APPLICATIONS OF THE FIRST LAW - PowerPoint PPT Presentation

1 / 30
About This Presentation
Title:

APPLICATIONS OF THE FIRST LAW

Description:

Without delving into the kinetic theory of gases, we can make ... readily deduced by considering an isothermal cyclic engine in contact with a single reservoir, ... – PowerPoint PPT presentation

Number of Views:116
Avg rating:3.0/5.0
Slides: 31
Provided by: andrew667
Category:

less

Transcript and Presenter's Notes

Title: APPLICATIONS OF THE FIRST LAW


1
  • APPLICATIONS OF THE FIRST LAW
  • Kinetic theory and internal energy predicting
    the specific heats of dry air
  • Applications of the first law
  • Kirchoffs theorem
  • Poissons equation
  • Potential temperature

KINETIC THEORY AND INTERNAL ENERGY Without
delving into the kinetic theory of gases, we can
make use of one of its results in order to obtain
some insight into internal energy. each molecule
possesses an amount of energy per degree of
freedom equal to where k is Boltzmanns
constant1.38?1023 JK-1 . We can think of the
number of degrees of freedom as the number of
variables required to fully describe the motion
of the molecules. So a monatomic molecule has 3
degrees of freedom (3 velocity components are
required to specify its translational motion), a
diatomic molecule has 5 degrees of freedom (3
translational plus 2 angular velocities), while a
triatomic molecule has 6 degrees of freedom (3
translational, 3 angular). Dry air consists
primarily of molecular Nitrogen (N2) and Oxygen
(O2) and so is essentially diatomic with 5
degrees of freedom.
2
Hence the internal energy per mole is given by
(4.1)
where n is the number of degrees of freedom and
NA is Avogadros number equal to NA6.02?1023
mol-1 . Note Boltzmanns constant may therefore
be thought of as the universal gas constant per
molecule. The internal energy per kilogram
(i.e., the specific internal energy) is therefore
given by
(4.2)
where M is the molecular weight of the gas. Thus,
we would predict that for dry air, and since
then . Moreover,
since cp-cvR, then we can also predict that for
dry air . Using R287.05 Jkg-1K-1
, our prediction becomes for dry air cp1005
Jkg-1K-1 . The table below shows that this
prediction compares rather well with observations.
3
-40oC 40oC
30 kPa 1004 1006
100 kPa 1006 1007
Table 4.1 Measured specific heat capacity at
constant pressure, cp, for dry air.
4
POISSONS EQUATIONS Despite the fact that energy
in the atmosphere/ocean system ultimately comes
from radiative heating, adiabatic processes in
the atmosphere are of interest for several
reasons. Often it is because real atmospheric
processes occur quickly in comparison with the
time scale for heat transfer, and so may be
considered to be approximately adiabatic. Alterna
tively, we may wish to make the adiabatic
assumption simply because we are ignorant of the
heat transfer and consequently must ignore it or
give up. Poissons equations describe relationship
s between the state variables T, p, and ? for
adiabatic processes. When ?q0 (adiabatic
process), the first version of the first law may
be written
(4.5)
Substituting for p from the ideal gas law,
dividing by cvT, and using Rcp-cv we have,
after some manipulation
(4.6)
where ?cp/cv (7/5 for an ideal gas). This
implies that, for an ideal gas undergoing an
adiabatic process
(4.7)
Eq. (4.7) is one of three Poisson equations.
5
  • Eq. (4.7) can be used to explain a number of
    atmospheric phenomena. For example
  • When you compress the air in a bicycle pump, v
    decreases and hence T increases and the
  • air warms. (Sometimes the bicycle pump is
    noticeably warmer after use.)
  • 2. A similar explanation can be offered for the
    heating of subsiding (i.e., descending) air that
  • gives rise to a Chinook.
  • 3. Conversely, in a rising, expanding thermal, v
    increases and so T falls. This is a dynamical
  • component that contributes to the decrease of
    temperature with height in the troposphere.
  • (The other component is the height dependence of
    absorption of longwave radiation emitted by
  • the Earth.)

Starting with the second version of the first
law, Eq. (3.8) and following a similar
manipulation, one may arrive at the second
Poisson equation for an adiabatic process in an
ideal gas
(4.8)
where ?R/cp (2/7 for an ideal gas). Finally,
by combining Eqs. (4.7) and (4.8), we arrive at
the third Poisson equation
(4.9)
Where ? cp / cv7/5. Eq. (4.9) is the most
familiar of Poissons equations, although they
are all equivalent.
6
POTENTIAL TEMPERATURE
Meteorologists use Eq. (4.8) to define a quantity
known as potential temperature (a thermodynamic
concept that seems to be unique to meteorology).
Suppose we start with a parcel of air in some
arbitrary initial state specified by T, p. Let us
move the air parcel adiabatically to a pressure
of 100 kPa and call the temperature which it
achieves the potential temperature, ?. By using
Eq. (4.8) in the initial and final states, it can
be easily shown that
(4.10)
Where p must be express in kPa (since we have
used 100 kPa in the numerator). Because
the potential temperature of an air parcel is
conserved under dry adiabatic processes, it may
be used as a tracer for air parcels. Isentropic
coordinates are ones in which potential
temperature is used as the vertical
coordinate instead of height. In such a
coordinate system, a parcel of dry air undergoing
only adiabatic processes will always remain on
the same coordinate surface. NOTES Why they are
called isentropic coordinates will become clear
when we discuss entropy and the Second Law of TD.
We will also take into account the presence of
water vapour. A similar concept is used in
oceanography. These are isopycnal
coordinates. The first law of thermodynamics
may be expressed in yet a third version using
potential temperature. Let us begin with the
definition of potential temperature, Eq. (4.10),
take logarithms and then total differentials
7
(4.11)
Multiplying by cp and using the ideal gas law and
the second version of the first law
(4.12)
This third version of the first law has several
useful features. It is somewhat simpler than the
previous two versions, it expresses the fact
that the potential temperature is constant
for adiabatic processes and varies for diabatic
processes, and it seems to imply that the
quantity on the right hand side is a function of
state (since the potential temperature is a
function of state). We will encounter ?q/T again
soon in conjunction with the Second Law of
Thermodynamics. By considering Eq. (4.12) along
with the first version of the first law for a
cycle, and remembering that a function of state
does not change in a cyclic process (e.g.,
), it is easy to show that The
integral on the right hand side is the net work
done during the cycle. Hence this
equation implies that on a graph with axes cpln?
and T, the area enclosed by a cycle will equal
the work done during the cycle. We will make use
of this fact later when we consider the Tephigram.
8
  • THE SECOND LAW OF THERMODYNAMICS
  • Thermodynamic surface for an ideal gas
  • Second Law
  • Entropy

THERMODYNAMIC SURFACE FOR AN IDEAL GAS The ideal
gas law is the equation of a surface in a
three-dimensional space whose coordinates are p,
v, and T. This surface constitutes the ensemble
of all states of an ideal gas that are permitted
by the ideal gas law. It is called the
thermodynamic surface for that gas. Any changes
in the gas state variables simply reflects
movement on this surface. The following diagram
illustrates this surface and on it are examples
of isobaric, isothermal, isosteric, and adiabatic
processes. Such processes may all be represented
in the form


(5.1) Where n0 is an isobar, n1 is an
isotherm, n? is an adiabat, and n?? is an
isostere. NOTE an isosteric process is one for
which the specific volume remains constant. Such
a process is also an isopycnic process because
the density is constant. Processes for which the
total volume of the air parcel remains constant
are called isochoric.
9
(No Transcript)
10
(No Transcript)
11
SECOND LAW Sometimes the use of the first law
will give rise to a seemingly impossible result.
How do we know that it is impossible? Because it
is entirely inconsistent with our collective
experience. As an example, consider the cooling
of bathwater. We will assume this to be an
isobaric process, so that the first law is
(5.2)
Let us imagine that 100 litres of water cool from
50oC to 20oC. If that heat is given to 10 kg of
air in the bathroom (assuming the bathroom to be
well insulated so that no heat is lost),
what will be the final temperature of the air?
If the warming of the air also occurs
isobarically then the first law for the air is
(5.3)
Equating the ?Qs in Eqs. (5.2) and (5.3), we can
solve for dTa using cp1.0?103 Jkg-1K-1 and
cw4.2?103 Jkg-1K-1. The temperature change of
the air is 1.3?103 K!!! We know this result is
impossible, but it follows from the first law, so
what is wrong? Clearly, in the final stages of
the process as we have envisaged it, the air will
be warmer than the bathwater. And yet we have
implicitly assumed that heat will continue to
flow from the water into the air to warm it.
Experience tells us that heat simply does not
flow spontaneously from a colder to a warmer
body. This is called the Clausius formulation of
the second law.
12
Rudolf Clausius (1822-1888) was a German
physicist who established the foundations
of modern thermodynamics in a seminal paper of
1850. He introduced the concepts of
internal energy and entropy (from the Greek for
transformation). His great legacy to physics was
the concept of the irreversible increase of
entropy (Die Energie der Welt ist constant
die Entropie strebt einen Maximum zu. (1865)).
13
A more precise expression of the Clausius
formulation is
It is impossible to construct a cyclic engine
that will produce the sole effect of transferring
heat from a colder to a hotter reservoir.
The term sole effect means that no work can be
done. Of course heat pumps exist, so it
is possible to transfer heat from a colder to a
warmer reservoir, but only by doing work. Oxford
physicist P.W. Atkins in his book entitled The
Second Law states it in yet another way (p. 9)
although the total quantity of energy must be
conserved in any process.the distribution of
that energy changes in an irreversible manner.
He also talks about a fundamental tax, stating
that Nature accepts the equivalence of heat and
work, but demands a contribution whenever heat is
converted into work. (p. 21). Note, however,
that there is no tax when work is converted into
heat (for example, by friction). Since we may
think of heat as disordered motion, and work as
ordered motion, it would appear that the second
law also has something to say about a fundamental
disymmetry between order and disorder. In order
to formulate the second law mathematically, we
will take another look at the bathtub problem,
invoking a cyclic engine to transfer the heat
between the room air and the water in the tub, as
in the following diagram.
14
We will focus our attention this time on the
thermodynamics of the cyclic engine. Let
us suppose that Tw?Ta . Let ?qa be the heat added
to the engine by the room air, and let ?qw be
the heat added to the engine by the bath water.
Our second law experience says that the former
will be positive and the latter negative (that is
the air, being warmer, will give up heat to the
engine and the bathtub water will gain heat from
the engine). Moreover, the first law requires
that the magnitudes of these heat transfers must
be the same. Hence we may write ?qa ?q -?qw
where we insist that the quantity?q must be
positive. Let us now consider . Why we do
this is not immediately obvious, although you
should recall from last lecture that ?q/T is a
function of state. In this case
15
The inequality in Eq. (5.4) holds in general for
real (irreversible) processes. The equality in
Eq. (5.4) will only hold as the two temperatures
become equal. Heat transfer under such
near-equilibrium conditions will be very slow
and essentially reversible. If we define entropy,
s, as follows
(5.5)
then, generally (dropping the line integral)
(5.6)
where the inequality holds for irreversible
processes and the equality for reversible processe
s. Clearly for adiabatic irreversible processes,
the entropy must always increase, leading to the
previously mentioned statement by Clausius.
NOTE See also Stephen Hawkings A Brief
History of Time for a discussion of entropy and
the arrow of time (Ch. 9). For adiabatic
reversible processes, entropy remains
constant. Formally, we can think of entropy as a
function of state whose increase gives a measure
of the energy of a system which has ceased to be
available for work during a given process. Note
also from last lecture that dscpdln? for
reversible processes. Therefore, processes
in which potential temperature is conserved are
also isentropic processes.
16
Although it is important to be aware of the
second law of thermodynamics, the field of
atmospheric thermodynamics makes little use of
it. We tend to assume that atmospheric processes
occur reversibly. However, since our atmosphere
is in actuality an enormous heat engine that
transports heat from the tropics to the poles,
the concepts and ideas that arise from the second
law are of some use.
SUMMARY OF REVERSIBLE AND IRREVERSIBLE
PROCESSES From Zemansky and Dittman,
Macroscopic Physics A reversible process is
one that is performed in such a way that, at the
conclusion of the process, both the system and
the local surroundings may be restored to their
initial states, without producing any changes in
the rest of the universe. For example, imagine a
weight on a pulley. Suppose that the weight
is lowered by a small amount so that work is done
and a transfer of heat takes place from
the system to its surroundings. If the system can
be restored to its original state, I.e., the
weight is lifted back up and the surroundings
forced to part with the heat that they had gained
during the lowering, then the original process is
reversible. A little thought should convince
you that no real process is reversible. The
abstraction of reversible processes, however,
provides a clean theoretical foundation for the
description of real world, irreversible
processes.
17
What real world processes makes things
irreversible? Dissipative effects, such as
viscosity, friction, inelasticity, electric
resistance, and magnetic hysteresis,
etc. Processes for which the conditions for
mechanical, thermal, or chemical equilibrium,
i.e., thermodynamic equilibrium are not
satisfied. In atmospheric and oceanic sciences,
it is common to assume that there are no
dissipative effects (particularly for flows at
large spatial scales) and that motion of the
atmosphere and oceans is isentropic. Viscous and
frictional dissipation do, of course, occur
(particularly in the atmospheric and oceanic
boundary layers) but a good understanding of the
dynamics can be obtained from inviscid theory.
18
  • APPLICATIONS OF THE SECOND LAW
  • Carnot cycle
  • Applications of the second law
  • Combined first and second law
  • Helmholtz and Gibbs free energies

CARNOT CYCLE The Carnot cycle is an ideal heat
engine cycle that offers insights into other heat
engines, including the atmosphere. It is a
reversible cycle, performed by an ideal gas, that
consists of two isothermal processes linked by
two adiabatic processes, as illustrated below
A
1
4
B
2
D
C
3
19
Nicolas Leonard Sadi Carnot (1796-1832) led a
short but interesting life. He was able to pursue
his scientific research as a result of his
appointment to the army general staff. He died of
scarlet fever followed by cholera, and all his
papers were burned after his death. Only three
major scientific works survive. His work on heat
engines led to the ideal heat engine and cycle
that now bear his name. Interestingly, he was
able to make significant scientific advances
while still believing in the erroneous caloric
theory of heat, in which heat is taken to be a
function of state.
20
For each of the four processes in the Carnot
cycle, we will consider the change in
internal energy, the heat added to the working
gas and the work done by the gas (i.e., the
components of the first law).
Isothermal expansion at temperature T1 du10 ?w1?q1
Adiabatic expansion between T1 and T3 du2cv(T3-T1) ?q20 ?w2-?u2 ds20
Isothermal compression at T3 du30 ?w3?q3
Adiabatic compression between T3 and T1 du4cv(T1-T3) ?q40 ?w4-?u4 ds40
Entire Carnot cycle du30 ?q?q1?q3 ?w ?q1 ?q3 ds0
Table 1 Thermodynamics of the Carnot cycle
Notes 1) ?q3 is negative because we consider ?q
to be the heat added to the system. 2) The net
work done in a Carnot cycle must equal the
difference between the heat input and the heat
exhausted. This work is also equal to the area
contained within the Carnot cycle on a p-v
diagram. 3) In order to show that ds0, we must
show that vB/vAvC/vD. This can be done by
comparing the Poisson equations for the two
adiabatic processes.
21
The efficiency of the Carnot cycle is defined to
be the ratio of the mechanical work done to
the heat absorbed from the hot reservoir. That is
(6.1)
It can be shown that this is the maximum
efficiency of any cyclic engine working between
the same two temperatures. This is known as
Carnots theorem. Note that the efficiency of
the Carnot cycle can only be unity (i.e., 100)
if the cold reservoir (T3) has a temperature
of absolute zero. In order to increase the
efficiency of an engine, one would like to
minimize T3/T1. EQUIVALENCE OF HEAT AND WORK,
EFFICIENCY, AND THE FIRST LAW Recall the first
law du?q- ?w. If we consider an engine cycle in
a p-v diagram then the path taken (clockwise for
a heat engine, counterclockwise for a
refrigerator) may be expressed as the line
integral of the first law around the cycle
Furthermore, since du is an exact differential
its line integral vanishes, leaving us with
(6.2)
22
i.e. there is an equivalence of net heat
gained/lost and net work done by/on the
system. Efficiency, E, may also be considered to
be the ratio of Work Out to Heat In.
Consider the following idealized engine
cycle The net heat input is QHQ12Q23.
The net heat output is QCQ34Q41. The net heat
in is therefore QQH-QC and we now know that this
must be the net work done, i.e., W QH-QC . The
efficiency may then be expressed in terms of the
heat as
Important Note QH and QC here are MAGNITUDES of
heat, and are therefore always positive.
23
Example Suppose that the working fluid is an
ideal gas. From the first law we have that
dU?Q- ?W. The incremental work done is related
to pressure and volume by ?WpdV. Also, the
change in internal energy dU along any branch of
the cycle is equal to dUCvdT(Cp-nR)dT. NOTE
we are not using molar values here, so
Cp-CvnR. Branch 1?2 Volume constant
So heat is absorbed.
Branch 2?3 Pressure constant Since we are
dealing with an ideal gas for which
we can express the change in volume in
terms of the change in temperature as So
the first law along this branch gives us
?qdupdV(cp-nR)dTnRdTcpdT from which we get
that
So, again, heat is absorbed.
24
Branch 3?4 Volume constant
So heat is rejected.
Branch 4?1 Pressure constant
So, again, heat is rejected.
The efficiency of this engine, E, is therefore
Note that since we want the MAGNITUDE of QC, the
signs in the numerator are such that it is a
positive quantity. What would an efficiency of
100 mean in this case? Well, we can substitute
in TipiVi/nR for i1?4 and show that in order
for E to equal 1 we must have that
25
where weve cancelled out the nRs. Now note that
V3V4, V1V2, p1p4, and p2p3 to show
which can only be true if p3?p4 and V4?V1. In
other words, 100 efficiency is only possible if
the cycle collapses to a single point on the p-V
diagram, and no work is done!
Whoever said that doing no work is
inefficient???????
In class example Efficiency of the gasoline
engine.
26
The atmosphere as a heat engine Our atmosphere
is heated in the tropics (say Ttropics313K) and
cools at the poles (say Tpoles233K). If we were
to imagine our atmosphere as an enormous heat
engine that converts net incoming heat into work
(kinetic energy) then the efficiency of this
engine is only 26. In fact, the actual
efficiency is much lower than this. Sandströms
theorem (see Houghton, The Physics of
Atmospheres) states that a steady circulation in
the atmosphere can be maintained only if the heat
source is situated at a higher pressure than the
heat sink. Some reflection on the Carnot cycle
will reveal that the theorem must be valid.
Because the atmosphere of the Earth is
transparent to solar radiation, the requirements
of the theorem are satisfied for Earth. What
about Venus?
27
  • APPLICATIONS OF THE SECOND LAW
  • Impossible processes
  • Kelvin formulation of the Second Law
  • Combined First and Second Laws
  • Helmholtz and Gibbs Free Energies
  • Impossible Processes
  • Impossible processes are ones that violate the
    second law of thermodynamics, i.e., ones for
  • which

(Recall that for adiabatic reversible processes
ds0. These processes are therefore
called isentropic and have many applications to
the atmosphere and oceans. Adiabatic
irreversible processes have dsgt0, since adiabatic
processes have ?q0.)
2. Kelvin Formulation of the Second Law This may
be postulated as follows It is impossible to
construct a cyclic device that will produce work
and no other effect than the extraction of heat
from a single heat source. This result can
be readily deduced by considering an isothermal
cyclic engine in contact with a single
reservoir, and examining the consequences of the
second law (which will demonstrate that, in fact,
heat must be lost from the engine to the
reservoir, and hence that work must be done on
the engine).
28
If it were not for the second law, as expressed
by Kelvin, our energy worries would vanish. We
could simply extract as much work as we needed
from a very large heat source such as the oceans
or the interior of the Earth.
3. Combined First and Second Laws If we combine
the first law in the form
(6.2)
with the second law in the form
(6.3)
then the combined laws take the form
(6.4)
4. Helmholtz and Gibbs Free Energies The
Helmholtz and Gibbs free energies are functions
of state. i.e., they are perfect
differentials. They are defined, respectively, by
(6.5)
29
Using them, the combined first and second laws
(Eq. 6.4) may be written as where, again,
the equality hold for reversible processes and
the inequality holds for irreversible
processes. The Helmholtz free energy is
particularly useful when considering chemical
reactions that occur isothermally. Youll note
that the change of the Helmholtz energy during a
reversible, isothermal process equals the work
done on the system, and for a reversible,
isothermal, and isochoric process the Helmholtz
energy is conserved. The Gibbs free energy is
particularly useful when considering phase
changes, which are isothermal and isobaric (and
can be conceived of as occuring reversibly).
Hence during such phase changes the Gibbs free
energy remains constant. The Gibbs energy can be
used to uniquely determine the triple point
temperature and pressure of a substance, since g
must be the same for the solid, liquid, and
vapour phases (2 equations, 2 unknowns
(T,p)). For irreversible processes, the Gibbs
free energy decreases as the substance approaches
its new equilibrium. It is therefore clear that
an equilibrium state is one at which the Gibbs
energy is at a minimum.
(6.6)
30
Hermann von Helmholtz (1821-1894) had become the
patriarch of German science by 1885. He began his
career as an army doctor. As an MD and professor
of physiology, he published a massive volume of
work on physiological optics and acoustics. He
also invented the ophthalmoscope. At the same
time he began to publish work unrelated to
physiology--a paper on the hydrodynamics of
vortex motion in 1858, and an analysis of the
motion of violin strings in 1859, for example. In
1871, Helmholtz turned his attention full-time to
physics, and made significant contributions in
areas ranging from electrodynamics
to thermodynamics and meteorology. Josiah
Willard Gibbs (1839-1903) was a Yale graduate and
professor, who turned his attention to
thermodynamics in the early 1870s.
He contributed to the use of geometrical methods
and thermodynamic diagrams. Perhaps his most
significant contribution was to the
understanding of thermodynamic equilibrium,
which he viewed as a natural generalization of
mechanical equilibrium, both being characterized
by minimum energy. He also made a
substantial contribution to statistical mechanics.
Write a Comment
User Comments (0)
About PowerShow.com