Ciencia  y  Tecnología,  27  (1  y  2):  45-­‐‑52,  2011  
ISSN:  0378-­‐‑0524  
 
  THE  LEGACY  OF  VAN  DER  WAALS    
  
  J.  F.  Ogilvie*  
  
Escuela  de  Quimica  y  CELEQ,  Universidad  de  Costa  Rica,  Ciudad  Universitaria  Rodrigo  Facio,  San  Pedro  de  
Montes  de  Oca,  San  Jose  11501-­‐‑2060,  Costa  Rica;  ogilvie@cecm.sfu.ca    
  
Recibido  17  de  marzo,  2011;  Aceptado:  10  de  diciembre,  2011  
  
Abstract  
  
The   physical   implications   of   the   form   and   nature   of   the   van   der   Waals  
equation   of   state   are   explained.   The   relation   of   this   equation   to   so-­‐‑called  
van  der  Waals   forces  and  radii   is  discussed.  The  conclusion   is  drawn  that  
the   van   der   Waals   equation   of   state   is   practically   worthless   and   its   use  
should  be  abandoned.  
  
Resumen  
  
Se   explican   las   implicaciones   físicas   de   la   forma   y   la   naturaleza   de   la  
ecuación  de  estado  de  van  der  Waals.  Además,  se  discute  la  relación  entre  
esta  ecuación  y   las   llamadas   fuerzas  y   radios  van  der  Waals.  Se   llega  a   la  
conclusión  de  que  la  ecuación  de  estado  de  van  der  Waals  no  es  útil  y  que  
no  debería  utilizarse.  
  
  
Key  words:   equation   of   state,   corresponding   states,   intermolecular   forces,  
weakly  bound  molecules,  van  der  Waals.  
  
Palabras   claves:   ecuación   de   estado,   fuerzas   intermoleculares,   van   der  
Waals.  
  
  
  
  
  
  
  
                                                
*  Corresponding  author:  ogilvie@cecm.sfu.ca  
J.F. OGILVIE 
Ciencia  y  Tecnología,  27(1  y  2):  45-­‐‑52,  2011  –  ISSN:  0378-­‐‑0524  
 
46 
  
   On   the   occasion   in   2010   of   the   centenary   of   Johannes   Diderik   van   der  Waals  
(1837-­‐‑1923)  being  awarded  Nobel   laureate   in  physics,  Tang  and  Toennies  published  a  
paean   of   praise   [1]   that   astonishingly   proclaimed   that   Dutchman’s   achievement   of  
allegedly   “triggering   a   revolution   in   the   understanding   of   the   molecular   physics   of  
liquids,   gases   and   their   mixtures,   and   laying   the   foundations   of   modern  
thermodynamics  and  statistical  mechanics”.     Although  van  der  Waals  might  certainly  
have  exerted  a  formative  influence  on  Dutch  science  and  scientists  during  his  life,  and  is  
undoubtedly   mildly   important   as   an   historical   figure   in   physics,   the   validity   of  
association   of   some   concepts   with   his   name   must   be   challenged:   that   objective,   a  
critique  of  his  equation  of  state  and  the  validity  of   the   term  ‘van  der  Waals  molecule’  
constitute  the  purpose  of  this  essay.    
  
   Despite  the  cautions  that  from  time  to  time  appear  in  the  literature  emphasizing  
not   to  attach   too  great  significance   to  conclusions  drawn  from  its  application,   the  van  
der  Waals  equation  of  state  maintains  a  remarkably  resilient  position  in  many  textbooks  
of  general  and  physical  chemistry.  In  part  following  Duclaux  [2],  we  present  here  some  
considerations   of   this   equation   and   its   implications,   after   reviewing   the   historical  
development  and  related  background.    
  
   One  seeks   to  specify  completely   the  state  of  unit  quantity   (one  mole)  of  a   fluid  
according  to  some  characteristic  equation,    
 
   f(p,  Vm,  T)    =    0,       (1) 
 
in  which  p  =  pressure,  Vm  =  volume  per  mole  and  T  =  absolute  temperature.  According  
to  the  kinetic-­‐‑molecular  theory  of  gases,  the  ideal  gas  law,  
 
   p  Vm  /  T    =    constant    =    R,        (2) 
    
is  derivable.  To  the  extent  –  considerable  excepting  extreme  conditions  of  pressure  and  
temperature   –   that   a   real   gas   has   properties   resembling   those   of   an   ideal   gas,   this  
equation   is   useful.   One   pursues,   however,   a   more   accurate   representation   of   the  
properties   of   real   fluids   to   take   account   of   the   possibility   of   universal   liquefaction   at  
sufficiently   small   temperatures,   and   other   deviations   at   large   pressures.   The   first  
attempt  to  allow  for  non-­‐‑ideal  behaviour  was  made  by  Hirn  (1867)  who  wrote  
 
     p  (Vm  −  b)    =  R  T,                     (3)    
 
in  which  b   is  a  characteristic  parameter  called   the  covolume.   In  his  dissertation  at   the  
University  of  Leyden  (1873),  van  der  Waals  introduced  a  second  correction  to  obtain    
The  legacy  of  van  der  Waals 
Ciencia  y  Tecnología,  27(1  y  2):  45-­‐‑52,  2011  –  ISSN:  0378-­‐‑0524  
 
47 
 
   (p    +  a  /Vm2  )  (Vm  − b)  =  R  T                  (4)  
 
The  correction  a   /Vm2  was  intended  to  allow  for  attractive  forces  between  molecules   in  
the   gaseous   sample.   Direct   experimental   evidence   of   such   attractive   forces   between  
gaseous  molecules  had  been  previously  obtained  and  recognised  by  Joule  and  Thomson  
(Lord  Kelvin)  as  a  result  of  their  ‘porous-­‐‑plug’  experiment,  1853.    
  
   Interactions   of   three   classes      –      electrostatic,   inductive   and   dispersive   –      arise  
between  simple  molecules  at  sufficiently  large  separations  [3];  orientational  factors  are  
neglected.  The  Keesom  interaction,  an  attraction   that  arises  between   two  molecules  of  
separation  R  each  having  permanent  electric  dipolar  moment  µμ,  has  this  energy,    
 
   EK    =  − 2  µμ4  /  (4π  ε0  )2  (3  R6  k  T  )     (5) 
 
A  molecule  with   a   permanent   electric   dipolar  moment   invariably   induces   an   electric  
dipolar  moment  in  a  neighbouring  non-­‐‑polar  molecule  of  polarizability  α,  producing  an  
attraction  known  as  the  Debye  interaction,  equal  to    
 
                                    ED    =  −  α  µμ2  /  (4  π  ε0)2  R6         (6) 
 
Another   interaction,  which  arises   from   fluctuating   instantaneous  dipoles   in  otherwise  
non-­‐‑polar  molecules,  has  the  London  energy,    
 
   EL  =  − 3 I α2 / (4 π ε0)4 (4 R6 )     (7) 
 
in  which  the  like  molecules  have  first  ionization  energy  I  and  polarizability  α.  All  these  
interactions  have  energies  that  signify  attraction  by  their  negative  signs,  and  that  are  of  
appreciable  magnitude   for  only  small   separations,  as   indicated  by   the  dependence  R-­‐‑6  
on  separation.  At  even  smaller  separations,   repulsion  becomes  more   important.  These  
two  effects  are   taken   into  account   in  a  model   function   for  potential  energy  associated  
with  Lennard-­‐‑Jones,  
 
   V(R)  =  4 ε [ (σ/R)12  −  (σ/R)6 ] ;     (8) 
          
in  this  equation  ε  is  the  binding  energy  or  depth  of  the  well  of  potential  energy,  and  σ  is  
the   distance   of   closest   approach   or   molecular   diameter.   As   force   is   the   gradient   of  
energy,   the   force  of  attraction  has   then  a  dependence  R-­‐‑7   appropriate   to   the  attraction  
energy  as  R-­‐‑6.  With  this  background,  we  proceed  to  explore  the  implications  of  the  van  
der  Waals  equation  of  state.    
  
J.F. OGILVIE 
Ciencia  y  Tecnología,  27(1  y  2):  45-­‐‑52,  2011  –  ISSN:  0378-­‐‑0524  
 
48 
   We  examine  first  the  pressure  term.  The  correction  a/V  2  is  obtainable  [2]  on  being  
guided  by  no  theory  whatsoever,  but  merely  on  developing  a  series  in  negative  powers  
of  Vm  and  terminating  with  Vm-­‐‑2,  
 
   p  =  R T / (Vm − b)  −  a / Vm2      (9) 
 
What   is   the   physical   significance   of   a?  As   expressed   by  Moelwyn-­‐‑Hughes   [4]   on   the  
basis  of  a  dimensional  argument,  “the  molecular  model  with  which  the  equation  of  van  
der  Waals   is   consistent   is   that   of   a   system   of   incompressible   spheres,   of   diameter   σ,  
which   attract   each   other   with   a   force   varying   inversely   as   the   fourth   power   of   their  
distance  apart”.    As  we  see  above,  the  various  interactions  as  forces  that  arise  between  
non-­‐‑polar  or  weakly  polar  molecules  all  have  dependences  according  to  R-­‐‑7,  not  R-­‐‑4,  on  
separation.     As   attractive   forces  were   not   discovered   by   van  der  Waals,   and   as   these  
forces   according   to  his   equation   lack   the   correct  dependence  upon  R,   it   consequently  
seems   inappropriate   to   associate   collectively   the   Keesom,   Debye   and   London  
interactions  with  van  der  Waals.  We   can,   for   the   same   reason,   expect   correction   term  
a/Vm2   to   yield   inaccurate   results   of   calculations.   The   typical   quasi-­‐‑derivation   of   this  
correction  is,  in  any  case,  most  dissatisfying,  as  the  derivation  implies  that  the  pressure  
near  a  boundary  of  the  gas  is  less  than  in  the  bulk  because  of  unbalanced  forces  tending  
to  pull  back  the  molecule  from  the  boundary  [5].  In  any  real  situation,  the  boundary  is  
constituted  of  a  dense  material,  on  the  surface  of  which  there  are   invariably  adsorbed  
layers  of  molecules  of  the  gas  providing  that  sufficient  sample  is  available;  in  that  case  
the   pressure   must   increase   toward   the   boundary,   relative   to   the   bulk   gas.   The  
adsorption   of   part   of   the   gas  would   cause   a   bulk   pressure   decreased   by   comparison  
with   an   hypothetical   ideal   gas   that  were   not   adsorbed.   This   effect  might   perhaps   be  
encompassed,   or   avoided,   in   an   alternative   derivation   of   the   corrective   term   for  
pressure,   but   its   presence   until   that   achievement  makes   the   offered   derivations   seem  
spurious.    
  
   Regarding  covolume  parameter  b,  its  significance  is  that  it  represents  the  average  
forbidden  molar  volume  [4]  of  molecules  for  which  only  binary  collisions  are  important.  
The  magnitude  of  b  is  thus  four  times  the  intrinsic  volume  of  a  mole  of  incompressible  
spheres.  The  circumscribed  volume  of  a  collection  of  close-­‐‑packed  spheres  is,  however,  
1.350  times  the  intrinsic  volume  of  the  spheres.  It  is  thus  entirely  feasible  for  the  volume  
of  a  fluid  during  compression  to  approach,  to  become  equal  to,  and  then  to  become  less  
than  b  that  is  almost  three  times  the  minimum  attainable  volume,  i.e.  the  circumscribed  
volume  of  close-­‐‑packed  spheres.  Under  these  conditions,  it  would  be  necessary  to  admit  
of   negative   pressures,   positive   and   negative   infinities   of   pressure,   and   even   zero  
pressure  during  compression,  as  Duclaux  has  deduced  [2].  van  der  Waals,  apparently  
aware  of  the  first  consequence,  above,  of  his  equation,  justified  it  by  stating  that  a  liquid  
The  legacy  of  van  der  Waals 
Ciencia  y  Tecnología,  27(1  y  2):  45-­‐‑52,  2011  –  ISSN:  0378-­‐‑0524  
 
49 
can   support   a   negative   pressure:   this   phenomenon   has   never   been   observed.   These  
difficulties,   and  others,   arise  because  of   the  nature  of   the   subtractive   correction  b,   the  
covolume,   which   really   regresses   to   Hirn'ʹs   formulation.   A   covolume   is   a   common  
feature   of   three   state   equations   containing   two   parameters,   those   of   van   der  Waals,  
Berthelot  and  Dieterici.     To  prevent  absurd  consequences  resulting  from  the  covolume  
correction,  it  is  merely  necessary  to  define  b  to  be  the  circumscribed  volume  of  the  close-­‐‑
packed  collection  of  molecules,   in  which  case  for  pressure  to  tend  to   infinity  when  Vm  
approaches  b   is  physically  acceptable.     The  effects  of   ternary  collisions,  which  become  
significant   at   large   pressures,   and   consequently   the   effects   of   conditions   at   large  
pressures  or  densities,  are  formally  neglected  in  these  equations  of  state.  
  
   The   attraction   of   the   van   der   Waals   theory   is   such   that   the   magnitudes   of  
parameters  a  and  b  of  his  equation  of  state  are  considered  constant,  even  though  that  at  
least   a   varies   with   temperature   has   been   long   known   (Clausius,   1880).   To   test   the  
constancy   by   numerical   calculation   is   easy:     Duclaux   [2]   presented   data   that   indicate  
that   both   a   and   b   for   non-­‐‑polar  molecules   dinitrogen   and   dioxygen   vary   by   about   a  
factor  two  for  a  temperature  range  of  a  similar  factor  up  to  300  K;  a  and  b  thus  vary  in  
inverse   proportion   to   the   absolute   temperature.   This   effect   is   expected   as   the  
resemblance   of   these   gases   to   an   hypothetical   ideal   gas   must   increase   as   the  
temperature  increases  far  above  the  critical  temperature.  In  compilations  of  quantities  a  
and  b  for  various  substances,  for  instance  in  editions  of  standard  handbooks  [6],  there  is  
indication  of  neither   the   temperature   for  which   the  stated  values  are  appropriate,  nor  
even   that   a   dependence   on   temperature   exists;   users   are   thus   exposed   to   significant  
error.  For  instance,  for  dioxygen  at  15  MPa  and  423  K  -­‐‑-­‐‑    neither  extreme  conditions  nor  
near  the  critical  point,  with  standard  values  of  a  and  b  the  correction,  relative  to  an  ideal  
gas,  for  pressure  even  has  an  incorrect  sign.      
  
   As   the   covolume   parameter   is   generally   taken   to   be   related   to  molecular   size,  
further  complications  appear.  According   to  certain  questionable  assumptions,  one  can  
pragmatically  derive  molecular  radii,  or  radii  of  atoms  or  groups  of  atoms  in  molecules;  
compilations  of   so-­‐‑called  van  der  Waals   radii   exist   [7].   In   the   singular,   the   specifying  
term   implies   an   effective   radius   constituting   half   the   distance   of   closest   approach  
between   like   atoms   deemed   not   covalently   bonded.   One   might   attempt   formally   to  
calculate  these  radii  from  covolume  parameter  b,  but  to  consider  molecules  to  become  
smaller   as   temperature   increases   would   be   then   necessary.   In   fact,   just   the   reverse  
occurs,   as   molecules   enlarge   slightly   with   increasing   temperature   because   the  
occupancy   of   excited   vibrational   states   becomes   thermally   enhanced,   with  
correspondingly  increased  mean  amplitudes  of  oscillation.  Above  the  dissociation  limit  
of  a  diatomic  molecule,  the  effective  radius  of  each  separate  atomic  fragment  decreases  
with   increasing   temperature,   but,   for   two   interacting   strongly   bound   diatomic  
OGILVIE: The legacy of van der Waals 
 
Ciencia y Ciencia y Tecnología, 27(1 y 2): 45-52, 2011 – ISSN: 0378-0524 
 
50 
molecules   within   a   weakly   bound   tetratomic   molecule,   the   effective   radius   of   each  
fragment  again  increases  with  increasing  temperature.  The  association  of  van  der  Waals  
with   these  non-­‐‑bonded  radii   is  hence  unjustified;  neither  his  equation  of  state  nor   the  
principle  of   corresponding  states  provides  direct  evidence   for   the  existence  of  weakly  
bound  molecules,   which   are   adequately   characterised   in   contemporary   spectrometric  
experiments.    The  association  of  van  der  Waals  with  these  radii  because  of  the  attractive  
(and   repulsive)   forces,  which  are   responsible   for   the  proximity  with  which  molecules  
can  approach  one  another,   is   equally   spurious.     Various  weakly  bound  molecules  are  
characterised,  with  4He2  and  Ar2  as  diatomic  instances  involving  atomic  fragments,  and  
(HF)2   involving  diatomic   fragments;   their  properties  and  characteristics  are  so  diverse  
as   to   lack  much   in   common  other   than   their   small   energies  of  dissociation   into  either  
atoms   or   strongly   bound  molecules.      As   the   energies   of   dissociation   of   the   strongest  
‘hydrogen  bonds’  are  not  much  less   than  those  of  bonds  considered  ordinary,  such  as  
that  of  diiodine,  I2  ,  in  its  electronic  ground  state,  there  is  nearly  a  continuous  gradation  
of  bonds  from  the  weakest,  in  4He2,  to  the  strongest,  in  CO,  for  diatomic  molecules  for  
instance.      There  has   been  proffered  no   suggestion   that   van  der  Waals   ever  discussed  
such  a  diversity,  nor  that  his  understanding  of  their  molecular  physics  exceeded  that  of  
Joule   and   Kelvin   who   originated   an   experimental   proof   of   the   existence   of  
intermolecular   forces   in   a   gaseous   context;   it   is   significant   that   the   Joule-­‐‑Thomson  
coefficient,   (∂T/∂p)H,   is   the   only   thermodynamic   function   of   a   real   gas   that   fails   to  
approach  the  corresponding  function  of  an  ideal  gas  as  the  gaseous  density  approaches  
zero.  
 
 Rowlinson  stated  [2]  in  1958,  “The  best  known  equation  of  state  ....  is  that  of  van  
der  Waals,   and   for  many  years   the  principle   [of   corresponding   states]  was   associated  
solely   with   this   equation”.      According   to   this   principle,   pressure,   volume   and  
temperature  are  expressed  in  terms  of  ratios  with  their  respective  values  at  the  critical  
point,  so  pr  =  p  /  pc,  Vr  =  Vm  /  Vm  c  and  Tr  =  T  /  Tc  ,  and  the  compression  factor  Z  =  P  Vm  /RT  
is  the  same  for  all  gases,  but  differing  markedly  from  unity  –  becoming  as  small  as  0.2  
near   the   critical   temperature   [8].      Guggenheim   stated   [2]   in   1945,   “The   principle   of  
corresponding  states  may  safely  be  regarded  as  the  most  useful  by-­‐‑product  of  the  van  
der  Waals  equation  of  state.  Whereas  this  equation  of  state   is  nowadays  recognised  to  
be   of   little   or   no   value,   the   principle   of   corresponding   states,   correctly   applied,   is  
extremely   useful   and   remarkably   accurate”.   Contrary   to   popular   belief,   the   van   der  
Waals  equation  of  state  fails  to  establish  the  physical  reality  of  corresponding  states  [2],  
because  it  leads  to  invalid  mathematical  definitions  instead  of  physical  definitions.  The  
mathematical   definitions   are   contained   in   these   well   known   relations   for   the   critical  
parameters,  obtained  on  differentiation:    
  
         Vm  c    =    3  b,                              (10)  
Ciencia  y  Tecnología,  27  (1  y  2):  45-­‐‑52,  2011  
ISSN:  0378-­‐‑0524  
 
Ciencia y Ciencia y Tecnología, 27(1 y 2): 45-52, 2011 – ISSN: 0378-0524 51 
         Tc    =    8  a  /  (27  b  R),     
         pc    =    a  /  (27  b2  )  
  
Here   is  exposed  an   internal  contradiction  of   the  van  der  Waals  equation:  whereas   the  
minimum  circumscribed  volume  of  a  collection  of  spheres  is  0.337  b,  the  critical  volume  
is  nine  times  as  great,  an  unreasonably  large  factor.  These  relations  would  be  acceptable  
if   parameters   a   and   b   were   independent   of   temperature.   Even   when   a   and   b   for   a  
temperature  slightly  greater  than  the  critical  point  are  inserted  into  these  relations.  the  
resulting  critical  parameters  for  dinitrogen  are  inaccurate  [2].  If  this  ‘law’  were  true,  the  
critical  ratio,  
  
         pc  Vm  c  /  R  Tc    =    3/8                     (11)  
  
would   be   the   same   for   all   gases,   whereas   this   ratio   varies   between   1/5   and   1/3   for  
common  gases  [8].    In  any  case,  all  equations  of  third  degree  in  V,  as  is  that  of  van  der  
Waals,   yield,   if   they  deviate  not   too   severely   from  experience,   a  mathematical   critical  
point.  All  equations  with  two  parameters,  such  as  a  and  b,  yield  reduced  variables  and  
corresponding  states.  There  is  hence  no  special  virtue  of  the  van  der  Waals  equation  as  
far  as  corresponding  states  are  concerned:   it   leads  to  nonsensical  situations  because  of  
the  abstract  nature  of  critical  parameters  determined  from  a  and  b.    
  
   Barker  and  Henderson  stated  [9],  "ʺIt   is  a  tribute  to  the  insight  of  van  der  Waals  
that,   in   the   nineteenth   century,   he   was   able   to   give   an   essentially   correct   physical  
description  of  the  liquid  state,  and  that  much  of  the  recent  progress  has  been  to  put  his  
ideas  on  a   rigorous  mathematical  basis"ʺ.     They   then  proceeded   to  derive   some  useful  
results  with  which  to  prove  their  point  [9],  but  they  achieved  these  results  by  imposing  
on   their   systems   the   6-­‐‑12  potential-­‐‑energy   function  of  Lennard-­‐‑Jones,   as   in   formula   8  
above,  whereas  the  van  der  Waals  equation  is  consistent  with  an  inverse  fourth-­‐‑power  
attractive   force   [4].   This   approach   seems   to   be   an   instance   of   what   Duclaux   [2]  
contended  when   he   stated   that   a  mystical   atmosphere   surrounds   the   van   der  Waals  
theory  such  that  men'ʹs  eyes  are  closed  to   the  manifest  errors  and   inadequacies  of   this  
equation  of  state.  As   it  expresses  or  verifies  no  single   theory,  but   is   instead  consistent  
with  many  theories,  it  is  possible  to  read  into  this  equation  diverse  properties  and  hence  
to  be  misled  as  to  its  significance.    
  
   In  summary,  because  of   its  demonstrated   faults,   the  van  der  Waals  equation  of  
state,   for   both   practical   and   pedagogical   purposes,   “is   of   little   or   no   use”,   after  
Guggenheim.   It   can   of   course   be   used   merely   as   an   interpolation   formula   for  
moderately   accurate   calculations   of   pressure   within   a   limited   range   at   a   particular  
temperature   for   which   parameters   a   and   b   are   optimal,   but,   for   this   purpose,   other  
OGILVIE: The legacy of van der Waals 
 
Ciencia y Ciencia y Tecnología, 27(1 y 2): 45-52, 2011 – ISSN: 0378-0524 
 
52 
empirical   equations   of   state   yield   superior   results;   for   instance   in   the   vicinity   of   the  
critical  point,  the  Dieterici  equation  of  state  is  remarkably  accurate  and  “is  the  best  all-­‐‑
round   analytic   two-­‐‑constant   equation   of   state”   [3].   Never   having   had   any   sound  
theoretical   foundation,   the   van   der   Waals   equation   must   be   considered   only   an  
historical  relic.   In  agreement  with  Duclaux  [2],  we  conclude  that  this  equation  of  state  
can   be   entirely   abandoned,   without   regret.      During   the   past   sixty   years,   numerous  
papers   on   the   van   der   Waals   equation   of   state,   mostly   alleging   some   pedagogical  
significance,  appeared  but  can  likewise  be  ignored  as  mere  distraction.    The  association  
of   weakly   bound   molecules   and   of   forces   with   dependence   R-­‐‑7   on   intermolecular  
distance  with  the  name  van  der  Waals  is  equally  unjustifiable  and  worthy  of  abandon.    
Tang  and  Toennies  [1]  noted  that  the  number  of  publications  that  mention  the  name  van  
der  Waals  in  the  title,  abstract  or  key  words  is  currently  about  2000  per  annum;  in  their  
vast  majority   these  papers  refer  not   to   the  worthless  equation  of  state  nor  even  to   the  
separate   and   worthy   principle   of   corresponding   states,   but   rather   to   van   der   Waals  
forces  or  molecules  for  which  there  is  at  most  only  a  superficial  basis  to  associate  with  
van  der  Waals  the  physicist.    
 
  
References    
  
1   Tang,  K.-­‐‑P.;  Toennies,   J.   P.,  Angewandte  Chemie   (international   edition),  2010,  49,  
9574  –  9579    
2   Duclaux,  J.,  Journal  de  Chimie  Physique,  1967,  64,  1614–1678    
3   Hirschfelder,  J.  O.;  Curtiss,  C.  F.;  Bird,  R.  B.,  Molecular  Theory  of  Gases  and  Liquids,  
Wiley:  New  York,  NY  USA,  1961,  p.  987  -­‐‑  990.  
4   Moelwyn-­‐‑Hughes,  E.  A.,  Physical  Chemistry,  2a  ed.,  Pergamon:  Oxford,  UK,  1964,  
p.  594  
5   Tyler,  F.,  Intermediate  Heat,  Arnold:  London,  UK,  1955,  p.  127  
6   for   instance,   Handbook   of   Chemistry   and   Physics,   forty-­‐‑fifth   edition,   Chemical  
Rubber  Publishing  Co.,  Cleveland,  OH  USA,  1964;  Handbook   of  Chemistry,   tenth  
edition,   McGraw-­‐‑Hill,   New   York,   NY   USA,   1961;  American   Institute   of   Physics  
Handbook,   second   edition,   McGraw-­‐‑Hill,   New   York,   NY   USA,   1963,   and   their  
subsequent  editions  
7   for   instance,   Harvey,   K.   B.;   Porter,   G.   B.,   Introduction   to   Physical   Inorganic  
Chemistry,  Addison-­‐‑Wesley:    Reading,  PA  USA,  1963,  p.  245-­‐‑247    
8   Moore,  W.  J.,  Physical  Chemistry,  fourth  edition,  Prentice-­‐‑Hall,  Englewood  Cliffs,  
NJ  USA,  1972,  p.  21  
9   Barker,  J.  A.;  Henderson,  D.,  Journal  of  Chemical  Education,  1968,  45,  2  –  5