physical chemistry 2q

 can any of you do both questions perfect within 12 hrs , i am attaching related chapter of the book as well. 

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.