Ch. 5 2nd Law of Thermodynamics

1
Ch. 5 2nd Law of Thermodynamics

• 2nd Law Summary
• Entropy is defined as a state function and (as
  internal energy) has an arbitrary additive
  constant, but we are generally interested only in
  changes in entropy.
• Formulations (with always applying to
  reversible)
• Finite process
• Adiabatic process
• Isentropic process
• Finite isothermal process
• cycle

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Ch. 5 2nd Law of Thermodynamics

• 2nd Law Summary
• We also have
• Isochoric process
• Isobaric process
• Of course entropy can be related to the 1st Law
  in general, by dividing by T, to get

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Ch. 5 2nd Law of Thermodynamics

• 2nd Law Summary
• We can also use the variables p and T using the
  other form of the 1st Law to get a slightly less
  useful expression, i.e.,
• These equalities apply to reversible processes.

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Ch. 5 2nd Law of Thermodynamics

• Entropy and the 1st Law
• If we write TdS ? ?Q we can include entropy in
  the 1st Law, which will now be generalized for
  reversible and irreversible processes (the ?
  sign).
• by virtue of the fact that dU CVdT. We can
  rewrite this
• Similarly

5
Ch. 5 2nd Law of Thermodynamics

• Free Energy Functions
• In these two expressions, U and H appear as
  functions of S and V, and S and p, respectively,
  as independent variables.
• It is often convenient to have general
  expressions where pairs T, V and T, p appear as
  independent variables.
• This leads to the introduction of two new state
  variables, the Helmholtz function (free energy)
  and the Gibbs function (free energy/enthalpy).
  They are defined as

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Ch. 5 2nd Law of Thermodynamics

• Free Energy Functions
• Both of these functions are sometimes referred to
  as thermodynamic potentials.
• We can then differentiate these expressions to
  get relationships involving pairs T, V and T, p.
• Which, using dU ? TdS pdV we get

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Ch. 5 2nd Law of Thermodynamics

• Free Energy Functions
• By the same procedure the Helmholtz function
  becomes
• leading to, (using dH ? TdS Vdp)
• Interpretation
• For isothermal process dF ? ??W and F is the
  energy that can be converted to work
• For isothermal-isobaric process (phase change) dG
  0 so G is conserved

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Ch. 5 2nd Law of Thermodynamics

• Free Energy Functions
• The table below summarizes the characteristic
  functions and their fundamental equations.

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Ch. 5 2nd Law of Thermodynamics

• State Functions
• Taking the characteristic functions in the table
  and using the sign we can manipulate the 1st
  and 2nd equations as
• Similarly we get, through combining the other
  pairs of equations (noting that these are true
  regardless of whether the process is reversible
  or irreversible).

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Ch. 5 2nd Law of Thermodynamics

• Maxwell Relations
• All the thermodynamic functions (U, H, S, F, G)
  are State Functions (exact differentials) and are
  subject to the mathematical characteristics we
  discussed earlier. That is
• Using this and the expressions derived on the
  previous slide we can get the so-called Maxwell
  equations
• Start with the 1st equation
• And substitute for T and p from the previous slide

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Ch. 5 2nd Law of Thermodynamics

• Maxwell Relations
• We also have
• But we also know that, for an exact differential
• Which can be verified by the equation above.
  Knowing that we can take the appropriate
  differentials of T and p to get the 1st Maxwell
  relationship

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Ch. 5 2nd Law of Thermodynamics

• Maxwell Relations
• The other Maxwell relations can be derived in a
  similar manner and are given by

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Ch. 5 2nd Law of Thermodynamics

• Some Discussion
• dS ? ?Q/T is a general statement of the 2nd Law.
• Max heat that can be absorbed by a system is TdS.
• Any spontaneous, irreversible process occurring
  within an isolated system, not involving external
  forces has Sf gt Si.
• The state of maximum entropy is a state of
  stable equilibrium.
• This is where the oft-quoted statement about the
  entropy of the universe increasing comes from.

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Ch. 5 2nd Law of Thermodynamics

• Non-Compensated Heat
• Rather than deal with the inequality, we could
  write
• With this convention, we could write the
  fundamental equations without the inequality,
  e.g.,
• In all cases ?Q? is called the non-compensated
  heat.
• Gives a measure of the irreversibility of the
  process, i.e., the smaller ?Q?, the closer the
  process is to being reversible.

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Ch. 5 2nd Law of Thermodynamics

• Non-Compensated Heat
• Normally, T and p are defined (for the usual
  reversible process) as the equilibrium values in
  the system.
• When the process is irreversible (inequality
  holds)
• T represents temperature of sources in contact
  with system
• p represents external pressure on system
• Condition of irreversibility depend on
  differences between these and equilibrium values,
  i.e.,
• (T - Teq) measures thermal irreversibility
• (p - peq) measures mechanical irreversibility

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Ch. 5 2nd Law of Thermodynamics

• Non-Compensated Heat
• Note that when T Teq and p peq, the system is
  in equilibrium (and therefore is reversible).
• We can therefore write
• Since the internal energy change of the system is
  the same, we have
• This expresses the total degree of
  irreversibility of the process. At equilibrium
  ?Q? goes to zero.

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Ch. 5 2nd Law of Thermodynamics

• Entropy and Potential Temperature
• The potential temperature is given by
• The enthalpy form of the 1st Law is given by
• If we divide by T we get
• Now if we divide by the mass, md, of the air
  parcel, we get

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Ch. 5 2nd Law of Thermodynamics

• Entropy and Potential Temperature
• We now have s, the specific entropy and cpd, the
  specific heat of dry air at constant pressure.
• Take logarithmic derivatives of the potential
  temperature equation to get
• The middle term on the right is zero. If we
  multiply through by cpd, we get

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Ch. 5 2nd Law of Thermodynamics

• Entropy and Potential Temperature
• The right hand sides of our equations for
  potential temperature and specific entropy are
  the same, so we have
• Leading to the final result
• Changes in entropy (for a reversible process) are
  directly proportional to changes in potential
  temperature.
• Adiabatic (reversible) processes are isentropic.

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Ch. 5 2nd Law of Thermodynamics

• Entropy and Potential Temperature
• We could write a similar equation for the
  extensive variable, S, as
• What about an irreversible adiabatic process with
  ds gt 0?
• For any adiabatic process, d? 0 ? ds for an
  irreversible process.
• Entropy increase is due to irreversible work
  (uncompensated heat), such as frictional
  dissipation.
• Isentropic equals adiabatic, but not always the
  other way.