Lab. 12 - Determine the ? value of air ??12:?? ? ????

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Lab. 12- Determine the ? value of air??12?? ?
????

• I. Object Measure the ? ratio of air using
  Clement Desorms method.
• ????????????????????
• cp/cv ???????????,
• -- ?????????????

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Adiabatic process (????)

• Broilge Law at constant T,an ideal gas with the
  fixed particles obeys PV nRT constant.
• But real gas isnt a good heat conductor, it need
  time to reach a new thermal equilibrium.
• As P and V change too rapidly, e.g., the
  propagation of sound, the energies between the
  parts of gas can not exchange with each other at
  once.
• So it is impossible for the isothermal process
  (????), the realize reaction shall be an
  adiabatic process.
• Adiabatic process
• If gas is compressed (expanded), all work done by
  the surrounding becomes (dissipate) the internal
  energy of gas, thus the P and T of gas will
  simultaneously increase (decrease).
• The slope of p(V) curve in adiabatic process must
  steeper than in the isothermal process.

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Principle (??)

• ???????(????)???? U ??? dU
• (1st thermodynamical law energy conservation
  law)
• ?/?????(endothermic/exothermic reaction do
  positive work) dU dQ dW
• ?/?????(endothermic/exothermic reaction do
  negative work) dU dQ pdV
• ??????(molar heat capacity at constant volume)
• ?????? (molar heat capacity at constant
  pressure)
• ?????(T 300 K)????????? (Air at 300 K can be
  as idea gas)
• pV NkT , pV nRT
• N ??? (Particle number)
• k 1.38 x 10-23 J/K ?????(Boltzman
  constant)
• n ? N/NA ?????(Molar no.)
• NA 6.02 x 1023 /mol
  ??????(Advogadro constant)
• R ? kNA 8.31 J/mol-K ???? (Gas
  constant)

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Derive the ? value of idea gas
Ideal gases pV nRT ? pdV Vdp
nRdT Constant Pressure dp 0 (???) dQ dU
pdV dU nRdT

• cP cv R
• ? cp/cv (cV R)/cv

• ????(Equipartition of Energy) kT/2 per degree of
  freedom
• ??????????N2 78, O2 21,???(T 300 K)
• ? ?3???????,2???????,??????
• ? ???????5???,?????? kT/2 ???
• ? ???Total U 5nRT/2
• ???? cv 5 R/ 2 ???? cp cv
  R 7 R/ 2
• ??????????? ? ? cp/cV 7/5

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Internal Energy of an ideal Gas ???????
(19 11)
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• Cv 3/2R 12.5 J/mol.K
• Cp 5/2R 20.8 J/mol.K
• Theoretical values of CV, CP, and ? are in
  excellent agreement for monatomic gases.

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Molar Specific Heat for an Ideal gas

• Several processes can change the temperature of
  an ideal gas.
• Since DT is the same for each process, DEint is
  also the same.
• The heat is different for the different paths.
• The heat associated with a particular change in
  temperature is not unique.

• Specific heats are frequently defined at two
  processes
• Constant-pressure specific heats cp
• Constant-volume specific heats cv
• Using the number of moles, n, defines molar
  specific heats for these processes.

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Adiabatic expansion dQ 0 (????)
For an ideal gas with n 1

• Here both A and A are constant values.
• The slope of p(V) curve in adiabatic process must
  steeper than in the isothermal process.
• One can determine the ? value from the adiabatic
  process.

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(19 16)
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Block Diagram of Clement-Desorms Experiment
1
2
3
P1 T1 To V1
P2 Po T2 V2
P3 T3 To V3 V2
(1)
(2) P1V1 P3V2
(1) Adiabatic expansion (2) Isothermal expansion
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Clement-Desorms Configuration???-???????

1. ?????(spherical glassware)
2. ??(Cupper Tip, ??????????????????)
3. ??? (rubber tube)
4. ???? (glass valve)
5. ??? (gas charging ball)
6. U-???(U-shape soft tube, ??????????)

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Clement-Desorms Experiment Configuration ???-????
??????
Initial condition of system at room
temperature, T0 300 K and under ambient
atmosphere, p0 1 atm (?????????,???????????)
(1) ?A????,?B????? ??????????????????,
????????p1, ??????? ????, ?????????,
?U?????????????h1, ?? p1 p0
h1dg d ???????, g ????? T1
T0
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• (2) ?B?, ???????????, ?????????, ?B?.
• p2 p0
• p1V1? p2V2? (????, ?????????)
• T2 lt T1 T0 (???????)
• (3) ????, ????????????
• T3 T0 (???)
• ???????h3,? p3 p0 h3dg
• (A) p1V1 n1RT0 (T1 T0)
• (n1/n3)p3V3 (T0 T3)
• p3V3
  (n1 n3)
• (B) p1V1? p0V2? (p2 p0)
• p0V3?
  (V2 V3, ?????????)
• (A)?/(B) p1?-1 p3?/p0 (p1,
  p3??)
• or ? h3/(h1 - h3) (???? ?
  7/5 1.4 ??)

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? Value derived in Classical Mechanics
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Molar Specific Heat

• Specific heats are frequently defined at two
  processes
• Constant-pressure specific heats cp
• Constant-volume specific heats cv
• Using the number of moles, n, defines molar
  specific heats for these processes.
• Molar specific heats
• Q nCvDT for constant volume processes
• Q nCpDT for constant pressure processes
• Q (in a constant pressure process) must account
  for both the increase in internal energy and the
  transfer of energy out of the system by work.
• Q(constant P) gt Q(constant V) for given values of
  n and DT

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Ideal Monatomic Gas- contains only one atom per
molecule

• When energy is added to a monatomic gas in a
  container with a fixed volume, all of the energy
  goes into increasing the translational kinetic
  energy and temperature of gas.
• There is no other way to store energy in such a
  gas.
• Eint 3/2nRT, in general, the internal energy of
  an ideal gas is a function of T only.
• The exact relationship depends on the type of gas
• At constant volume, Q DEint nCvDT, applies to
  all ideal gases, not just monatomic ones.

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Specific Heat of Monatomic Gases

• At constant volume, Q DEint nCvDT
• applies to all ideal gases, not just monatomic
  ones.
• Cv 3/2R 12.5 J/mol.K
• Be in good agreement with experimental results
  for monatomic gases.
• In a constant pressure process, DEint Q W
• Cp Cv R ? Cp 5/2R 20.8 J/mol.K
• This also applies to any ideal gas
• Define a Ratio of Molar Specific Heats
• Theoretical values of CV, CP, and ? are in
  excellent agreement for monatomic gases or any
  ideal gas.

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Adiabatic Process for an Ideal Gas

• At any time during the process, PV nRT is valid
• None of the variables alone are constant
• Combinations of the variables may be constant
• The pressure and volume of an ideal gas at any
  time during an adiabatic process are related by
  PVg constant
• All three variables in the ideal gas law (P, V,
  T) can change during an adiabatic process

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• ??
• ??????

Adiabatic compression process
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Equipartition of Energyfor Complex Molecules

• Other contributions to internal energy must be
  taken into account
• Translational motion of the center of mass 3
  degrees of freedom
• Rotational motion about the various axes 2
  degrees of freedom for x and z axes
• ? To neglect the rotation around the y axis since
  it is negligible compared to the x and z axes.
• Eint 5/2nRT
• CV 5/2R 20.8 J/mol.K
• Cp 7/2R 29.1 J/mol.K
• predicts that g 7/5 1.40

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Equipartition of Energy

• The molecule can also vibrate
• There is kinetic energy and potential energy
  associated with the vibrations
• This adds two more degrees of freedom
• Eint 7/2nRT, Cv 7/2R 29.1 J/mol.K
• g 1.29

• This doesnt agree well with experimental
  results.
• A wide range of temperature needs to be included.
• For molecules with more than two atoms, the
  vibrations are more complex. ? The number of
  degrees of freedom is larger
• The more degrees of freedom available to a
  molecule, the more ways there are to store
  energy, and the higher molar specific heat.

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Agreement with Experiment-Molar specific heat of
a diatomic gas (acts like a monatomic gas) is a
function of T.
at high T CV 7/2R
at about room T Cv 5/2R
at low T Cv 3/2R
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Quantization of Energy

• Classical mechanics is not sufficient to explain
  the results of the various molar specific heats,
  we must use some quantum mechanics.
• The rotational and vibrational energies of a
  molecule are quantized

The energy level diagram of the rotational and
vibrational states of a diatomic molecule.

• The vibrational states are separated by larger
  energy gaps than are rotational states.
• 1. At low T, the energy gained during collisions
  is generally not enough to raise it to the first
  excited state of either rotation or vibration.
• Even though rotation and vibration are
  classically allowed, they do not occur.

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DVD ??????(Engine of Nature)(MU46)
??????? ?????? dU dQ dW (??????? ????
????(work)) dU TdS pdV (S ?dQ/T
??(entropy)) ??(???)(heat engine) ??(Tinput)??
Qi, Si Qi/Ti (Ti constant) ??(Toutput)?? Qo,
So Qo/To ???? W Qi Q0 ??(efficiency) e ?
W/Qi 1 Qo/Qi ????(Carnot cycle)/????(Carnot
engine) 1.??????(isothermal expansion) Ti
constant, ??Qi 2.????(adiabatic expansion) dQ 0
(??), Ti ? To (??) 3.??????(isothermal
compression), To constant, ??Q0 4.????(adiabatio
c compressiion) dQ 0, To ? Ti (??) ? W/Qi
1 To/Ti (Si So, ????/ideal heat engine)