Ch. 4

1
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• For simple gases like N2, O2, and Ar, the
  experimental data are nearly constant for all
  temperatures and pressures of interest, so the
  temperature variation is not considered.
• From earlier we have, for monatomic gases, the
  total internal energy is U (3/2)NkT, which
  leads to CV (3/2)nR and cv (3/2)R.
  Similarly, Cp (5/2)nR and cp (5/2)R.
• For diatomic gases, where there are more degrees
  of freedom, so we get CV (5/2)nR and cv
  (5/2)R. Similarly, Cp (7/2)nR and cp
  (7/2)R. The ratios Cp/CV cp/cv ?.

2
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• The table below shows the values of cpd for
  various temperatures and pressures. Note the
  slight variation.

3
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• Dry air is considered to be a diatomic gas, so
  the second form applies.
• The ratio, cp/cv ? 1.4.
• We then attach the subscript, d, to the specific
  heats to indicate dry air.
• This leads to cvd 718 J kg-1 K-1 (171 cal kg-1
  K-1), cpd 1005 J kg-1 K-1 (240 cal kg-1 K-1),
  and Rd cpd cvd 287(.05) J kg-1 K-1.

4
Ch. 4 1st Law of Thermodynamics

• More Forms of the 1st Law
• Using the above expressions for (specific) heat
  capacities, we get more useful forms of the 1st
  Law, two of which are particularly useful
• and

5
Ch. 4 1st Law of Thermodynamics

• Special Cases
• For an isothermal transformation
• For an isochoric (constant volume) transformation
• For an isobaric transformation

6
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• An adiabatic transformation is one in which no
  heat is exchanged with the environment (?Q 0).
• Considering our 2 primary forms of the 1st Law
• We can write

7
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• Dividing by T we have
• If we introduce the ideal gas law (pV nRT), we
  have

8
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• Integrating the logarithmic derivatives, we have
• If we differentiate the ideal gas law (pV
  nRT), we have
• Now divide through by the gas law and integrate
  to get

9
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• Substitute for dlnT into either form to get
• Work with the first form and note that Cp Cv
  nR

10
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• To summarize, the forms of the 1st Law for an
  adiabatic transformation can be any one of the
  following 3 equations

11
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• Look at the first form
• We can integrate this to give

12
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• If we define ? Cp/Cv, we get
• We can do similar integration of the other 2
  forms, yielding

13
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• These 3 expressions are called Poissons
  relations for adiabatic processes. They are
  different from the ideal gas equation. During an
  adiabatic process, all three state variables
  change according to these relationships.
• When a parcel ascends and its volume increases,
  the temperature must decrease and this decrease
  is due to the work of expansion done by the
  parcel on its environment (no heat exchange).

14
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation
• Since, for an isothermal process, the ideal gas
  equation reduces to pV const, and the isotherms
  on a (p, V) diagram are equilateral hyperbolas.
• The adiabats on a (p, V) diagram are also
  equilateral hyperbolas, but since ? gt 1 (Cp/Cv
  1.4 for air), they are steeper than the
  isotherms.
• Similar conclusions can be drawn for (p, T) and
  (T, V) diagrams.

15
Ch. 4 1st Law of Thermodynamics

• Adiabatic Transformation

16
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• Look at the equation between two states
• and simplify
• we have
• We usually write ? nR/Cp R/cp (? 1)/?.

17
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• Then we have
• If we choose p0 1000 mb, T0 becomes ?, the
  potential temperature, and we have
• In the case of calculating ratios of pressure, it
  is alright to use mb as the unit, since the ratio
  cancels the units. Only when you take ratios of
  pressure can you use mb.

18
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• The potential temperature of an ideal gas is the
  temperature that it would have if we compressed
  (or expanded) it adiabatically from its current
  pressure to 1000 mb.
• The importance of potential temperature is
  directly related to the fundamental role of
  adiabatic processes in the atmosphere.

19
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• Restrict the discussion to dry air.
• Only radiative processes cause addition or
  abstraction of heat from a parcel of air.
• More generally, we have to deal with bulk
  (averaged) properties of the atmosphere.
• Such open systems have 3D turbulent mixing causes
  movement of environmental air into an out of a
  parcel moving with the bulk flow.

20
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• So, in reality, we have to add turbulent
  diffusion of heat to the list of non-adiabatic
  processes (along with radiative effects).
• Having said that, except in the lowest 100 mb,
  such non-adiabatic processes are relatively
  unimportant (slow time scales).
• In general, we can treat changes of state in the
  atmosphere as adiabatic (or quasi-adiabatic) on
  shorter time scales.

21
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• The potential temperature, ?, is constant for an
  adiabatic process (for a fixed composition,
  insulated system/parcel).
• We can use ? const to mark a parcel of air.
• As long as it behaves adiabatically, the value of
  ? will not change.
• Potential temperature is conserved during an
  adiabatic process.
• Used to trace the origin and history of air
  parcels (tracers).

22
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• Most atmospheric air movement is along isobaric
  surfaces.
• With no external heat source acting, T const.
• The small components moving across isobaric
  surfaces are very important.
• Cross isobaric flow (convection) leads to
  pressure changes and the temperature must change
  to keep ? constant.
• Adiabatic compression/subsidence
  (expansion/ascent) is accompanied by warming
  (cooling).

23
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• If ascending air contains H2O vapor, as it
  ascends the relative humidity increases
  saturation can be reached.
• At this point condensation begins and latent heat
  is released warming the air parcel (it is cooling
  during ascent).
• Net result, the parcel cools more slowly than dry
  adiabatic ascent and ? begins to increase.
• Still an adiabatic process because the heat
  source is internal.

24
Ch. 4 1st Law of Thermodynamics

• Potential Temperature
• But this shows that our formulation of the 1st
  Law is not valid when phase changes occur.
• Potential temperature is not conserved when
  evaporation or condensation occurs.
• Presence of water vapor does not upset the
  conservancy of ?, just the process of phase
  changes.
• Later we will derive an expression, similar to ?
  that is conserved during phase changes.