THERMODYNAMICS 3 laws, Study Guides, Projects, Research of Thermodynamics

Download THERMODYNAMICS 3 laws and more Study Guides, Projects, Research Thermodynamics in PDF only on Docsity!   1   THERMODYNAMICS.  An  alternative  to  the  textbook  version.     By   Håkan  Wennerström,  Division  of  Physical  Chemistry,  Department  of  Chemistry,  Lund   University,  Sweden     Abstract   We  present  an  alternative  description  of  the  basic  aspects  of  the  thermodynamic  theory.   The  purpose  is  firstly  to  highlight  the  assumptions  about  the  physical  reality  that  one   makes  in  the  theory  and  secondly  to  describe  the  connection  to  other  physical  theories   like  (quantum)  mechanics  and  statistical  mechanics.  The  most  essential  parts  of  the   treatment  is:  to  consider  the  concept  of  equilibrium  as  an  explicit  fundamental  part  of   the  theory;  to  use  the  equation  of  state  as  a  central  theoretical  concept;  consider  the   “first  law”  of  the  conventional  theory  as  primarily  a  definition  of  heat;  to  eliminate  from   the  theory  components  like  “quasi-­‐static  processes  “  and  the  “zeroth  law”  that  have  their     basis  in  operationalism.    The  theory  is  described  in  terms  of  three  basic  components;   “fundamental  concepts”,  “propositions”  (German  “ansatz”)  and  “laws”         2   Introduction   Thermodynamics  was  developed  during  the  19th  century  with  Carnot  as  the  most   important  pioneer.  The  theory  was  improved  and  put  into  a  self-­‐contained  unit  by   Clausius,  Helmholtz  and  several  others.  Towards  the  end  of  the  century  important   contribution  ware  made  by  Gibbs,  who,  for  example,  described  the  important  coupling   between  macroscopic  perspective  of  thermodynamics  and  the  microscopic  atomistic   description  of  matter  that  developed  into  statistical  mechanics.  The  theory  established   by  1900  was  based  on  two  basic  laws;  the  first  law,  which  prescribes  the  energy  to  be   constant  and  the  second  law  that  states  what  processes  are  “spontaneous”.  During  the   early  1900s  to  additional  laws  entered  the  theory.  The  third  law  stated  that  the  entropy   was  zero  at  the  zero  point  for  the  temperature  and  the  zeroth  law  gives  a  statement   about  the  equilibria  between  three  separate  systems  (If  A  is  in  equilibrium  with  B  and  B   is  in  equilibrium  with  C  then  A  and  C  are  also  in  equilibrium).     The  way  the  thermodynamic  theory  is  built  has  occasionally  been  presented  as  an  model   for  how  physical  theories  preferably  should  be  constructed.  It  is  not  based  on   quantitative  equations,  but  rather  on  more  general  principles  that  can  be  communicated   through  the  ordinary  language.  The  theory  appears  as  “simple”  by  this  criterion.   However,  teachers  know  that  it  is  difficult  to  get  the  theory  across  to  the  students.  It  is  a   common  remark  by  experienced  scientists  that  they  came  to  “understand”   thermodynamics  first  when  they  had  penetrated  the  statistical  mechanical  theory.  One   reason  why,  in  my  opinion,  it  is  difficult  to  teach  thermodynamics  is  that  the   conventional  textbook  version  of  the  theory  contains  logical  flaws  at  the  same  time  as  it   has  pretentions  of  being  formally  strict.     In  currently  used  elementary  thermodynamics  textbooks  one  usually  proceeds  in  line   with  the  classical  formulations  of  the  theory  in  terms  of  the  first,  second  and  usually   third  law.  However,  there  is  a  modern  discussion  of  the  topic  that  contains  a  critique  of   the  classical  theory.  This  critique  is  typically  based  on  two  different  circumstances.  The   most  obvious  is  that  our  understanding  of  the  microscopic,  atomistic,  world  has  changed   qualitatively  during  the  last  hundred  years.  It  then  seems  reasonable,  if  for  no  other   reasons  than  didactic,  to  encompass  this  understanding  also  into  the  thermodynamic   theory.  The  theory  is  based  on  a  macroscopic  perspective,  but  it  is  more  often  than  not     5   such  conflicts  are  typically  solved  through  compromises.  In  undergraduate  teaching   there  are  good  pedagogical  reasons  to  avoid  the  difficulties  that  are  created  by  these   compromises.  However,  a  big  danger  with  this  approach  is  that  students,  who  proceeds   to  a  research  career  brings  with  them  too  much  of  indoctrination  and  too  little  of  a   critical  analysis  of  the  difficulties  that  are  always  present.  One  important  purpose  with   this  text  is  to  unravel  some  of  the  difficulties  that  might  have  been  hidden  and  as  far  as   possible  provide  suggestions  of  how  these  could  be  handled.   It  is  of  considerable  help  to  specify  the  conceptual  framework  when  analyzing  the   foundations  of  a  theory  like  thermodynamics.  Below  we  will  discuss  the  theory  as  being   built  from  three  basic  ingredients.  These  are  central  concepts,  essential  propositions  and   basic  laws.  To  do  so  is  a  subjective  choice.  Central  concepts  are  quantities  that  are  there   to  connect  the  theory  with  the  observable  physical  reality.  The  concepts  can  be  more  or   less  useful,  but  they  are  not  true  or  false.  An  essential  proposition  is  a  condition  we  put   on  the  theory.  Typically  propositions  are  formal  requirements  on  the  theory.    Some  are   tacit,  like  the  one  that  the  laws  of  logic  and  mathematics,  are  obeyed  by  all  physical   theories.  Below  we  only  discuss  propositions  that  are  reasonably  specific  for   thermodynamics.  Like  the  concepts  basic  propositions  should  not  be  considered  true  or   false.  The  basic  laws  constitute  the  heart  of  the  theory.  They  express  relations  that  are   not  obvious  and  sometimes  unexpected.  Their  value  is  judged  by  comparing  empirical   observations  with  theoretical  predictions.  In  this  sense  the  laws  can  be  considered  true   or  false.  However,  it  is  important  to  realize  observations  are  influenced  by  theory  so  it  is   in  the  end  a  complex  matter  to  decide  whether  or  not  a  theory  is  compatible  with  certain   observation.  With  the  strategy  clearly  stated  the  next  step  is  to  explicitly  describe  the   thermodynamic  theory.     The  foundations  of  thermodynamics   Classical  mechanics  predates  thermodynamics  by  approximately  two  hundred  years.   From  a  current  viewpoint  one  can  say  that  also  quantum  mechanics,  statistical   mechanics,  electromagnetism  and  the  theory  of  relativity  is  a  part  of  the  general   framework  of  physical  theories  within  thermodynamics  also  plays  a  role.  The  purpose  of   thermodynamics  is  not  to  replace  these  theories  but  to  provide  an  alternative  way  of   describing  and  understanding  reality.  The  robustness  of  thermodynamics  can  be  seen   from  the  fact  that  the  theory  survived  intact  during  the  transition  from  classical     6   mechanics  to  quantum  mechanics/theory  of  relativity  that  occurred  during  the  first   decades  of  the  20th  century.  Thermodynamics  is  based  on  mechanics  and  it  includes  all   the  quantities/concepts  of  mechanics  that  can  be  applied  to  macroscopic  systems.  Such   quantities  include  amounts  of  matter,  volume,  pressure,  work,  and  energy.  The   usefulness  of  the  theory  is  then  accomplished  by  introducing  quantities  specific  to   thermodynamics.  Typical  examples  are  temperature,  entropy,  heat,  and  free  energy.   These  latter  are  specified  within  the  theory,  while  mechanical  quantities  are  considered   as  known.  The  “system”  is  a  central  concept  within  thermodynamics.  It  refers  primarily   to  a  (macroscopic)  part  of  reality  specified  by  quantities  known  from  mechanics   (amounts,  volume,  pressure,  energy).  In  the  developed  theory  systems  can  also  be  partly   specified  using  also  quantites  specific  to  thermodynamics.  We  now  have  the  background   to  present  the  theory  in  a  formalized  way:     Law  1  (Equilibrium  law):  For  a  given  macroscopically  specified  system  there  exists   a  state  of  equilibrium  with  unique  values  of  all  macroscopic  quantities.  All   systems  adopt  their  equilibrium  state  if  given  sufficient  time.  Once  the   equilibrium  state  has  been  reached  the  properties  of  the  system  remain  time   independent  as  long  as  the  defining  properties  remain  unchanged.   From  mechanics  we  know  that  microscopically  seen  a  system  can  adopt  a  large  number   of  physical  states.  It  is  an  essential  aspect  of  thermodynamics  that  this  multitude  of   states  can  be  captured  using  only  a  few  variables.  The  condition  of  equilibrium  is  the  key   feature  that  makes  it  possible  to  reduce  from  a  multitude  of  microscopic  states  to  the   macroscopic  description.  In  conventional  texts  on  thermodynamics  the  existence  of  the   equilibrium  state  is  tacitly  assumed,  while  in  this  presentation  this  existence  is  explicitly   considered  as  a  fundamental  aspect  of  the  theory.  The  theoretical  concept  of  equilibrium   is  practically  relevant  first  when  we  state  that  all  systems  reach  equilibrium  given   sufficient  time  to  relax.  It  is  not  possible  to  give  a  quantitative  criterion  for  “sufficient   time”.  Experience  have  shown  that  for  some  systems  this  time  is  very  short,  while  for   others  it  is  very  long  and  the  system  remains  in  a  non-­‐equilibrium  state  over  the   accessible  time-­‐scale.  In  this  latter  case  the  properties  of  the  system  can  be  analyzed  in   terms  of  a  conditional  equilibrium.  The  physical  reality  behind  this  is  that  the  system  is   captured  in  a  metastable  state  with  a  very  long  lifetime.  The  material  properties  of     7   diamond  are,  for  example,  well-­‐defined  in  spite  of  the  fact  that  graphite  is  the   equilibrium  state  under  normal  pressure.     Can  one  argue  that  the  law1  is  instead  a  proposition?  Is  it  possible  for  reality  to  behave   differently?  The  law  contains  one  feature  that  makes  thermodynamics  qualitatively   different  from  the  other  physical  theories.  By  stating  that  systems  relax  to  an   equilibrium  state  there  is  an  arrow  of  time  in  thermodynamics,  while  equations  in   (quantum)  mechanics  are  time  reversible.       Proposition  1:  The  equilibrium  condition  can  be  formally  described  through  an   equation  of  state.   Assume  that  a  system  at  equilibrium  can  be  characterized  by  N  variables  {Xi},  i=1,…..N.   The  equation  of  state  can  be  expressed  in  a  non  biased  way  relative  to  these  variables  in   the  form  f({Xi}=0.  An  alternative  is  then  to  choose  one  variable  Xj  as  dependent  so  that   Xj=fj({Xi≠j})  with  N-­‐1  independent  variables.  The  equation  of  state  has  a  central  role  in   many  practical  applications  of  thermodynamics.  It  is  usually  assumed  that  equation  of   state  can  be  differentiated.  However,  it  turns  out  that  this  does  not  apply  to  certain   subspaces  of  the  total  variable  space.  This  “anomalous”  behavior  is  usually  discussed  in   terms  of  phase  equilibria.       Proposition  2:The  variables  in  the  equation  of  state  are  either  extensive  or   intensive.   An  extensive  quantity,  Ex,  changes  linearly  with  the  size  of  the  system  so,  for  example,   Ex(µV,p,µn)=µEx(V,p,n),   Where  V  is  volume,  p  pressure  and  n  amount.  An  extensive  quantity  has  furthermore  the   property  that  the  value  for  the  total  system  is  the  sum  of  the  values  of  the  parts  even  if   thay  are  not  in  equilibrium  with  one  another.  An  intensive  quantity,  I,  on  the  other  hand,   is  independent  of  the  size  of  the  system  so  that   I(µV,p,µn)=I(V,p,n).   The  proposition  2  provides  a  significant  limitation  on  the  character  of  the  variables  that   enter  the  equation  of  state.  It  is  in  certain  applications  desirable  to  include  surface   effects  in  the  description  of  a  bulk  system.  However,  the  surface  doesn’t  scale  linearly   with  system  size  and  we  have  a  deviation  from  the  basic  requirement.  A  consistent  way   to  analyze  the  total  system  is  to  divide  it  into  a  bulk  part  and  a  surface  part.  These  can     10   The  equlity  sign  applies  when  the  subsystems  were  in  equilibrium  also  initially  so  that   the  final  and  initial  states  are  the  same.  This  Law  2  is  can  be  considered  as  one  of  many   variations  of  the  conventional  «Second  Law»  of  thermodynamics.  It  specifies  a  central   property  of  the  equilibrium  state.  In  many  textbooks  the  second  law  is  said  to  state  that   the  entropy  increases  in  irrerversible  processes.  One  then  avoids  to  mention  the   complication  that  the  entropy  is  only  defined  for  equilibrium  states.  This  is  the  reason   why  it  is  more  precise  to  formulate  the  law  in  terms  of  the  relation  between  two   different  equilibrium  states.  Important  applications  of  Law  2  are  found  in  the   description  of  chemical  equilibria  and  of  heat  engines.  The  latter  application  lead  Carnot   to  the  first  version  of  the  Law.  However,  to  make  an  explicit  statement  about  heat   engines  it  is  necessary  to  formulate  an  additional  proposition.     Proposition  5:  For  a  non-­‐isolated  system  the  heat,  q,  taken  up  by  the  system  is  the   difference  between  the  change  in  internal  energy,  ∆U,  and  the  work,  w,  done  on   the  system;  q=∆U-­‐w.   This  relation  is  normally  presented  as  the  first  law  of  thermodynamics,  while  in  this  text   it  is  seen  as  basically  a  definition  of  heat.  The  energy  concept  is  central  to  mechanics,   quantum  mechanics  and  theory  of  relativity.  In  these  theories  one  makes  use  of  explicit   coordinate  systems.  If  one  requires  that  the  predictions  of  the  theories  should  be   independent  of  the  choice  of  the  coordinate  systems  it  follows  that  there  exists  an   invariant  scalar  and  this  is  called  the  energy.  The  constancy  of  the  total  energy  is  in  the   theories  with  explicit  coordinate  systems  a  consequence  of  a  basically  mathematical   invariance  property.  There  are  no  explicit  coordinate  systems  in  thermodynamics.  To   make  the  theory  compatible  with  the  other  physical  theories  one  has  to  explicitly   require  the  total  energy  to  be  constant.  This  is  the  role  of  the  proposition  5.  The  new   concept  of  heat  is,  in  the  language  of  theory  of  science,  introduced  «ad  hoc»  to  ensure   agreement  with  other  theories.  The  proposition  6  divides  an  energy  change  into  two   categories.  Work  is  well-­‐defined  in  mechanics  (or  in  electromagnetism),  while  heat  is  a   quantity  specific  to  thermodynamics.  It  is  possible  to  provide  a  quantum  statistical   mechanical  illustration  of  the  division  into  work  and  heat  for  small  energy  changes  at   equilibrium.  In  the  expression  dU=∂w+∂q  the  work  represents  energy  changes  due  to   changes  in  the  energy  of  the  states,  while  ∂q  is  the  energy  change  from  changes  in   population  of  states  at  given  value  of  the  energies  of  the  states:     11   𝑼 = 𝒑𝒊 𝒆𝒒𝑬𝒊𝒊 ;𝒅𝑼 = 𝒑𝒊 𝒆𝒒𝒅𝑬𝒊 + 𝑬𝒊𝒊𝒊 𝒅𝒑𝒊 𝒆𝒒 = 𝝏𝒘+ 𝝏𝒒                        (1).   (The  symbol  ∂  is  used  to  denote  a  small  quantity  in  general  while  d  as  a  symbol  denotes   a  proper  differential.)     In  mechanics  and  related  theories  the  energy  is  constant  in  isolated  systems  and  there  is   no  reference  to  equilibrium  conditions.  Based  on  this  observation  it  is  then  possible  to   widen  the  concept  of  heat  within  thermodynamics  to  be  valid  for  all  changes  so  that   ∆E=w+q,  where  E  denotes  the  energy  of  the  system  irrespective  of  possible  equilibrium   conditions.  We  reserve  the  notation  U  for  the  energy  of  equilibrium  states.     In  teaching  thermodynamics  it  is  a  challenge  to  reconcile  the  fact  that  real  processes   occur  under  non-­‐equilibrium  conditions  while  the  theory  is  focused  on  equilibrium   conditions.  One  basis  for  using  the  theory  is  that  it  is  possible  to  identify  initial  and  final   states  that  are  both  at  equilibrium.  The  discussion  of  processes  is  in  typical  text-­‐books   based  on  three  concepts;  reversible,  irreversible  and  quasi-­‐static  processes.  Normally   these  terms  are  described  within  the  conceptual  framework  of  operationalism.  It  is  thus   stated  that  a  reversible  process  can  proceed  in  either  direction,  an  irreversible  one  only   goes  in  one  direction  and  a  quasi-­‐static  can  be  made  to  reverse  by  a  small  change  in   external  conditions.  It  is  an  ambition  of  the  present  text  to  reveal  that  operationalism   creates  more  problems  than  it  solves  in  thermodynamics.  The  division  of  these  three   classes  of  processes  provides  an  illustration  of  this  opinion.  Here  we  give  the  following   alternative  interpretations  of  the  concepts.  A  reversible  process  is  characterized  by  the   fact  that  the  equation  of  state  is  satisfied  throughout  the  change.  This  is  a  theoretical   concept  and  it  is  not  coupled  to  any  specific  real  operation.  An  irreversible  process  is  a   real  process  where  the  properties  of  the  system  are  changing  in  time.  The  irreversible   process  ultimately  ends  in  an  equilibrium  state  according  to  proposition  1.  The  quasi-­‐ static  process  is  introduced  to  handle  the  transition  between  the  theoretical  (ideal)   reversible  to  the  real  irreversible  process.  If  one  ignores  the  program  of  operationalism   there  is  no  need  to  specify  a  real  operation  that  leads  from  the  initial  state  to  the  final   one  and  the  concept  of  a  quasi-­‐static  process  becomes  superfluous.       In  thermodynamics  we  describe  intrinsically  very  complex  systems  using  only  a  few   variables.  One  price  paid  for  such  a  simplification  is  that  one  refrains  from  explicitly   describe  the  processes  that  lead  from  one  equilibrium  state  to  another.  It  is  solely  based     12   on  the  description  of  equilibrium  states.  The  area  of  “irreversible  thermodynamics”   provides  an  approach  to  enlarge  the  applicability  of  the  thermodynamic  concepts  also  to   the  explicit  description  of  dynamic  events.  The  use  of  the  terms  “reversible”  versus   “irreversible”  poses  pedagogical  difficulties.  They  can  be  seen  as  inherited  from   operationalism,  but  it  is  probably  difficult  to  implement  a  change  of  terminology.  It   would  be  a  more  distinct  to  replace  “irreversible  process”  with  “real  process”  and   “reversible  process”  with  “process  at  equilibrium”  stressing  that  this  is  a  theoretical   construction.  Using  such  a  terminology  one  avoids  the  (now  irrelevant)  question   whether  or  not  an  irreversible  process  can  be  made  to  reverse.    This  question  has   usually  not  a  simple  answer.     Some  basic  results.   Above  we  have  presented  an  abstract  theory  by  introducing  a  number  of  concepts,   stated  five  propositions  and  three  laws.  To  connect  to  reality  we  have  to  relate  empirical   observation  to  variables  in  the  theory.  This  is  in  general  a  profound  epistemological   problem,  but  here  we  take  the  approach  to  consider  this  problem  solved  for  mechanics   and  other  physical  theories  based  on  a  similar  framework.  To  relate  the  thermodynamic   theory  to  reality  we  simply  adopt  the  observation  criteria  from  these  other  theories.   This  places  us  in  the  position  to  apply  the  thermodynamic  theory  to  a  concrete   description  of  reality.   1.Heat  transfer   Consider  an  isolated  system  consisting  of  two  parts.  These  are  initially  isolated  from   each  other  and  they  are  in  internal  equilibrium  with  temperatures  T1  and  T2,   respectively.  What  happens  when  the  two  parts  are  brought  in  contact  so  that  a  transfer   of  energy  in  the  form  of  heat  can  occur,  but  there  is  no  work  involved  and  no  change  in   volume  in  the  process.  We  have   𝑈! + 𝑈! = 𝑈!"! = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡   since  the  total  system  is  isolated.  Thus  the  energy  change  ∆U1  in  one  part  of  the  total   system  has  to  be  balanced  by  the  energy  change  ∆U2  in  the  other  part  and   ∆U1=-­‐∆U2.   From  proposition  5  we  have  that  the  heat  transferred  between  the  two  subsystems   match  so  that  q1=-­‐q2.  The  question  we  now  ask  what  determines  the  sign  of  the  heat  q  or   energy  ∆U  transferred  between  the  subsystems.  Consider  first  the  onset  of  the  process     15   thermodynamics  have  the  same  temperature.  The  larger  system  not  only  has  twice  as   large  energy  as  the  smaller,  but  it  also  has  twice  as  large  density  of  states.     In  certain  presentations  of  thermodynamics,  where  one  has  the  ambition  to  be  more   rigorous,  one  introduces  as  zeroth  law.  The  statement  is  typically  that  if  subsystems  A   and  B  are  in  equilibrium  and  this  is  also  true  for  subsystems  B  and  C  the  law  states  that   A  and  C  are  also  in  equilibrium.  Note  that  if  one  in  the  example  above  on  heat  transfer   introduces  a  third  subsystem  the  statement  of  the  zeroth  law  actually  follows  from   Carnot’s  Law.  The  reason  for  introducing  the  zeroth  law  is  found  in  the  measurability   criterion  required  in  operationalism,  since  the  temperature  concept  obtains  a  meaning   first  in  relation  to  specified  measuring  process.  When  one,  as  in  the  present  text  discard   “operationalism”  and  its  definitions  through  measuring  processes,  there  is  no  longer  a   reason  to  introduce  the  zeroth  law.   2  Isothermal  expansion/compression  of  an  ideal  gas   The  equation  of  state  for  an  ideal  gas  is  pV=nRT.  One  can  show  that  this  also  implies  that   the  energy  U  is  only  dependent  on  T  so  that  U(T,V)=U(T,p)=U(T).   Thus  the  energy  U  is  unchanged  in  an  isothermal  change.  However,  there  is  in  general  a   heat  transfer  between  system  and  surrounding  to  compensate  the  work  done  on  the   system.  Combining  proposition  3  and  the  equation  of  state  we  have   dS=(p/T)dV=nRlnV     For  a  change  between  initial  state  i  and  final  state  f  the  entropy  change  is   ∆S=Sf-­‐Si=nln(Vf/Vi).   It  is  less  straightforward  to  determine  the  amount  of  heat  transferred  to  the  system.   There  is  no  unique  answer  since  there  is  a  dependence  on  how  the  process  is  actually   done.  In  one  limit  the  external  pressure  has  throughout  matched  the  pressure  of  the   system,  which  corresponds  to  a  reversible  process.  Then   𝑤 = 𝑤!"# = − 𝑝𝑑𝑉 =!! !! -­‐nRT 𝑑𝑙𝑛𝑉 = −𝑛𝑅𝑇𝑙𝑛(𝑉!/𝑉!) !! !!   From  mechanics  we  know  that  to  have  a  compression,  dV<0,  the  external  pressure  pext   needs  to  be  larger  than  the  pressure  of  the  system  and  conversely  dV>0  when  pext<p.   Thus  for  an  expansion  the  work  has  a  negative  sign  and  the  work  performed  by  the   system  is  less  negative  for  a  real  process  such  that  w>wrev     This  implies  that  for  an  isothermal  expansion  q≤T∆S,       16   where  the  equality  sign  is  for  reversible  process.  For  an  expansion  the  work  has  a   positive  sign  but  the  inequalities  remain  the  same.  For  a  real  gas  the  equation  of  state  is   more  complex,  but  the  inequalities  remain  valid.  This  also  goes  for  other  types  of  work.   The  important  conclusion  is  that  to  maximize  (expansion)  or  minimize  (compression)   the  work  and  the  corresponding  transfer  of  heat  it  is  desirable  to  have  the  real  process   occur  as  close  as  possible  to  the  equilibrium  path.       3  Adiabatic  expansion  of  an  ideal  gas   In  an  adiabatic  process  there  is  no  heat  exchange  with  the  surrounding.  The  initial  and   final  states  differ  in  volume,  pressure  and  temperature.  Since  q=0  it  follows  that  ∆U=w.   Even  when  we  know  V,T  and  p  for  the  initial  and  V  for  the  final  state  the  work   performed  is  unspecified  and  thus  ∆U.  It  follows  that  also  T  and  p  are  unknown.  In  one   limit  the  change  follows  the  path  given  by  the  equation  of  state   -­‐pdV=CVdT  ;  dV/V=-­‐(CV/nR)(dT)/T   𝑤 = 𝑤!"# = − 𝑝𝑑𝑉 = 𝐶!(𝑇! !! !! − 𝑇!)  ;  ln  (𝑉!/𝑉!) = −(𝐶!/𝑛𝑅)ln  (𝑇!/𝑇!)   In  an  expansion  the  temperature  of  the  system  decreases.  When  in  reality  the  expansion   occurs  relative  to  a  pressure  that  is  lower  than  the  equilibrium  pressure  of  the  work   performed  by  the  system  is  smaller  and  final  temperature  is  higher.  This  is  in  qualitative   analogy  with  the  relation  valid  for  isothermal  processes.  In  the  extreme  limit  of  zero   external  pressure  no  work  is  performed  and  there  is  no  change  in  temperature  for  an   ideal  gas.   One  can  construct  a  cycle  for  the  system  based  on  two  isothermal  and  two  adiabatic   processes.  This  is  the  so  called  Carnot  cycle  and  the  expressions  above  can  be  used  to   calculate  the  total  work,  the  total  amount  of  heat  transferred  at  given  values  of  the   temperatures  of  the  thermal  baths  for  the  two  isothermal  processes.  Such  an  analysis   provides  the  basis  for  applying  thermodynamics  to  the  performance  of  heat  engines.     4  Chemical  isomerization  equilibrium   The  description  of  chemical  equlibria  is  an  important  application  of  thermodynamics.     Let  us  consider  the  simple  example  of  an  isomerization  process  𝐴  𝐵  in  the  gas  phase.   Assume  also  that  the  components  can  be  described  as  ideal  gases  so  that  Dalton’s  law   applies.  For  an  isolated  system  the  differential  of  the  entropy  is  (proposition  3)     17   𝑑𝑆 = 𝑑𝑈 𝑇 − 𝑝𝑑𝑉 = 𝐶 𝑇 𝑙𝑛𝑇 − 𝑛𝑅𝑑𝑙𝑛 𝑝   where  C(T)  is  the  heat  capacity.  In  the  second  equality  we  have  used  the  ideal  gas  law   and  its  consequence  that  U  is  only  a  function  of  T.  The  two  terms  on  the  right  hand  side   are  independent  and  they  can  thus  be  integrated  separately  to  yield  an  expression  for   the  entropy   𝑆 𝑇,𝑝 !"#$%  !"# = 𝐶𝑑𝑙𝑛 𝑇 − 𝑛𝑅𝑙𝑛 𝑝 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ! ! = 𝑛 𝐶! ! ! 𝑑𝑙𝑛 𝑇 + 𝑅𝑙𝑛 𝑝 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑛𝑆!(𝑇,𝑝)   Here  the  expression  between  curly  brackets  is  the  entropy  per  mole  of  the  substance  SM.   When  this  expression  is  applied  to  a  mixture  of  ideal  gases  there  is  a  hidden  subtlety.   There  is  a  question  of  what  pressure  to  use.  According  to  Dalton’s  law  the  total  pressure   p  is  the  sum  of  the  partial  pressures  pi  and  p=pA+pB.  The  entropy  of  the  total  system  of  A   and  B  mixed  is   Stot=nASMA+nBSMB   provided  one  uses  the  partial  pressure  for  SMi.  However  if  we  consider  A  and  B  to  be   identical  such  a  choice  is  improper  (lnp≠lnpA+lnnpB).  This  is  called  Gibb’s  paradox  and   the  problem  will  be  discussed  further  in  example  6.     According  to  Carnot’s  law  the  entropy  has  a  maximum  with  respect  to  variations  in  the   amount  of  A  and  B.  For  an  isomerization  reaction  dnB=-­‐dnA    so  that   0 = 𝑑𝑆!"! 𝑑𝑛! = 𝜕𝑆!"! 𝜕𝑛! + 𝜕𝑆!"! 𝜕𝑛! 𝑑𝑛! 𝑑𝑛! + 𝜕𝑆!"! 𝜕𝑝! 𝑑𝑝! 𝑑𝑛! + 𝜕𝑆!"! 𝜕𝑝! 𝑑𝑝! 𝑑𝑛! + 𝜕𝑆!"! 𝜕𝑇 𝑑𝑇 𝑑𝑛!   Now  we  make  the  further  simplifying  assumption  that  the  two  isomers  have  the  same   ground  state  energy  and  also  the  same  molar  heat  capacity.  In  such  a  case  there  are  no   energy  canges  involved  in  the  isomerization  and  the  temperature  of  the  system  is   independent  of  the  degree  of  transformation  between  A  and  B.  Thus  the  fifth  term  on  the   right  hand  side  is  zero.  An  explicit  evaluation  reveals  that  the  third  and  the  fourth  term   cancel  and  the  resulting  equilibrium  condition  is  that   SMA-­‐SMB=0   Which  in  turn  implies   pA=pB   so  that  the  two  forms  occur  in  equal  amounts  at  equilibrium  as  intuitively  expected.     20   where  Ci  denotes  a  concentration  measured  as  moles  per  volume  or  equally  well  as  mole   fraction.  This  form  of  the  equilibrium  condition  was  originally  derived  by  Guldberg  and   Waage  on  the  basis  of  an  intuitive  kinetic  argument.  Such  an  argument  is  easier  to  accept   for  a  reaction  in  the  gas  phase,  but  it  is  not  generally  valid.  In  the  late  nineteenth  century   van’t  Hoff  realized  that  for  solutes  in  a  dilute  solution  the  concentration  dependence  of   the  chemical  potential,  𝜇 = !" !"  ,  can  be  approximately  described  as  ln(Ci).  This  relation   can  be  seen  as  an  approximate  equation  of  state  based  on  empirical  considerations.  To   arrive  at  a  theoretically  more  satisfactory  argument  it  is  necessary  to  go  beyond  the   thermodynamic  theory  and  use  an  argument  from  statistical  mechanics.  Boltzmann’s   expression  for  the  entropy  𝑆 = 𝑘𝑙𝑛(𝑛𝑢𝑚𝑏𝑒𝑟  𝑜𝑓  𝑝𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠)  one  finds  that  the  entropy   of  mixing  objects  A  and  B  is  S=-­‐{nAln(XA)+nBln(XB)}  for  the  case  when  the  probability  for   occupying  a  certain  position  is  equal  for  A  and  B.  This  is  called  ideal  mixing.  For  the  case     in  example  5  such  ideal  mixing  conditions  are  ensured  by  Dalton’s  law.  In  solution  there   is  typically  a  non-­‐random  mixing  due  to  selective  intermolecular  interactions.  However,   these  have  lesser  effects  the  higher  the  degree  of  dilution  so  the  ideal  mixing  expression   can  be  used  as  an  equation  of  state  for  sufficiently  highly  dilute  systems.  In  this  limit  one   obtains  the  Guldberg-­‐Waage  expression  for  the  equilibrium  in  solution  also.The   conventional  way  to  correct  for  the  interactions  manifested  at  higher  concentrations  is   to  introduce  “activity  coefficients”  in  the  expression  for  the  equilibrium.     7  Phase  transitions   It  is  an  empirical  observation  that  systems  can  show  abrupt  changes  in  their  properties   even  at  very  small  changes  in  the  surrounding.  This  implies  that  the  corresponding   equations  of  state  contain  discontinuities  or  divergences.  The  phenomenon  is  called  a   phase  transition  and  it  is  not  predicted  by  the  thermodynamic  theory.  The  basic   assumption  that  the  equation  of  state  is  continuous  and  differentiable  has  to  be   modified.  It  is  not  valid  for  a  part  of  the  region  of  definition.  However,  assuming  the   existence  of  phase  transitions  leads  to  a  very  useful  relation  called  the  “Gibbs’  phase   rule”.     For  an  equation  of  state  with  N  variables  we  can  chose  N-­‐1  as  independent.  At  least  one   of  these  has  to  be  extensive.  Choose  such  a  representation  with  one  extensive,ve,  and  N-­‐2   intensive  independent  variables.  Then  the  equation  of  state  can  be  written  as     21   Y=vekfY({vi})   Here  k=1  if  Y  is  extensive  and  k=0  if  it  is  intensive.  There  are  N-­‐2  independent  intensive   variables  vi  so  the  function  fY  is  defined  on  a  N-­‐2  dimensional  space.  In  thermodynamics   one  uses  the  terminology  that  there  are  f=N-­‐2  degrees  of  freedom.  The  number  of   degrees  of  freedom  can  be  reduced  by  imposing  additional  constraints  on  the  intensive   variables.  One  such  condition  is  that  there  is  coexistence  between  two  or  more  phases.   For  two  coexisting  phases  the  variables  in  the  equation  of  state  are  confined  to  a  surface   of  dimensionality  N-­‐3.  Then  the  number  of  degrees  of  freedom  is  reduced  to  f=N-­‐3.  If  f≥1   one  can  have  two  coexisting  phases  and  f=N-­‐4.  This  can  continue  until  one  reaches  f=0.   Combining  Gibbs’  law  with  the  constraints  implied  by  phase  coexistence  results  in  Gibbs’   phase  rule   f=2+c-­‐p   where  c  is  the  number  of  components  and  p  the  number  of  phases.  For  a  one  component   system,  c=1,  the  maximum  number  of  three  phases  that  can  coexist  at  equilibrium.  Zero   degrees  of  freedom  implies  a  point  in  the  variable  space  and  the  conditions  of  three   phases  at  equilibrium  is  called  a  triple  point.  For  water,  the  vapor,  the  liquid  and  the   solid  ice  coexist  at  equilibrium  at  T=273.16°K  and  p=610Pa.  It  is  a  very  clear  prediction   of  the  theory  that  for  a  pure  substance  like  water  three  phases  can  coexist  only  at  a   specific  combination  of  temperature  and  pressure.  This  is  far  from  intuitively  obvious   and  it  demonstrates  a  strong  point  of  the  theory.  The  Gibbs’  phase  rule  emerges  clearly   from  Law  1  (Gibbs’  law)  in  the  formulation  of  the  theory  presented  above.  If  one  instead   tries  to  derive  the  result  from  a  conventional  thermodynamic  theory  containing  the   zeroth,  the  first,  the  second  and  the  third  laws  it  is  difficult  to  see  how  Gibbs’  phase  rule   could  emerge  in  a  logical  way.     8  Concluding  remarks   The  presentation  above  of  the  thermodynamic  theory  has  two  parts.  In  the  first  half  the   formal  theory  was  laid  down  and  the  ambition  was  to  be  logically  consistent  and  present   all  essential  ingredients.  Whether  or  not  this  ambition  is  met  by  the  text  is  open  to   debate.  In  the  second  half  seven  examples  were  given  that  strive  to  illustrate  the  relation   between  the  abstract  theory  and  some  basic  applications  of  the  theory  to  problems  of   practical  interest.  Once  this  level  has  been  reached  textbooks  on  thermodynamics  are   typically  using  the  tools  in  a  consistent  way.       22     The  way  the  theory  is  presented  above  has  clear  similarities  with  the  way  Callen   describes  thermodynamics  in  his  book.  The  essential  features  of  the  formulation  of  the   theory  made  above  are:   The  epistemological  point  of  view  is  clearly  declared.   Thermodynamics  is  explicitly  related  to  other  physical  theories.   It  is  made  very  clear  that  the  thermodynamic  theory  is  describing  (conditional)   equilibrium  states.   The  concept  of  an  equation  of  state  is  given  a  more  central  role  than  is  customarily  done.   The  number  of  variables  in  an  equation  of  state  is  seen  as  emerging  from  a  fundamental   property  of  reality.     “The  first  law  of  thermodynamics”  is  primarily  seen  as  an  way  to  introduce  the  concept   of  heat  in  an  ad  hoc  way  to  achieve  agreement  with  other  physical  theories  based  on  the   explicit  use  of  coordinate  systems.   “The  third  law”  is  introduced  as  a  result  from  quantum  mechanics  and  statistical   mechanics.   Several  features  in  text-­‐book  presentations  of  the  theory  result  from  the  use  of   “operationalism”  as  a  basis  for  formulating  thermodynamics.  “Operationalism”  is  in   general  untenable  and  the  programme  has  little  value  for  understanding   thermodynamics  and  it  should  be  abandoned.  Thus  the  “zeroth  law”  is  superfluous  and   so  is  the  concept  of  a  quasistatic  processes.     Version  130528