2 More advanced topics on the ideal gas laws, calculations, kinetic particle model-theory and ideal and non-ideal real gas behaviour

Revision KS4 Science IGCSE/O level/GCSE Chemistry Information Study Notes for revising for AQA GCSE Science, Edexcel 360Science/IGCSE Chemistry & OCR 21stC Science, OCR Gateway Science  (revise courses equal to US grades 9-10)

EMAIL query?comment * The basic particle theory and properties of gases, liquids and solids, state changes & solutions are described on GCSE notes on particle models of gases-liquids-solids, describing and explaining their properties and advanced students should be familiar with ALL its contents before studying this page ....

Sub-index for Part II: Section 4 Ideal gas behaviour and the gas laws: Introduction-the kinetic particle theory of an ideal gas * 4a. Boyle's Law * 4b. Charles's-Gay Lussac's Law and the combined gas law equation * 4c. The ideal gas equation PV=nRT * 4d. Dalton's Law of partial pressures * 4e. Graham's Law of diffusion * Section 5. Non-ideal real gas behaviour and Van der Waal's Equation: 5a. The deviations of a gases from ideal behaviour and their causes * 5b. The Van der Waals equation of state * 5c Compressibility factors * 5d The Critical Point - The Critical Temperature and Critical Pressure *

For other calculations see the Calculations Index page including Mole definition and Avogadro Constant, molar gas volume and reacting gas volume ratios.

The Maxwell Boltzmann distribution of particle kinetic energies is discussed in the KINETICS (rates of reaction) pages.

4. Ideal gas behaviour and the gas laws

Introduction - the kinetic particle model of an ideal gas

• The (advanced) kinetic theory of gases is founded on the following six fundamental postulates:

  1. Gases are composed of minute discrete particles (usually molecules).

  2. The particles are in continuous chaotic motion moving in straight lines between very frequent collisions with each other and the sides of the container (approximately 10⁹/s).

  3. The bombardment of the container walls by the particles causes the phenomenon we call pressure (i.e. force of impacts/unit area).

  4. The collisions are perfectly elastic i.e. no energy loss on collision due to friction.

  5. At relatively low pressures the average distance between particles is large compared to the diameter of the particles and therefore the inter-molecular forces between the particles is negligible.

  6. The average kinetic energy of the particles is directly proportional to the absolute temperature on the Kelvin scale (K).

• When a gas behaves according to this model, the gas laws described in sections 4a to 4e are obeyed.

• However in real gases things are not so simple and this non-ideal behaviour is discussed in section 5.

4a. Boyle's Law for volume and gas pressure

• The particle theory of gas pressure was explained in Part 1 so this section concentrates on the gas law calculations involving pressure and volume.

• Boyle's Law states that for given mass of gas at a constant temperature, the product of the pressure multiplied by the volume is a constant.

• p x V = constant

• Therefore, for initial values of p₁ and V₁, which change to final values of p₂ and V₂, the following equation applies ...

• p₁ x V₁ = p₂ x V₂ (for fixed amount of gas at constant temperature)

• or p₂ = p₁ x V₁/V₂ or V₂ = p₁ x V₁/p₂

• The graph shows how the pressure and volume vary according to Boyles Law at two different temperatures.

• At lower temperatures the volume and pressure values are lower (see next section).

• You can use any volume or pressure units you like as long as both p's and both V's have the same units.

• Using particle theory and simple arithmetical values to explain Boyles Law.

  • If a gas is compressed to half its original volume the concentration or density of the gas is doubled. Therefore there will be twice as many collisions with the surface causing twice the impact effect i.e. double the pressure.

  • If the volume of a gas is increased by a factor of three, the concentration is reduced by the same factor, so the chance of particle collision with the container walls is similarly reduced, so the pressure decreases by a factor of three.

• Examples of Boyle's Law calculations (constant temperature assumed)

• Ex. Q4a.1

  • 240cm³ of air at a pressure of 100kPa in a bicycle pump is compressed to a volume of 150cm³.

  • What is the pressure of the compressed air in the pump?

  • p₁ x V₁ = p₂ x V₂ , rearranging to scale up for the new higher pressure

  • p₂ = p₁ x V₁/V₂  = 100 x 240/150 = 160 kPa

• Ex. Q4a.2

  • 10 m³ of butane gas at 1.2 atm was required to be stored at 6 atm pressure. To what volume must the gas be compressed to give the required storage pressure?

  • p₁ x V₁ = p₂ x V₂ , rearranging to scale down for the new lower volume

  • V₂ = p₁ x V₁/p₂ = 1.2 x 10/6 = 2.0 m³

• -

4b. Charles's Law/Gay-Lussac's Law for pressure/volume and temperature

and the combined gas law equation

• The particle theory of gas pressure was explained in Part 1 so this section concentrates on the gas law calculations involving pressure and volume and their variation with temperature.

• Charles's/Gay-Lussac's Law states that for a fixed mass of gas ...

  • (i) the volume of a gas is directly proportional to the absolute temperature (K) at constant pressure

    • V = constant x T (left graph), or

    • V/T = constant, or

    • V₁/V₂ = T₁/T₂  for conditions changing from 1 (initial) to 2 (final),

    • or V₁/T₁ = V₂/T₂  for constant pressure

    • V₂ = V₁ x T₂/T₁

    • or T₂ = T₁ x V₂/V₁

  • OR (ii) the pressure of a gas is directly proportional to the absolute temperature (K) at constant volume,

    • p = constant x T (right graph), or

    • p/T = constant, or

    • p₁/p₂ = T₁/T₂  for conditions changing from 1 (initial) to 2 (final),

    • or p₁/T₁ = p₂/T₂  for constant volume

    • p₂ = p₁ x T₂/T₁

    • or T₂ = T₁ x p₂/p₁

• In all calculations, the absolute or Kelvin scale of temperature must be used for T (K = ^oC + 273).

• If all the laws described in 4a and 4b are combined, you get the following general expression

• p x V/T = a constant (for a given mass of gas).

• This can be expressed in generalised form for calculations based on an initial set of conditions₁, changing to a new and final set of conditions₂ for a given mass of gas, giving the combined pressure-volume-temperature gas calculation equation ...

• p1 x V1		p2 x V2
  --------------	=	---------------
  T1		T2

• (p₁ x V₁)/T₁ = (p₂ x V₂)/T₂
• therefore the three permutations for problem solving are ...
• p₂ = (p₁ x V₁ x T₂)/(V₂ x T₁)
• or  V₂ = (p₁ x V₁ x T₂)/(p₂ x T₁)
• or  T₂ = (p₂ x V₂ x T₁)/(V₁ x T₂)

• Note:

  • If the temperature is constant you get Boyle's Law.

  • If p or V is constant you get Charles's/Gay-Lussac's Law.

  • You can use any volume or pressure units you like as long as both p's or both V's have the same units.

  • The graphs of p or V versus temperature become invalid once the gas has condensed into a liquid BUT when extrapolated back all the lines seem to originate from y = 0 (for p or V), x = -273^oC (for T).

  • This was part of the scientific evidence that led to the belief that -273^oC was the lowest possible temperature, though there is no theoretical upper limit at all.

  • This led to the devising of a new thermodynamic absolute temperature scale or Kelvin scale which starts at OK.

    • e.g. ice melts at 0^oC or 273K and water boils at 100^oC or 373K.

• Examples of P-V-T calculations

• Ex. Q4b.1

  • The pressure exerted by a gas in sealed container is 100kPa at 17^oC. It was found that the container might leak if the internal pressure exceeds 120kPa. Assuming constant volume, at what temperature in ^oC will the container start to leak?

  • 17^oC + 273 = 290K

  • p₁/T₁ =  p₂/T₂

  • rearranging to scale up to the higher temperature

  • T₂ = T₁ x p₂/p₁

  • T₂ = 290 x 120/100 = 348 K or 348 - 273 = 75^oC when the container might leak

• Ex. Q4b.2

  • A cylinder of propane gas at 20^oC exerted a pressure of 8.5 atmospheres. When exposed to sunlight it warmed up to 28^oC. What pressure does the container side now experience?

  • 20^oC = 273 + 20 = 293K, 28^oC = 273 + 28 = 301K

  • p₂ = p₁ x T₂/T₁

  • p₂ = 8.5 x 301/293 = 8.73 atm

• Ex. Q4b.3

  • A student was investigating the speed of reaction between limestone granules and different concentrations of hydrochloric acid. However after doing a whole series of experiments at different acid concentrations, there was no time to do the last planned experiment. The volume of carbon dioxide collected after 5 minutes in a 100cm³ gas syringe was used to determine the rate of reaction. All the experiments were done in one lesson at a temperature of 22^oC except for the last one. This was done in the next lesson, giving a carbon dioxide volume of 47.0 cm³ after 5 minutes, but at a higher temperature of 27^oC (when in Kelvin call this T₁, and the other temperature T₂).

  • To make the data analysis fair, all the gas volumes should be ideally measured at the same temperature, but a correction can be made for the last experiment.

  • (a) Calculate the volume the of 47.0 cm³ of gas at 27^oC,  would occupy at 22^oC.

    • V₁/V₂ = T₁/T₂ so V₂ = V₁ x T₂/T₁

    • V₁ = 47.0 cm³, T₁ = 273 + 27 = 300K, T₂ = 273 + 22 = 295K

    • V₂ = 47.0 x 295/300 = 46.2 cm³

  • (b) If the temperature was ignored, what is the % error in the rate of reaction measurement?

    • Volume error = 47.0 - 46.2 = +0.8 cm³, therefore ....

    • % error = 0.8 x 100/47 = +1.7% (so you would over calculate the reaction rate without this correction)

    • The % error in the volume would be the same as calculated for the rate e.g. in cm³/min.

  • (c) Should the calculated value for 22^oC be used in the rate calculation analysis? and are this still other sources of error?

    • The theoretical-calculated gas volume for 22^oC should be used for calculating the rate, it will improve the accuracy a little, BUT there is another problem!

    • If the reaction was unfortunately carried out at a higher temperature (i.e. 27^oC) there is a second source of error. At a higher temperature the reaction is faster, so you are bound to get a higher volume of gas formed in five minutes. Therefore you will calculate a faster rate of reaction e.g. in cm³ gas/minute at 27^oC, that would have occurred/been measured at 22^oC and so an unfair comparison with all the other results from the previous lesson.

    • So, although you can correct reasonably well the volume error due to an 'expanded' gas volume at the higher temperature, the gas volume will still be too high because of the faster rate of reaction at 27^oC and there isn't much you can do about that error except repeat the experiment at 22^oC, which is the best thing to do anyway!

      • Note that if the temperature of a rates experiment was too low compared to all the other experiments, the 'double error' would occur again, but this time the measured gas volume and the calculated speed/rate of reaction would be lower than expected.

  • (d) Would you need to do any correction for the volume of acid added to the limestone? Explain your decision.

    • No correction needed for this at all. Although liquids expand/contract on heating/cooling, the volume changes are far less compared to gas volume changes for the same temperature change. This is because of the relatively strong intermolecular forces between liquid molecules, which are almost absent in gases.

• Ex. Q4b.4

  • 25 cm³ of a gas at 1.01 atm. at 25^oC was compressed to 15 cm³ at 35^oC.

  • Calculate the final pressure of the gas.

  • p₁ = 1.01 atm, p₂ = ?, V₁ = 25 cm³, V₂ = 15 cm³,

  • T1 = 25 + 273 =  298 K, T2 = 35 + 273 = 308 K

  • (p₁ x V₁)/T₁ = (p₂ x V₂)/T₂

  • p₂ = (p₁ x V₁ x T₂)/(V₂ x T₁)

  • p₂ = (1.01 x 25 x 308)/(15 x 298) = 1.74 atm

4c. The Ideal Gas Equation of State pV = nRT

• A most 'compact molar' form of all the P-V-T equations is known as the ideal gas equation and is the simplest possible example of an 'equation of state' for gases (see also Van der Waals equation in section 5b). The explanation of the use of the word 'ideal' is explained in the 4. Introduction

• The equation is pV = nRT and requires a consistent set of units, so see below for the two most common examples, and take care!

• pressure p	volume V	n = mass g/Mr	Ideal gas constant R and its units	temperature T
  Pa (Pascal, 760mmHg = 1 atm = 101325 Pa)	m3 (1m3 = 106 cm3 or m3 = cm3/106)	mol	8.314 J mol-1 K-1	K (Kelvin = oC + 273)
  atm (atmospheres, 760 mmHg = 1 atm = 101325 Pa)	litre or dm3 (litre = dm3 = 1000 cm3 and dm3 = cm3/1000)	mol	0.08206 atm dm3 mol-1 K-1	K (Kelvin = oC + 273)

• The first set are becoming the 'norm' since they are the SI units, but the mass does not have to be in kg and can be in the more 'practical unit' of g as long as M_r is in g mol^-1.

• Examples of PV = nRT calculations

• Ex. Q4c.1

  • (a) Describe with the aid of a diagram a simple gas syringe method for determining the molecular mass of a volatile liquid.

    • 

    • A 100 cm³ gas syringe is mounted in an oven (ideally thermostated) but a humble bulb will do and the temperature is quite stable after an initial warming up period via internal convection. Some of the liquid (whose M_r is toe determined), is sucked into a fine 'hypodermic' syringe (e.g. 0.2cm³) and the syringe weighed. Quickly (to avoid evaporation losses), the liquid is injected into the gas syringe via a self-sealing rubber septum cap and the syringe re-weighed immediately. The difference in weighings gives the mass of liquid injected. When the gas volume has settled to its maximum value the volume is read (to the nearest 0.5cm³ if possible), together with the oven temperature and barometric pressure (mercury barometer for best accuracy i.e. in mmHg).

  • (b) In an experiment using the above apparatus the following data were recorded and the molecular mass of the volatile liquid calculated.

    • Mass of syringe + liquid = 10.6403 g

    • Mass of syringe after injection of liquid = 10.4227g

    • When volatilised the liquid gave 67.3 cm³ of gas.

    • The temperature of the oven = 81^oC, barometric pressure 752 mmHg.

    • Using the equation PV = nRT, calculate the molecular mass of the liquid.

      • R = 8.314 J mol^-1 K^-1

    • Mass of liquid injected = 10.6405 - 10.4227 = 0.2176 g

    • p = 101325 x 752/760 =  100258 Pa,

    • V = 67.3/10⁶ = 6.73 x 10^-5 m³, T = 273 + 81 = 354 K

    • PV = nRT, substituting for moles n gives PV = m/M_rRT

    • and then rearranging gives ...

    • M_r = mRT/PV

    • = (0.2176 x 8.314 x 354)/(100258 x 6.73 x 10^-5) = 94.9 (95 to 2sf)

  • (c) If the compound was formed from the reaction of bromine and a hydrocarbon, suggest a possible molecular formula for the compound.

    • bromomethane, CH₃Br, M_r = 12 + 3 + 80 = 95

  • (d) State very briefly, a method of determining the molecular mass of ANY compound that can be vapourised intact.

    • Mass spectrometer, from the molecular ion peak.

• Ex. Q4c.2

  • (a) What is the volume of 6g of chlorine at 27^oC and 101kPa (approx. 1 atm)?

    • pV = nRT, V = nRT/p

    • T = 273 + 27 = 300K,

    • n = 6/71 = 0.08451 mol chlorine, Mr(Cl₂) = 2 x 35.5 = 71

    • and p = 101 x 1000 = 101000 Pa.

    • V = 0.08451 x 8.314 x 300/101000 = 0.002087 m³

  • (b) What is the volume of the chlorine in dm³ and cm³?

    • 1 m³ = 1000 dm³ = 10⁶ cm³

    • V = 0.002087 x 1000 = 2.087 dm³

    • V = 0.002087 x 10⁶ = 2087 cm³

• Ex. Q4c.3

  • (a) A 5 litre container contained 0.5kg  of butane gas (C₄H₁₀). Assuming ideal gas behaviour calculate the pressure of the gas if the cylinder is stored at 25^oC.

    • M_r(C₄H₁₀) = (4 x 12) + 10 = 58, 0.5kg = 550g

    • moles of gas n = 500/58 = 8.621, T = 273 + 25 = 298K

    • R = 8.314 J mol^-1 K^-1 or 0.08206 atm litre mol^-1 K^-1

  • (i) SI units

    • PV = nRT, P = nRT/V, 5 litre = 5 dm³ = 5 x 10^-3 m³

    • P = 8.621 x 8.314 x 298/(5 x 10^-3)

    • P = 4271830 Pa = 4272 kPa = 4.272 MPa

  • (ii) 'old units'

    • P = 8.621 x 0.08206 x 298/5 = 42.16 atm

• For other gas calculations see Mole definition and Avogadro Constant, molar gas volume and reacting gas volume ratios.

4d. Dalton's law of partial pressures

• Dalton's Law of partial pressures states that at constant temperature the total pressure exerted by a mixture of gases in a definite volume is equal to the sum of the individual pressures which each gas would exert if it alone occupied the same total volume.

• For a mixture of gases 1, 2, 3 ... p_tot = p₁ + p₂ + p₃ ... where p₁, p₂ etc. represent the partial pressures.

• The partial pressure ratio is the same as the % by volume ratio and the same as the mole ratio of gases in the mixture.

• This means for a component gas z:

  • p_z = p_tot x %z/100 or

  • p_z = p_tot x mol z/total mol = p_tot x mol fraction z

• Examples of partial pressure calculations

• Ex. Q4d.1

  • In the manufacture of ammonia a mixture of nitrogen : hydrogen in a 1 : 3 ratio is passed over an iron/iron oxide catalyst at high temperature and high pressure.

    • N_2(g) + 3H_2(g) 2NH_3(g)

  • What are the partial pressures of nitrogen and hydrogen if the total pressure of the gases is 200 atm prior to reaction?

    • The 1 : 3, N₂ :H₂ ratio means that nitrogen forms ¹/₄ of the mixture, therefore

    • p_N2 = ¹/₄ x 200 = 50 atm and

    • p_H2 = p_tot - p_N2 = 150 atm (or from ³/₄ x 200)

• Ex. Q4d.2

  • Methanol can be synthesised by combining carbon monoxide and hydrogen in a 1 : 2 ratio.

    • CO_(g) + 2H_2(g) CH₃OH_(g)

  • In an experimental reactor experiment, 300^oC at a total pressure of 400kPa, the final equilibrium gaseous mixture contained 10% carbon monoxide.

  • (a) Calculate the % of hydrogen gas and % methanol vapour in the final mixture.

    • Whatever hydrogen is left, its % must be double that of carbon monoxide since they were both mixed and react in a 1 : 2 ratio, so there will 20% hydrogen in the equilibrium mixture.

    • Therefore there will be 100 - 10 - 20 = 70% methanol in the final mixture.

  • (b) Calculate the partial pressures of the three gases in the mixture.

    • p_CO = 0.1 x 400 = 40 kPa,  p_H2 = 0.2 x 400 = 80 kPa,  p_CH3OH = 0.7 x 400 = 280 kPa

  • (c) Calculate the value of the equilibrium constant, K_p, under these reaction conditions (use Pa pressure units).

    • Kp =	pCH3OH
      	-------
      	pCO pH22

    • Kp =	280000
      	--------------------- = 1.09 x 10-9 Pa-2
      	40000 x 800002

    • Note that although the equilibrium constant seems small for the 70% methanol, its to do with the relatively large numbers on the bottom line and a power of 2 as well.

• -

4e. Graham's law of diffusion

• Diffusion, or the 'self-spreading' of molecules, naturally arises out of their constant chaotic movement in all directions, though on a time average basis, more molecules will move in the direction of a region of lower concentration if such a situation exists e.g. initially 'pouring bromine vapour into air' in gas jars (see GCSE notes).

• Molecules of differing molecular mass diffuse at different rates.

• It has been shown that, assuming ideal gas behaviour and constant temperature, the relative rate of diffusion of a gas through porous materials or a mixture of gases or a tiny hole (effusion) is inversely proportional to the square root of its density.

• Since the density of an ideal gas is proportional to its molecular mass, the relative rate of diffusion of a gas is also inversely proportional to the square root of its molecular mass*.

  • * PV = nRT, PV = m/M_rRT, M_r = mRT/PV, since d = m/V, then M_r is proportional to density.

• r1		√d2		√M2
  ---	=	----	=	----
  r2		√d1		√M1

• Which is the mathematical ratio representation of Graham's law of diffusion for comparing two gases of different molecular masses.

• Graham's Law arises from the fact that the average kinetic energy** of gas particles is a constant for all gases at the same temperature.

  • **The formula for kinetic energy is KE = ¹/₂mu², where m = mass of particle, u = velocity. This means the average mu² is a constant for constant kinetic energy, so u is proportional to 1/√m and the m can be shown via the Avogadro Constant to be proportional to M_r, the molecular mass of the gas.

• Examples of diffusion rate calculations

• Ex. Q4e.1

  • 

  • Two cotton wool plugs are separately soaked in concentrated aqueous ammonia and hydrochloric acid solutions respectively and sealed in a long tube with rubber bungs.

  • Using a simple chemical equation and Graham's Law of diffusion, account for (a) the appearance of a 'white smoke ring' and (b) the fact the smoke ring occurs about ²/₃rds along from the ammonia end of the tube.

    • Atomic masses: N = 14, H = 1, Cl = 35.5

  • (a) The aqueous ammonia will give off ammonia fumes and the conc. hydrochloric acid gives off hydrogen chloride fumes which will diffuse down the tube towards each other. When the meet an acid base reaction gives fine crystals of the salt ammonium chloride.

    • NH_3(g) + HCl_(g) ==> NH₄Cl_(s)

  • (b) M_r(NH₃) = 17, M_r(HCl) = 36.5

  • If r is the relative rate of diffusion the following ratio applies,

  • rNH3		√MrHCl = √36.5		6.04
    ----	=	-------------	=	----
    rHCl		√MrNH3 = √17		4.12

  • and this shows that ammonia will diffuse about 50% faster than hydrogen chloride so the smoke ring will appear much nearer the HCl end of the tube.

• Ex. Q4e.2

  • Zeolites are silicate minerals that are porous at the molecular level and they are used as catalysts and 'molecular sieves' in the petrochemical industry in processes such as cracking and subsequent molecule separation.

  • (a) Calculate the relative rates of diffusion of pentane CH₃CH₂CH₂CH₂CH₃, hexane CH₃CH₂CH₂CH₂CH₂CH₃ and 2-methylpentane (CH₃)₂CH₂CH₂CH₂CH₃ into a zeolite mineral.

    • Atomic masses: C = 12, H = 1

    • Hexane and 2-methylpentane are structural isomers of C₆H₁₄ with the same molecular mass.

    • rC5H12		√MrC6H14 = √86		9.27
      -----	=	-------------	=	---- = 1.09
      rC6H14		√MrC5H12 = √72		8.49

    • Relative rate of diffusion is 1.09 : 1.00 for pentane : hexane/2-methylpentane

  • (b) In practice 2-methylpentane does not diffuse into the zeolite as fast as hexane or maybe not at all. Suggest a reason for this behaviour.

    • The 'methyl branching' in 2-methylpentane makes it a more bulky molecule that has greater difficulty fitting into zeolite minerals.

• Ex. Q4e.3

  • Enriching uranium means to increase the relative ratio of ²³⁵U/²³⁸U to produce uranium metal suitable for use as fuel rods in nuclear reactors. It is the ²³⁵U isotope that is very fissile (readily undergoes fission) but only occurs as a small % in uranium ores in which most uranium is the non-fissile ²³⁸U. To produce 'enriched' uranium metal it is first extracted by reduction from uranium oxide and then converted into gaseous uranium(VI) fluoride (uranium hexafluoride). The ²³⁵UF₆ is concentrated by a diffusion process in huge gas centrifuges before being converted back to uranium metal. Atomic mass of F = 19

  • (a) Calculate the relative rate of diffusion of the hexafluorides of the two uranium isotopes.

    • r235UF6		√Mr238UF6 = √352		18.76
      -----	=	-------------	=	----- = 1.004
      r238UF6		√Mr235UF6 = √349		18.68

  • (b) Suggest why the process must be repeated many times before enough enrichment has occurred.

    • For each diffusion 'run' only a very small enrichment occurs because of the similarity of the molecular masses of ²³⁵UF₆ and  ²³⁸UF₆ and hence the very similar rates of diffusion.

5. Non-ideal real gas behaviour and Van der Waal's Equation

5a. The deviations of a gases from ideal behaviour and their causes

• Certain postulates in the kinetic theory of gases (see 4.) are far from true in real gases, particularly at higher pressures and at 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 in 5c) 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 (V_molecules) is significant at high pressures i.e. the free space for random particle movement (V_ideal) is less than it appears from volume measurements.

    • V_real = V_ideal + V_molecules

    • 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 M_r).

  • 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 (p_ideal) by an amount (p_reduction).

    • p_real = p_ideal - p_reduction

    • 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 (see graph in 5c.) 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 p_ideal 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 -78^oC!

      • (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 5a. and 5c.

• 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.

5b. 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 (i) as

• (i)  (p + a') (V - b') = RT

• 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 in 5a.)

• 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 in 5a.)

• 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.

• Data	Van der Waals constants		critical values of the gas
  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 T_c

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

  • Critical pressure p_c

    • 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.

5c. 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 section 5a. 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 in 5a.
• 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

5d The Critical Point - The Critical Temperature and Critical Pressure

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 +ve charge x -ve charge divided by the distance squared (F α 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, 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 (T_c) 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 vapor of the substance cannot be liquefied, no matter how much pressure is applied.

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

• The vapor-liquid critical point denotes the conditions above which distinct liquid and gas phases do not exist. The point at the critical temperature and critical pressure is called the critical point of the substance.

• -

Website content copyright © Dr W P Brown 2000-2010 All rights reserved on revision notes, puzzles, quizzes, worksheets, x-words etc. * Copying of website material is not permitted * I do not personally endorse the adverts - but they do pay for the site!