1. Some notes on thermodynamics
1. Introduction to thermodynamics
2. Relevance for atmosphere
3. Global perspectives
4. Heat machines & Geophysics
From Wikipedia, the free encyclopedia
Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power)
is a branch of physics that studies the effects of changes in temperature, pressure, and
volume on physical systems at the macroscopic scale by analyzing the collective motion
of their particles using statistics. Roughly, heat means "energy in transit" and dynamics
relates to "movement"; thus, in essence thermodynamics studies the movement of energy
and how energy instills movement. Historically, thermodynamics developed out of the
need to increase the efficiency of early steam engines.
The starting point for most thermodynamic considerations are the laws of
thermodynamics, which postulate that energy can be exchanged between physical
systems as heat or work. They also postulate the existence of a quantity named entropy,
which can be defined for any system. In thermodynamics, interactions between large
ensembles of objects are studied and categorized. Central to this are the concepts of
system and surroundings. A system is composed of particles, whose average motions
define its properties, which in turn are related to one another through equations of state.
Properties can be combined to express internal energy and thermodynamic potentials are
useful for determining conditions for equilibrium and spontaneous processes.
With these tools, thermodynamics describes how systems respond to changes in their
surroundings. This can be applied to a wide variety of topics in science and engineering,
such as engines, phase transitions, chemical reactions, transport phenomena, and even
black holes. The results of thermodynamics are essential for other fields of physics and
for chemistry, chemical engineering, cell biology, biomedical engineering, and materials
science to name a few.
Quotes
• "Thermodynamics is the only physical theory of universal content which,
within the framework of the applicability of its basic concepts, I am
convinced will never be overthrown." — Albert Einstein
• "The law that entropy always increases - the Second Law of
Thermodynamics - holds, I think, the supreme position among the laws of
physics. If someone points out to you that your pet theory of the universe is
in disagreement with Maxwell's equations - then so much the worse for
Maxwell's equations. If it is found to be contradicted by observation - well,
1.1

2. these experimentalists do bungle things from time to time. But if your theory
is found to be against the Second Law of Thermodynamics I can give you no
hope; there is nothing for it but to collapse in deepest humiliation." — Sir
Arthur Eddington
• “Isn’t thermodynamics considered a fine intellectual structure, bequeathed by
past decades, whose every subtlety only experts in the art of handling
Hamiltonians would be able to appreciate?” Pierre Perrot, author: “A to Z
Dictionary of Thermodynamics”
• "Thermodynamics is a funny subject. The first time you go through it, you
don't understand it at all. The second time you go through it, you think you
understand it, except for one or two small points. The third time you go
through it, you know you don't understand it, but by that time you are so used
to it, it doesn't bother you any more." — Arnold Sommerfeld
1.2

3. 1. Introduction to (atmospheric) thermodynamics
1.1 On the atmosphere
The atmosphere is an ever changing fluent layer that circumvents our earth. The
atmosphere constantly exchange heat, forces, and mass between the solid earth surface
and the outer space.
The atmosphere consists of a number of different gases and materials. Many of these
play an important role for setting the structure of the atmosphere. The atmosphere mainly
1.3

4. consists of nitrogen gas, N2, Oxygen, O2, Argon, Ar, etc. To a smaller extent there is also
carbon dioxide, CO2, Methane, CH4. These latter gases are strongly affected by biological
activity, and by burning of fossil fuels and organic material. Accordingly, the
concentration of these gases varies significantly with time. Notably, they also play an
important role for the radiation budget of the earth although they are not important
directly for the thermodynamical properties of the atmosphere – which is the subject of
this course.
A material of special interest for this course is water, which exist in its all three phases
in the natural environment. Furthermore, the transition between water vapor and
water/ice is very important for many atmospheric processes and describing these
transition properties will be one of the main tasks in this course.
1.1.2 Specific for meteorology
There are few things which may be considered to be specific for the atmosphere and
how we deal with it from a thermodynamic viewpoint.
1. Mixture of gases, including water vapor.
2. Pressure decreases with height.
3. Is forced thermally at the ground and some heights (by radiation). Exchange with
ground (and space).
4. Is in constant movement (is not in equilibrium).
Thermodynamically the following point receive most attention
1. Dry air (torr luft)
2. Water in its three phases (aggregationstillstånd). Ice, water, gas (is, vatten, ånga).
3. Mixture of dry air and moisture (luft+ånga=fuktig luft)
4. A mixture of moist air and water droplets and/or ice crystals.
From Wikipedia, the free encyclopedia
In the physical sciences, atmospheric thermodynamics is the study of heat and energy
transformations in the earth’s atmospheric system. Following the fundamental laws of
classical thermodynamics, atmospheric thermodynamics studies such phenomenon as
properties of moist air, formation of clouds, atmospheric convection, boundary layer
meteorology, and vertical stabilities in the atmosphere. Atmospheric thermodynamic
diagrams are used as tools in the forecasting of storm development. Atmospheric
thermodynamics forms a basis for cloud microphysics and convection parameterizations
in numerical weather models, and is used in many climate considerations, including
convective-equilibrium climate models.
1.4

5. 1.1.3 This course
We will analyze some thermodynamic properties of the atmosphere highlighting the
influence of water – and its phase transitions – on the atmosphere. We will also consider
the main force balances of the atmosphere in this course and the vertical acceleration of
air parcels – this is referred to as “statik”.
1.2 Basic thermodynamic stuff
Thermodynamic principles are firmly rooted in experiments. Thermodynamics is a
science of measurable quantities rather than invisible constructors. It seeks no
explanation at a level below what we can observe directly with our coarse senses and
measuring instruments. Thus thermodynamics treats phenomena on a macroscopic scale.
Accessible to us and our instruments, as opposed to a microscopic scale, the atomic and
molecular scale. Thus, thermodynamics do not rely on the existence of invisible atoms
and molecules – they provide a way to explain processes in a different way but the
thermodynamic theory would essentially remain unchanged if we discover that the
prevailing theory of atoms and molecules were wrong. In some regards thermodynamics
is phenomenological in the way that it does not try to explain underlying causes in detail,
rather it deals the description and classification of phenomenon.
1.2.1 There are some important concepts in thermodynamics.
From Wikipedia, the free encyclopedia
An important concept in thermodynamics is the
“system”. A system is the region of the universe under
study. A system is separated from the remainder of the
universe by a boundary which may be imaginary or not,
but which by convention delimits a finite volume. The
possible exchanges of work, heat, or matter between the
system and the surroundings take place across this
boundary. There are five dominant classes of systems:
Isolated Systems – matter and energy may not cross the boundary.
Adiabatic Systems – heat may not cross the boundary.
Diathermic Systems - heat may cross boundary.
Closed Systems – matter may not cross the boundary.
Open Systems – heat, work, and matter may cross the boundary.
For isolated systems, as time goes by, internal differences in the system tend to even out;
pressures and temperatures tend to equalize, as do density differences. A system in which
all equalizing processes have gone practically to completion, is considered to be in a state
of thermodynamic equilibrium.
1.5

6. In thermodynamic equilibrium, a system's properties are, by definition, unchanging in
time. Systems in equilibrium are much simpler and easier to understand than systems
which are not in equilibrium. Often, when analyzing a thermodynamic process, it can be
assumed that each intermediate state in the process is at equilibrium. This will also
considerably simplify the situation. Thermodynamic processes which develop so slowly
as to allow each intermediate step to be an equilibrium state are said to be reversible
processes.
1. System: A system is a well defined volume that we intend to study.
a. Closed system: Is not in contact with the outside world??
i. There are no sharp and impermeable boundaries in nature. All such
boundaries are purely mathematic.
b. Open system: Interacts with the outside world, for instance can receive
energy.
2. Equilibrium: This is an important concept in thermodynamics. The postulate of
local equilibrium is just that, a postulate, not a law. More than a postulate it is a
hope that nature is kind to us.
a. Internal equilibrium (inre jämvikt.). A system that does to exchange any
properties with the surrounding is in equilibrium.
b. External equilibrium (yttre jämvikt). A system is in equilibrium with an
external contact.
c. Systems can be described relatively well even if there is not a true
equilibrium. Requires that a system is relatively close to equilibrium.
i. Thus we will only consider small infinitesimal disturbances.
ii. Packets that we will frequently refer to has scales
molecules<<package<<system.
3. Temperature: Can only be defined as something that is in equilibrium with an
external system. Temperature requires equilibrium to be defined and it is a
complicated variable to define and measure. There exist very many different
meteorological properties defined as temperature. Potential temperature, virtual
temperature, potential virtual temperature, dew point, etc.
4. Macro variables.
a. Mass (m or ∆ m); The total mass of the system is a key variable of the
system
b. Volume (V or ∆ V); In meteorology the volume is often expressed in a
different form namely as
i. Density, (ρ =M/V or ρ =∆M/∆V); Density is a commonly used
variable in dynamical meteorology.
1.6

7. ii. Specific volume (α =∆V/∆M). Is used in the thermodynamic
analysis of meteorology. Notably ρα=1. In principle we can
always use α instead of V in the thermodynamics relation provided
that we change appropriate constants.
c. Pressure (p); Pressure implies the presence of a force perpendicular to an
area A which equals
∆F
i. p= .
∆A
ii. Unit is Pa=1N/m2. Often used is mbar=102Pa=1hPa.
d. Temperature (T or θ ); Is relatively complicated. Requires equilibrium to
be defined in a proper way. Uses Kelvin scale (K) or Celsius (oC).
0oC=273.15 K.
i. There are a large number of temperature definitions in
meteorology.
5. Micro variables: Molecular speed, number of molecules etc. Not really considered
in thermodynamics, belongs to statistical mechanics.
6. Equation of state (tillståndsekvationen). An important pile stone of the
thermodynamics is that there exist some well described relations between the
physical quantities P, V, and T. (in meteorology, p, α , T).
a. pV = nR *T : n is number of moles, R*(=8.3144 J mol-1 K-1) universal gas
constant.
i. Alternative form pv = RT , v is molar volume (volume V per mole
n. Avogadros number v=22.414 m3/mol for p0=1 atm, T0=273 K).)
ii. Meteorology: pα = RT , or p = ρRT . R=R*/M (M=molmassa) is
the specific gas constant (specifika eller inviduella gaskonstanten).
 a 
iii.  p + 2 ( v − b ) = nR T . Van der Waals equation. a, b are
*
 v 
constants appropriate for each gas.
b. Daltons law: A mixture of ideal gases will behave as an ideal gas. In a gas
containing more than one component, each component add to the total
pressure (for an ideal gas).
i. p k V = n k R *T , thus Volume and Temperature are “global”
quantities.
*
T T n k Rk
ii. Total pressure p = ∑ p k = ∑ nk R = V n∑ n
*
k
k V k k
1.7

8. iii. For air: pα = Rd T , Rd=287.0 J/K. Specific molmass Md=R*/
Rd=28.97 kg/mol.
iv. Air composition is relatively constant up to 100 km, thereafter it
decreases with height.
Figure text: Mean molecular weight versus height for U.S. Standard Atmosphere.
c. An important concept is that the relation between state variables can be
written F(p, α, T)=0.
i. If two variables are known, the third can be calculated. We may
write p= p (α,T), α = α (p, T), T = T(p, α). The direct implication
is that all functions will varies independently on two of the states
variables only, that is U= U1(α,T), U2 (p, T), U3(p, α). Notably, U1,
U2, U3 are three completely different functions. However to avoid
naming an incomprehensive number of functions one usually skip
the index on the function.
1.3 Laws of thermodynamics
From Wikipedia, the free encyclopedia
In thermodynamics, there are four laws of very general validity, and as such they do not
depend on the details of the interactions or the systems being studied. Hence, they can be
applied to systems about which one knows nothing other than the balance of energy and
matter transfer. Examples of this include Einstein's prediction of spontaneous emission
around the turn of the 20th century and current research into the thermodynamics of black
holes.
The four laws are:
1.8

9. Zeroth law of thermodynamics, stating that thermodynamic equilibrium is an
equivalence relation.
If two thermodynamic systems are in thermal equilibrium with a third, they are
also in thermal equilibrium with each other.
First law of thermodynamics, about the conservation of energy
The increase in the energy of a closed system is equal to the amount of energy
added to the system by heating, minus the amount lost in the form of work done
by the system on its surroundings.
Second law of thermodynamics, about entropy
The total entropy of any isolated thermodynamic system tends to increase over
time, approaching a maximum value.
Third law of thermodynamics, about absolute zero temperature
As a system asymptotically approaches absolute zero of temperature all processes
virtually cease and the entropy of the system asymptotically approaches a
minimum value.
My view
• 0’th law: Two systems that are in thermal equilibrium with a third are in
thermal equilibrium with each other.
• 1’st law: Energy is conserved. For instance
 dv  d v2 dz
v ⋅ m = mge z  ⇒ m = mg ⇒
o  dt  dt 2 dt
,
d
( K + P ) = 0 ⇒ K + P = const
dt
 Energy is conserved in a mechanistic system, can change
between the kinetic energy of the point mass (energy of motion)
and the potential energy (energy of position).
 For system with many particles, it can be shown that we also
consider changes in kinetic and potential energy in the way the
mass centre moves (the kinetic energy of a fictious body with
mass equal the entire system of point masses and moving with
its centre of mass velocity) and in the way the particles move
randomly or disorganized, this thus also represent a kinetic
energy but may represent internal kinetic energy (or
temperature)
 Ordered mean motions are easily transferred to internal random
motions. Consider a balloon filled with water. If we drop it will
have a certain mean motion before it strikes the floor. However,
after hitting the floor the mean motion becomes zero and must
1.9

10. have been transferred to disorganized random motion with zero
net mass transfer. Conservation of energy implies that internal
motion must have increased. In other words mean motion
(external) have been transformed into disorganized motion
(heat). Thermodynamics is a way to describe these processes
with giving an exact description of the entire procedure.
o If a closed system is caused to change from an initial state to a final
state by adiabatic means only, then the work done on the system is
same far all adiabatic paths connecting the two states.
o If there is an exchange of properties between the system and the
outside word, the work will depend on the exact pathway.
o Internal energy.
dU
 = Q − W ; Q is heating rate, W working rate. (these are
dt
equal zero for a closed system).
dα
• In general the work is W = p which implies that we
dt
dU dα
often write =Q− p .
dt dt
• It should be noted that we require that ∆x ∆t << v s ,
where vs is the speed of molecules (i.e., roughly the
speed of sound) for the system to be in a reasonable
quasi-stationary state. This feature is always valid for
the atmosphere and is actually well fulfilled in a normal
engine.
 dU = dQ + dW
• We need to distinguish between exact differentials
(denoted d) which refers to state variables, and inexact
differentials (denoted d ) that refers to external forcing
parameters.
• 2’nd law: Entropy always increases. (Whatever than means)
The best way to understand energy and entropy – indeed, all concepts, scientific or
otherwise – is to use them in as many contexts as possible, proceeding from the familiar
to the unfamiliar.
From Wikipedia, the free encyclopedia
1.10

11. Thermodynamic processes
A thermodynamic process may be defined as the energetic evolution of a
thermodynamic system proceeding from an initial state to a final state. Typically, each
thermodynamic process is distinguished from other processes, in energetic character,
according to what parameters, as temperature, pressure, or volume, etc., are held fixed.
Furthermore, it is useful to group these processes into pairs, in which each variable held
constant is one member of a conjugate pair. The six most common thermodynamic
processes are shown below:
1. An isobaric process occurs at constant pressure.
2. An isochoric process, or isometric/isovolumetric process, occurs at constant
volume.
3. An isothermal process occurs at a constant temperature.
4. An isentropic process occurs at a constant entropy.
5. An isenthalpic process occurs at a constant enthalpy.
6. An adiabatic process occurs without loss or gain of heat.
Thermodynamic potentials
As can be derived from the energy balance equation on a thermodynamic system there
exist energetic quantities called thermodynamic potentials, being the quantitative measure
of the stored energy in the system. The four most well known potentials are:
Internal energy
Helmholtz free energy
Enthalpy
Gibbs free energy
Potentials are used to measure energy changes in systems as they evolve from an initial
state to a final state. The potential used depends on the constraints of the system, such as
constant temperature or pressure. Internal energy is the internal energy of the system,
enthalpy is the internal energy of the system plus the energy related to pressure-volume
work, and Helmholtz and Gibbs free energy are the energies available in a system to do
useful work when the temperature and volume or the pressure and temperature are fixed,
respectively.
1.11

12. 1.4 Changes due to heating
Lets us assume that we heat a certain volume of gas. The response may be considered
from the response in time or as ordinary differentials. Personally I do think that it is
easier to consider changes in time than in differentials. Let us write
dU dα
Q= +p
dt dt
The internal energy U for a simple closed system such as a gas may be considered s
function of the two independent variables, temperature T and volume V, while the third
variable p is related to these two by the ideal gas law.
1.4.1 Constant volume
Applying the chain rule simply provides
dU ∂U dT ∂U dα
= +
dt ∂T dt ∂α dt
Notably ∂U ∂α has the dimension of pressure and
is sometimes called the internal pressure (for an ideal
gas ∂U ∂α =0, Joules law and has been confirmed to
be small in experiments using real gases). Continuing
∂U ∂α = 0 for an ideal gas (pdV=0 for
∂U dT ∂U dα dα
Q= + +p above experiment)
∂T dt ∂α dt dt
∂U dT  ∂U  dα
= + + p
∂T dt  ∂α  dt
The first partial derivative on the right hand side of this equation appears with
sufficient frequency that is has acquired a name, heat capacity (at constant volume)
∂U
Cv =
∂T
Its relevance is clear from the following equation appearing for constant volume.
dT
Q = Cv , V=const.
dt
Notably we also find that
∂C v ∂ ∂U ∂ ∂U
= = ≈0
∂V ∂V ∂T ∂T ∂V
This relation holds for an ideal gas where interactions between molecules are negligible.
Cv depends on the total mass of the system and sometimes it is convenient to deal with
the specific heat capacity instead, thus
1.12

13. cv
Cv = .
m
1.4.2 Constant pressure
To find how the system respond to heating under constant pressure we need to find an
expression which has dp/dt. Starting as before
dU dα
Q= +p
dt dt
dα d  T  R dT RT dp
using that pα = RT we can write =R  = − 2
dt dt  p  p dt
  p dt
dU  R dT RT dp  dU dT RT dp
Q= + p
 p dt − p 2 dt  = dt + R dt − p dt

dt  
dU dT dT RT dp
= +R −
dT dt dt p dt
under constant pressure we find
dT dT
Q = ( Cv + R ) = Cp
dt dt .
C p = Cv + R
We thus find that the heat capacity (or resistance to become warm when heated) is
larger under constant pressure than under constant volume. To show that cp greater than
cv makes physical sense, consider an ideal gas confined to a cylinder fitted with the usual
frictionless nut tightly fitted piston. Fix the piston in place and heat the gas for a certain
amount of time. The temperature of the gas rises in this constant volume process. Now let
the piston move freely so that gas pressure is constant. Heat the gas for the same amount
of time a before. Again the temperature increases, but in this process the piston rises; thus
work is done by the gas and consequently its internal energy doesn’t increase as much as
before. This implies that the temperature increase isn’t as great. Stated another way, cp is
greater than cv, which is consistent with what we derived.
1.13

14. For liquids and solid material the compressibility (implying that pdα≈0) is essentially
zero and thus
C p ≈ Cv .
This can be shown but I am too lazy 
1.4.3 Constant pressure 2
The derivations for the temperature response to heating under constant volume and
constant pressure took different paths. However, let us redo the calculation in another
way, it is convenient to introduce enthalpy, defined as
H = U + pα
dU dα
The first law Q = +p can now be written
dt dt
dH dp
Q= −α
dt dt
the chain rule gives
dH ∂H dT ∂H dp
= + .
dt ∂T dt ∂p dt
We thus have
∂H dT ∂H dP dP
Q= + −α
∂T dt ∂P dt dt
∂H dT  ∂H  dp
= + −α
∂T dt  ∂V  dt
1.14

15. For constant pressure we have that the heat capacity (at constant volume) is
∂H
Cp =
∂T
Its relevance is clear from the following equation appearing for constant volume.
dT
Q = CP , p=const.
dt
Using the definition of entalphy (and conidering the independent variables to be T and
p)
H = U + pα
H (T , p ) = U (T , α ) + pα
∂H dT ∂H dp ∂U dT ∂U dα dp dα
+ = + +α +p
∂T dt ∂P dt ∂T dt ∂α dt dt dt
dα ∂α dp ∂α dT
α = α ( p, T ) ⇒ = +
dt ∂p dt ∂T dt
 ∂H ∂U ∂U ∂α ∂α  dT  ∂H ∂U ∂α dα  dp
 − − −p  +
 ∂P − V ∂α ∂p − p dp  dt = 0

 ∂T ∂T ∂α ∂T ∂T  dt  
thus
∂U ∂α ∂α
C p = Cv + +p
∂α ∂T ∂T ∂U
, ideal gas = 0 (Joules law).
∂α ∂α
C p = Cv + p = Cv + R
∂T
Note that the intermolecular forces are negligible in an ideal law. Joules law can be
showed in experiments where a chamber is split in two parts. The left part has a certain
pressure and the right volume has no pressure. If the wall is removed gas will go from left
to the right side. There is no exchange of heat, no work done and thus the internal energy
must remain constant.
∂U ∂α ∂α
Joules law and the expression C p = C v + +p will be considered in later
∂α ∂T ∂T
sections. However, to pave the way forward we need to introduce the concept entropy,
which is done in section 3.
From Wikipedia, the free encyclopedia
In thermodynamics, the quantity enthalpy, symbolized by H, also called heat content, is
the sum of the internal energy of a thermodynamic system plus the energy associated
1.15

16. with work done by the system on the atmosphere which is the product of the pressure
times the volume. The term enthalpy is composed of the prefix en-, meaning to "put
into", plus the Greek suffix -thalpein, meaning "to heat".
Enthalpy is a quantifiable state function, and the total enthalpy of a system cannot be
measured directly; the enthalpy change of a system is measured instead. A possible
interpretation of enthalpy is as follows. Imagine we are to create the system out of
nothing, then, in addition to supplying the internal energy U for the system, we need to
do work to push the atmosphere away in order to make room for the system. Assuming
the environment is at some constant pressure P, this mechanical work required is just PV
where V is the volume of the system. Therefore, colloquially, enthalpy is the total amount
of energy one needs to provide to create the system and then place it in the atmosphere.
Conversely, if the system is annihilated, the energy extracted is not just U, but also the
work done by the atmosphere as it collapses to fill the space previously occupied by the
system, which is PV.
Enthalpy is a thermodynamic potential, and is useful particularly for nearly-constant
pressure processes, where any energy input to the system must go into internal energy or
the mechanical work of expanding the system. For systems at constant pressure, the
change in enthalpy is the heat received by the system plus the non-mechanical work that
has been done. In other words, when considering change in enthalpy, one can ignore the
compression/expansion mechanical work. Therefore, for a simple system, with a constant
number of particles, the difference in enthalpy is the maximum amount of thermal energy
derivable from a thermodynamic process in which the pressure is held constant.
1.16