Chemistry  252   Problem  Set  2

1. Calculation  of  average  energies  from  partition  functions       Consider  a  system  that  has  only  three  possible  energy  states  with  E  =  0,  E0  and  2E0.

a) Write  a  general  expression  for  the  partition  function  q.

b) From  your  expression  in  (a),  derive  an  expression  for  the  average  energy  E  as   a  function  of  temperature  T.

c) Derive  an  expression  for  the  specific  heat  Cv.

d) Suppose   that   E0   =   100   cm-­‐1.   Estimate   the   energy   needed   to   raise   the

temperature  of  the  system  from  300  K  to  400  K  in  units  of  KJ/mol.       2. Specific  heats  of  real  gaseous  molecules     A   review   of   sections   4-­‐6   and   4-­‐7   of   the   textbook   will   be   useful   in   solving   this   problem.

a) Study  equation  4.39  of   the   text  and   the  resulting  partition   function  derived   for   a   gaseous   diatomic   molecule   in   example   4-­‐5   of   the   text.   Use   these   expressions  for  Q(N,V,T)  to  derive  an  expression  for  the  specific  heat.

b) Interpret   each   of   the   terms   in   the   expression   you   derived.   This   is   actually

done   in   the   text   (and   we   reviewed   it   in   class)   so   it   should   be   easy   but   I   thought  it  was  so  important  that  I’d  like  you  to  put  what  they  say  into  your   own  words.

c) Now,   I’d   like  you   to  generalize   the  expression   for   the   specific  heat  Cv   to  be

appropriate  to  a  polyatomic  molecule,  specifically  H2O(g).  Assume  that  we’ll   be   working   in   the   temperature   range   from   300   –   800   K   where   we   can   consider   ourselves   to   be  way   above   the   rotational   temperature  Θrot   so   you   can  just  approximate  the  rotational  contribution  to  specific  heat  to  be  3R/2   per  mole   corresponding   to   R/2   for   each   of   the   three   rotational   degrees   of   freedom.  The  main  modification  you  need  to  make   is   to  consider  that   there   are   4   degrees   of   vibrational   freedom   for  water   (asymmetric   stretch   ħω1   =   3756   cm-­‐1,   symmetric   stretch   ħω2   =   3652   cm-­‐1,   bending  mode   ħω3   =   1595   cm-­‐1  and  another  bending  mode  ħω4  =  1595  cm-­‐1).

d) Use   your   expression   from   c)   to   plot   Cv   versus   temperature   over   the

temperature   range   specified   above   (calculating   a  point   every  100  K   should   be  enough).  Compare  your  result  to  experimental  values  from  the  literature.

  e) As  is  evident  from  Figure  4.7,  there  is  excellent  agreement  of  calculated  and

measured  specific  heats.  The  text  notes  (page  160)  that  the  agreement  can  be   improved  still  further  if  we  refine  the  harmonic  oscillator  model  to  consider   anharmonicity   –   i.e.   the   fact   that   the   potential   is   not   really   harmonic.   (For   reference,   see   problem   1-­‐27   and   1-­‐31   of   the   text   on   page   34).   Given   that   accounting  for  anharmonicity  decreases  the  spacings  between  energy  levels   relative   to   what   they   would   have   been   in   a   completely   harmonic   system,   reason  as  to  whether  making  a  correction  for  anharmonicity  would  increase   or   decrease   the   values   of   your   calculated   specific   heat.   Explain   your   reasoning.

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