Class VIII Sample Systems and Second Derivatives

1
Class VIII Sample Systems and Second Derivatives

• Contents
• Summary of formal structure (p.63 in Callens)
• Simple Ideal Gas and Multi-component Simple Ideal
  Gases
• Entropy of Mixing
• The Ideal van der Waals Fluid
• Electromagnetic Radiation/Rubber Band/Magnetic
  Systems
• Second Derivatives
• Molar Specific Heat of an Ideal Gas and Solids

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Summary of Formal Structure

• The fundamental equation U(S,V,N) contains all
  thermodynamic information about a system
• The first partial derivatives of this equation
  are the intensive parameters (e.g.,
  )
• The fundamental equation implies three equations
  of state TT(S,V,N)T(s,v), PP(S,V,N)P(s,v),µµ
  (S,V,N)µ(s,v)
• If all three equations of state are known they
  may be substituted in the Euler relation and one
  recovers the fundamental equation
• If two equations of state are known the
  Gibbs-Duhem relation can be integrated to obtain
  the third-but the equation of state thus found
  will contain an undetermined integration constant

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Summary of Formal Structure

• An equivalent but more direct method to obtain
  the fundamental equation when two equations of
  state are known is by integrating the molar
  relation duTds-Pdv, but again we will have one
  undetermined constant of integration

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Simple Ideal Gas and Multicomponent Simple Ideal
Gases

• A simple ideal gas is characterized by the
  following two equations of state

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Simple Ideal Gas and Multicomponent Simple Ideal
Gases

• Working out equation 7.6 we obtain

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Simple Ideal Gas and Multicomponent Simple Ideal
Gases
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Entropy of Mixing

• Visualize that we have a container divided into
  two compartments. In one compartment we have n1
  moles of an ideal gas, gas 1, at pressure, P and
  temperature, T. In the other compartment we have
  n2 moles of another ideal gas, gas 2, at the same
  P and T.
• If we remove the partition the gases will begin
  to diffuse into each other and the system will
  eventually reach the state where both gases are
  uniformly distributed throughout the container.
  This is clearly an irreversibly process so that
  we would expect that the entropy will increase.

• To calculate the entropy change we must find a
  reversible path to carry out the process, even if
  the path is fictitious. For example imagine that
  we can devise a process that will expand one gas
  reversibly and isothermally, but leave the other
  gas undisturbed. We know how to calculate the
  change in entropy for the reversible isothermal
  expansion of an ideal gas.

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Entropy of Mixing

• Recall that for an isothermal expansion dU 0.
• Equation 7.22 is of the same form that Shannon
  found for theEntropy of a Message.
• Equation 7.22 can be expanded to incorporate
  more ideal gases.
• The definition of an ideal gas/solution is one in
  which the entropy of mixing is given by equation
  7.22 and the enthalpy of mixing is 0.
• Experimentally, equation 7.22 is found to
  account well for the entropy of mixing for
  molecules that only interact via van der Waals
  forces.

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Entropy of Mixing

• Since the mole fractions in equation 7.23 are
  always less than unity, the ln terms are always
  negative, and the entropy of mixing is always
  positive. Its variation with concentration is
  shown in the diagram below.
• For the ideal gas/ solution, as defined above,
  the Gibbs free energy from equation 7.24 is
  always negative and becomes more negative as the
  temperature is increased (see diagram). The
  decrease in free energy on mixing is always a
  strong force promoting solubility.

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Entropy of Mixing

• Example molar entropy of mixture The composition
  of dry air is approximately 78 N2, 21 O2, and
  1 Ar by volume (which is the same as mole
  percent). What is the molar entropy of mixing of
  air?
• If we look at the processes which have positive
  entropy changes we can see that in each case an
  increase in entropy is associated with an
  increase in disorder.
• An isothermal expansion gives the molecules more
  room to move around in, the molecules are less
  localized.
• Increasing the temperature increases the average
  speeds of the molecules. The molecules are said
  to be more disordered in "velocity space" (or
  momentum space).
• Mixing gases (or liquids) intersperses the
  molecules among each other increasing the
  disorder.
• Phase changes, such as going from a solid to a
  liquid or a gas, or from a liquid to a gas,
  increase the entropy because gases are more
  disordered than solids or liquids and liquids are
  more disordered than solids.

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The Ideal van der Waals Fluid

• Ideal gas PVNRT
• Gas molecules are perfect spheres.
• Gas molecules when colliding with selves or
  vessel walls, lose no energy by friction.
• Volume occupied by gas molecules in vessel,
  negligibly small part of total vessel volume.
• Van der Waals (1873)
• Gas molecules occupy more than negligible volume
  assumed by ideal gas law.
• Gas molecules exert long-range attractive forces
  on one another.

7.25
Where b . Accounts for finite volume occupied
by molecules a/V2 accounts for forces of
attraction between molecules
Note When a and b 0, ideal gas law results
PV RT

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The Ideal van der Waals Fluid
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The Ideal van der Waals Fluid

• We want to supplement this equation with a
  thermal equation of state of the form 1/T
  f(u,v).
• Equation 7.32 can be rewritten as

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The Ideal van der Waals Fluid

• In the line drawing below, as the volume
  decreases the pressure begins to rise steeply. 
  This relationship holds for relatively large
  volumes of gas and relatively warm temperatures. 
  The Ideal Gas Law is a good approximation in many
  applications.  However, the figure implies that
  if you could lower a gas to absolute zero
  temperature either it would exert zero pressure
  or it wouldnt take up any space!   We know
  intuitively that there must be a minimum volume a
  gas can occupy, since the molecules themselves
  cannot be compressed infinitely.  Even near this
  minimum volume we know that a gas exhibits other
  behaviors - like changing phase into a liquid or
  solid.  In reality, molecules have a volume of
  their own and that inherent molecular size limits
  the volume to which a gas can be compressed. 
  Molecules also have attraction toward one another
  and towards the boundaries of their volume. 
  These attractions increase the experienced
  pressure depending on the type of gas being
  studied.  These additional factors were taken
  into account by Dutch physicist Johannes van der
  Waals in 1873.  Van der Waals equation is a
  closer approximation to observed properties of
  gases

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The Ideal van der Waals Fluid
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Electromagnetic Radiation/Rubber Band/Magnetic
Systems

• Electromagnetic Radiation the electromagnetic
  radiation is governed by a fundamental equation
  of the form SS(U,V) in which there are only two
  rather than three independent extensive
  parameters. A truncated Euler relation can be
  used ( empty cavity -- no particles to be counted
  by a parameter N). (Callen page 79)
• Rubber band stretching. Use again the
  requirement that the mixed second-order partial
  derivatives should be equal to build the
  fundamental equation.
• The extensive parameter that characterizes the
  magnetic state is the magnetic dipole moment I
  (Joules/Tesla or J/T). Here the fundamental
  relation takes on the form UU(S,V, I, N). The
  intensive parameter conjugate with the magnetic
  moment is Be (Tesla,T) , the external magnetic
  field that would exist in the absence of the
  system.There are no walls restrictive with
  respect to magnetic moment. (Callen page 81)

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Second Derivatives

• We have seen that the first derivatives of the
  fundamental equation have important physical
  significance.
• What we learn now is that the various second
  derivatives are all linked to important materials
  properties.
• The three primary second derivatives and the
  materials property they define are presented on
  the right.
• All other second derivatives can be expressed in
  terms of these three. These relationships are
  simply based on the fact that to be a perfect
  differential, it is required that the mixed
  second-order partial derivatives are equal. See
  example developed in equation 7.36.

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Second Derivatives

• Maxwell Relations page 182

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Molar Specific Heat for an Ideal Gas

• An ideal gas is taken from one
• isotherm at temperature T to
• another at temperature TDT
• along three different paths
• From the first law
• The heat Q is different for each path because the
  work W (W the negative of the area under the
  curves) is different
• We define the molar specific heats for two
  processes

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Molar Specific Heat for an Ideal Gas

• For a given value of n and DT demonstrate that
  Cp gt Cv

P constant
V constant
(gas expands)
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Molar Specific Heat for an Ideal Gas

• Consider an ideal monatomic gas such as He, Ne,
  Ar

Uint of an ideal gas is a function of T only
For a constant-volume process ( i to f ) no work
is done on the system
This applies to all ideal gases (monatomic and
poly-atomic)
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Molar Specific Heat for an Ideal Gas

• In the limit of infinitesimal changes

• For the monatomic gas

For all monatomic gases
Real gases weak intermolecular interaction gt
small variations of CV from predicted
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Molar Specific Heat for an Ideal Gas

• Consider a process Pconst ( i to f ' )

The first law of thermodynamics
lt applies to any ideal gas
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Molar Specific Heat for an Ideal Gas
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Molar Specific Heat for an Ideal Gas

• An adiabatic process is one in which no energy is
  transferred by heat between a system and its
  surroundings
• An adiabatic expansion
• of an ideal gas Equation of state
• PV nRT is valid
• Show that P and V of an ideal gas at any time
  during an adiabatic process are related by
  expression

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Molar Specific Heat for an Ideal Gas

• A gas is compressed adiabatically in a thermally
  insulated cylinder

• Because Uint (T) gt

• The total differential of the equation of state

eliminate dT
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Molar Specific Heat for an Ideal Gas

• Substituting R CP - CV and dividing by PV
  we obtain

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Molar Specific Heat for an Ideal Gas

• Uint of a gas includes contributions from the
  translational, rotational, and vibrational
  motion of the molecules

• From Statistical Mechanics for a large number
  of particles obeying the laws of Newtonian
  mechanics the available energy is shared equally
  by each independent degree of freedom (on the
  average!)

• Each degree of freedom contributes 1/2 kBT of
  energy per molecule

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Molar Specific Heat for an Ideal Gas

• Diatomic gas molecules have the shape of a
  dumbbell

• There are five degrees of freedom for translation
  and rotation

• The center of mass of the molecule can translate
  in the x, y and z directions

• The molecule can rotate about the x and z axes
• Neglect the rotation about the y axis moment of
  inertia Iy ltlt Ix, Iz

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Molar Specific Heat for an Ideal Gas

• The internal energy Uint for a system of N
  diatomic molecules
• 3 translations 2 rotations (vibrations
  ignored)

• The molar specific heat at constant volume CV

• The molar specific heat at constant pressure CP
  

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Molar Specific Heat for an Ideal Gas

• The internal energy Eint for a system of N
  molecules
• 3 translations 2 rotations 2
  vibrations

• The molar specific heat at constant volume CV

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Molar Specific Heat for an Ideal Gas

• Experiment molar CV of molecular diatomic
  hydrogen H2 as a function of temperature

• Three plateaus are at values predicted by the
  used classical theory

• The motion of molecules is governed by Quantum
  Mechanics. Energies are quantized. Some degrees
  of freedom may be frozen out gt do not
  contribute

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Molar Specific Heat for an Ideal Gas
diatomic molecule

• The lowest allowed state is called the ground
  state

• Vibrational states are separated by larger energy
  gap than are rotational states

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Molar Specific Heat for an Ideal Gas

• At high temperatures T gt 300K

• To explain T-dependence of CV at high temperature
  for solids use equipartition theorem each atom
  has six degrees of freedom harmonic motion in
  the x, y, and z directions

Low temperature Quantum Mechanics