3.4. Isentropic exponent - Physikalisches Anfängerpraktikum
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3.4 Isentropic exponent 253
3.4. Isentropic exponent
General remark: Throughout this text symbols for physical quantities are used which are
common in German scientific literature even if other symbols are more common in English
scientific literature. This is done because the German and English versions of this manual
are intended to be as compatible as possible and the dominant language in the student
lab is German.
Goal
The isentropic exponent κ of various gases is determined experimentally using two different
methods.1 The experiment is meant to illustrate the connection between the isentropic
exponent, the structure of gas particles and their degrees of freedom. Additionally the
ideal gas law and some of its special cases are treated.
Hints for preparation
You should know the answers to these questions before performing the experiment. They
are the basis for the discussion with your tutor before the experiment. Information on
these topics can be found in the literature listed at the end of this text.
1. • What does isentropic mean? What is an isentropic process?
• What is the isentropic exponent?
• What is the difference in meaning between cp and cV ? How can one of these
values be calculated from the other?
• Why is the isentropic exponent always greater than 1?
• What is the first law of thermodynamics?
2. • How many degrees of freedom do the gases have which are used in this expe-
riment?
• What are the state variables2 used to describe the state of a gas?
• Name four processes that can change the state of a gas.
• What is the equation of state of an ideal gas?
• What is the content of the Poisson equation?
1
The isentropic exponent is also often called adiabatic exponent although in a strict sense only adiabatic
reversible processes are isentropic and the equation in which the exponent κ occurs is only valid for
such adiabatic reversible processes but not for adiabatic irreversible processes.
2
engl.state variable = dt.Zustandsgröße
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Parts of the setup
General:
• compressed gas cylinders with pressure regulator, pressure gauge and connecting
hoses for each of the gases under investigation, i. e. argon (Ar) and carbon dioxide
(CO2 )
• pressure gauge and connecting hoses for nitrogen (N2 ) from central gas supply (wall
outlet)
Measurement according to Rüchardt and Flammersfeld:
For this experiment two slightly different setups are available. One of the setups is marked
with blue plastic pieces at the ends of the glass tube (“left” setup), the other one with red
plastic pieces (“right” setup). Throughout this text the setups are referred to as “blue”
and “red” setup.
Please make sure to note which setup you use for the measurements!
• glass bulb with attached glass tube and tightly fitted metal cylinder inside the tube
– effective volume of the bulb up to the hole in the tube wall:
setup blue“: 2182.0(1.0) cm3
”
setup red“: 2257.0(1.0) cm3
”
– properties of oscillating metal cylinders:
∗ radius 4.960(10) mm,
∗ mass: see adhesive badge at the corresponding setup
Please note down the values and do not move the oscillating cylinders between
the setups!
• photoelectric switch with electronic counter
• stopwatch
Measurement according to Clément and Desormes:
• big glass “bottle” with rubber plug and three connections:
– via shut-off valve to gas supply
– via shut-off valve to ambient air
– liquid column manometer3 with water filling
Basics
General remark: Some parts of this section are currently only available in German.
3
engl.liquid column manometer = dt.U-Rohr-Manometer
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State variables
The state of a gas can be unambiguously described by a set of so called state variables
p : pressure,
V : volume,
n : amount of substance
T : absolute temperature
The connection between those quantities is in the simplest case given by the ideal gas law
pV = nRT , (3.4.1)
where R ≈ 8.314 molJ K is the universal gas constant (see also table 1.0.5). Gases which
follow that law are called ideal gases. For “real” gases this is not exactly true but in most
cases can be considered a reasonable approximation at least far enough above the boiling
point.
Heat capacity
The heat capacity or thermal capacity of an object is the ratio between the thermal
energy added to the object and the temperature rise of the object caused by that energy.
Often the thermal capacity of a substance is given per mass (specific heat capacity) or
per amount of substance (molar heat capacity).
In general the heat capacity depends on the conditions under which the thermal energy
is added, e. g. the pressure p or the volume V may be kept constant. If the pressue is
kept constant, an ideal gas will expand upon temperature rise and deliver mechanical
work to the environment. Therefore less energy is left over to increase the internal energy
and more thermal energy needs to be added to achieve the same temperature rise as it
would result for constant volume. This means that the molar heat capacity cp at constant
pressure is always higher than the molar heat capacity cV at constant volume.
Special thermodynamic processes
There is a number of thermodynamic processes which can be especially easily described
because one of the state variables stays constant throughout the process. table 3.4.1 gives
an overview of these processes. For each of them there are different laws for the dependence
between the state variables pressure p, volume V and temperature T .
• For an isothermal process the temperature T does not change, i. e. T = const.
With n as the amount of substance of the gasand R the universal gas constant, the
ideal gas law is given by p · V = n · R · T , and we obtain: p · V = const. This law way
discovered in 1662 by R. Boyle and again independently in 1679 by E. Mariotte.
• For an isochoric process, the volume stays constant. The ideal gas law is of course
also valid and we find: Tp = const. This law was probably first found in measurements
by J. A. C. Charles, which were later continued by J. L. Gay-Lussac.
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• During an isobaric process the pressure is constant leading to VT = const. This law
was described in 1703 by G. Amontons and in 1802 by J. L. Gay-Lussac.
• We call a process isentropic if the entropy remains constant (dS = const.). Such a
process follows the laws found by S. D. Poisson (see table 3.4.1), e. g. p · V κ = const.
The exponent κ (or γ mainly in english literature) is called “isentropic exponent”.
In the p(V ) diagram the line of an isentropic process looks similar to an isotherm
but is steeper. This is because the temperature drop during an isentropic expansion
results in an additional pressure drop.
Isentropic processes are quite common because they can often be realized at least in
good approximation by fast reversible processes leaving no time for heat exchange
δQ with the environment. If the process is also reversible, then due to dS = δQ T
it
4
is isentropic.
Degrees of freedom
There is a number of terms in the total energy function of the particles of a substance.
Single atoms of a gas can e. g. move in all three coordinate directions. This results in three
quadratic terms of kinetic energy and is called three degrees of freedom of translation.
If the particles are more complex than a single atom, there are additional degrees of
freedom, e. g. resulting from rotation around each of the three coordinate axes or from
different vibrational modes within the particle. Not all of these movements are observed
at all temperatures, because some of them need high energies to be activated. In such
cases the movements might not be observed at all or at least not at room temperature.
This is true e. g. for rotations of linear molecules like N2 oder CO2 around the long axis.6
For vibrations there is kinetic energy of the atoms as well as potential energy resulting
from their interaction (like a “spring” between the atoms). This means that a vibration
can contribute up to two degrees of freedom if it is activated.
4
This is why κ is often called “adiabatic exponent” with adiabatic meaning that no heat exchange takes
place. An adiabatic reversible process is necessarily always isentropic.
5
The names of the laws are not consistently used in literature and do not necessarily correspond to the
history of discoveries. To be on the safe side one should preferably use the terms isothermal, isochoric,
isobaric and isentropic.
6
This is a frequent cause of misconceptions:
The rotation along the longitudinal axis is not unimportant because “nothing changes during such a
rotation” or “there is too little energy in such a rotation”, as one could naively assume. Indeed just
the opposite is true: The minimum energy needed for activating such a rotation would be higher than
the energy available from thermal motion as is easy to prove:
If a body with moment of inertia Θ rotates at an angular frequency ω, this leads to an angular
momentum L = Θ · ω and a rotational energy Erot = 12 Θω 2 . According to the rules of quantum
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Table 3.4.1.: Some thermodynamic processes of an ideal gas.5
In the diagram with the isentropic processes the dashed line corresponds to
κ = 5/3, the dotted line to κ = 9/7. For comparison reasons the corresponding
isotherm (κ = 1) is also shown as a thin line.
thermodynamic condition p(V ) diagram laws
process
isothermal T = const. p · V = const.
p “discovered” by:
R. Boyle (1662)
E. Mariotte (1679)
V
p
isochoric V = const. T
= const.
p “discovered” by:
J. A. C. Charles
J. L. Gay-Lussac
V
V
isobaric p = const. T
= const.
p “discovered” by:
G. Amontons (1703)
J. L. Gay-Lussac (1802)
V
isentropic S = const.
or p·V κ = const.
p T · V κ−1 = const.
δQ = 0
p1−κ · T κ = const.
“discovered” by:
S. D. Poisson
V
mechanics the angular momentum cannot have arbitrary values but is limited to integer multiples of
. We obtain
L = Θ·ω = n· (3.4.2)
n·
=⇒ ω= (3.4.3)
Θ
1
=⇒ Erot = · Θ · ω 2 (3.4.4)
2
1 n2 2
= ·Θ· 2
2 Θ
2
2
1
=n · · . (3.4.5)
2 Θ
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Table 3.4.2.: Degrees of freedom7 for various gases, corresponding theoretical isentropic
exponent κtheo and experimentally determined literature value κlit at T =
20 ◦ C [Kuc88].
Stoff effective degrees of freedom κtheo κlit
translation rotation vibration sum
Ar 3 – – 3 5/3≈ 1.666 1.648
N2 3 2 – 5 7/5= 1.4 1.401
CO2 3 2 ≈ 1.857 ≈ 6.857 ≈ 8.857/6.857 ≈ 1.292 1.293
The minimum energy necessary for starting the rotation (corresponding to n = 1) is therefore higher
for smaller inertial moment Θ. For particles with all nuclei located on the rotation axis (i. e. atoms
and linear molecules), the distance of the rotation masses from the rotation axis is very small. As the
inertial moment depends quadratic on this distance, the minimum energy is in this case so high that
1
2 kB T , the thermal energy per degree of freedom, is not sufficient to start the rotation and thus the
rotation is “frozen in”.
7
The value ≈ 1.857 for the vibrational degrees of freedom in CO2 results from the fact that the CO2
molecule has four possible so called normal vibration modes (symmetric and antisymmetric valence
vibration, double degenerate deformation vibration – this one can take place with identical energy
in two independent planes, see e. g. [BM99]), which are only partly activated (to different fractions)
at room temperature due to their different energies. Each vibration can contribute two degrees of
freedom at maximum, as kinetic and potential energy have both to be considered. It is possible to
calculate the temperature dependent contribution of each vibration to the heat capacity using the
partition function (dt. Zustandssumme) of the vibration (see e. g. [Atk01]). Defining υ = kBE·ST as
the ratio between “energy quantum” ES = h · ν of a vibration with frequency ν and “thermal energy”
(product of Boltzmann constant kB und thermodynamic temperature T ), we find for the “effective
degrees of freedom” of this vibration:
2
e−υ/2
fS = 2 · υ· . (3.4.6)
1 − e−υ
From infrared spectroscopy measurements the wave numbers (and thus the frequencies and energies)
of the vibrational modes are known (siehe e. g. [Atk01] p. 662) allowing to calculate the effective
degrees of freedom at T = 20 ◦ C = 293.15 K:
• double degenerate deformation vibration: 667.4 cm−1 , 0.876 effective degrees of freedom each ,
• symmetric valence vibration: 1388 cm−1 , 0.102 effective degrees of freedom
• antisymmetric valence vibration: 2349 cm−1 , 0.003 effective degrees of freedom,
• sum: 2 · 0.876 + 0.102 + 0.003 = 1.857.
This finally leads to a combined value of ≈ 1.857 effective degrees of freedom for all vibrations. Adding
the contributions of translation and rotation we get ≈ 3 + 2 + 1.857 = 6.857 effective degrees of
freedom in total. The temperature dependence of the effective number of degrees of freedom can be
observed by e. g. measuring the temperature dependence of the velocity of sound.
The effective degrees of freedom for rotations show a more complicated temperature dependence (see
e. g. [Fli99] p. 227), even though diagrams shown in literature (see e. g. [Atk01] p. 668 or [HRW03] p.
585) suggest a quite similar behaviour.
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Measurement according to Rüchardt and Flammersfeld
In 1929 E. Rüchardt described a method to determine the isentropic exponent by using
the oscillation of a steel ball on a gas spring. The setup mainly consists of a big glass bulb
and a precision glass tube [Rüc29]8 .
Figure 3.4.1.: Experimental setup for determination of the isentropic exponent from the
oscillation frequency of a “gas spring pendulum” according to Rüchardt and
Flammersfeld.
Unfortunately the oscillation is strongly damped which makes it difficult to observe more
than about 10 periods. In 1964 E. M. Hafner and J. G. Duthie described a method to
overcome this problem [HD64]. The main component is a slightly conical glass tube with
a diameter that increases from one end to the other. If one puts the narrow end at the
bottom one can apply a continuous stream of gas and obtain undamped oscillations. The
steel ball will be pushed up stronger by the gas flow if it is at a lower and thus narrower
position inside the tube. In the 1960s some glass tubes “accidentally” had such a conical
form due to the fabrication process. Later on it became more difficult to find conical
tubes.
In the revised method of A. Flammersfeld replaces the accidentally conical tube by a
cylindrical tube with a small hole intentionally drilled near the middle of the tube [Fla72]
8
Long before that there were similar experiments with an oscillating mercury column in a U shaped
tube (Assmann 1852 [Ass52] and P. A. Müller 1883 [Mül83]). These older experiments were much
more difficult to carry out.
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and the steel ball by a steel cylinder. A sketch of the setup is shown in figure 3.4.1. It
is important to note that the basic idea of the measurement according to Rüchardt is
unchanged. Like in the older setup the rapid pressure variations of the gas in the new
setup arise mainly from the upward and downward movement of the steel object. The
hole is too small to immediately release the overpressure when the steel cylinder passes
the hole in upward direction and the gas supply also needs some time to build up the
pressure again after the steel cylinder has passed the hole in the downward direction. The
small amount of gas passing through the setup mainly compensates friction losses.
There is no net force on the oscillating cylinder if the pressure p in the glass bulb equals
the sum of the air pressure p0 outside the bulb in the lab and the additional pressure
caused by the gravitational force of the oscillating cylinder:
mS · g
p = p0 + (3.4.7)
π r2
with
p = gas pressure in equilibrium,
p0 = air pressure in the lab,
mS = mass of oscillating steel cylinder,
g = acceleration due to gravity,
r = radius of glass tube and oscillating cylinder9 .
To simplify the calculation we consider this equilibrium position as x = 0. For small
displacements x from this position we find a volumne change V = π r2 · x and a
pressure change p. As the process is fast enough, it can be treated as isentropic and we
can write
p · V κ = const.
=⇒ p = V −κ · const.
dp 1
=⇒ = −κ · · p
dV V
p 1
=⇒ ≈ −κ · · p
V V
p
=⇒ p ≈ −κ · · V (3.4.8)
V
p
= −κ · · π r2 · x . (3.4.9)
V
The repulsive force10 caused by the pressure change accelerates the oscillating cylinder.
We find
mS · ẍ = π r2 · p . (3.4.10)
9
As the oscillating cylinder needs to fit very tight into the glass tube, the two radii may considered
equal in good approximation.
10
For an upward displacement with positive sign the pressure decreases. The pressure change is therefore
negative and the resulting force points downward.
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Inserting equation (3.4.9) in equation (3.4.10) leads to
p
mS · ẍ + π r2 · κ π r2 · x ≈ 0
V
κ p π2 r4
=⇒ mS · ẍ + · x ≈ 0 . (3.4.11)
V
For the chosen zero point of the x axis and for small displacements we can write
κ p π2 r 4
mS · ẍ + ·x = 0 (3.4.12)
V
which is the equation of motion of the oscillating cylinder. This is the differential equation
of a harmonic (i. e. sinusoidal) oscillation. For its circular frequency ω and oscillation
period T we find
κ p π2 r4
ω2 = (3.4.13)
mS · V
4π 2
4 mS · V
T2 = 2 = . (3.4.14)
ω κ p r4
For best results the acceleration of the mass mG of the gas inside the tube (approximately
from the bulb to the hole – for the experimental setup used here this length is ≈ 60 cm)
needs to be considered. This means that in equations (3.4.10) through (3.4.14) the mass
mS needs to be replaced by
m = mS + mG (3.4.15)
with
m = accelerated total mass,
mS = masse of oscillating cylinder,
mG = mass of accelerated gas .
The mass of the accelerated gas may be calculated from volume, temperature, pressure
and molar mass of the gas using the ideal gas law.
For equation (3.4.7) similar considerations could be done. But the correction in this case is very small, because the gas
pressure at the bottom end of the glass tube is not much different from the pressure at the hole. Otherwise we would also
have to tell whether the air pressure p0 in the lab is given for e. g. the floor level or the lab table level.
Solving for κ finally yields
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' $
4·m·V
κ= (3.4.16)
T 2 · p · r4
with
m = accelerated total mass,
V = effective volume up to the hole,
T = oscillation period,
p = gas pressure in equilibrium position given by equation (3.4.7),
r = radius of glass tube.
& %
Measurement according to Clément and Desormes
The isentropic exponent can be determined by an expansion experiment. The experimental
setup used here is shown in figure 3.4.2. In the following section p0 is the ambient pessure
and T0 the ambient temperature. The measurement involves a sequence of steps:
1. First the big glass bottle with volume V1 is filled with the gas under examination
using a pressure that is slightly higher than the ambient pressure. After waiting for
the gas to reach ambient temperature T1 = T0 , the overpressure p1 = p1 − p0 is
determined from the height difference h1 of the water columns in a liquid column
manometer (p1 = · g · h1 , where is the density of water and g the gravitational
acceleration of the earth).
2. The gas outlet is briefly opened to release the overpressure. The pressure thus drops
to the ambient pressure p2 = p0 , the volume increases to V2 , and the gas delivers
mechanical work against the ambient pressure.
Of course from now on a certain amount of gas is no longer inside the glass bottle making it impossible to observe
its further processes. This does not perturb the measurements because only specific quantities (“per mole”) are
considered.
As the expansion process doesn’t take much time to complete and due to the rela-
tively small pressure difference there isn’t much turbulence as well, the process can
in good approximation be considered isentropic. Thus the Poisson equation
T κ · p1−κ = const. (3.4.17)
is valid during the process and temperature T2 is reached. We find:
1−κ
κ
p1
T2 = T1 · . (3.4.18)
p2
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3. As soon as the pressure difference is zero the outlet valve is closed again and the
gas starts to heat up until it reaches ambient temperature T3 = T0 . During this
isochoric process the law of Charles
p
= const. (3.4.19)
T
is valid and a final pressure p3 = p0 + p3 is reached. We find:
T3
p3 = p2 · . (3.4.20)
T2
4. The isentropic exponent κ can be calculated from the pressure differences p1 und
p3 . This is done by inserting equation (3.4.18) in equation (3.4.20) and solving for
κ:
T3
p3 = p2 · 1−κ
p1 κ
T1 p2
T0
= p2 · 1−κ (from T1 = T3 = T0 )
p1 κ
T0 p2
T0
= p0 · 1−κ (from p2 = p0 and p1 = p0 + p1 )
p0 +p1 κ
T0 p0
κ−1
p1 κ
= p0 · 1 +
p0
p1 κ − 1
≈ p0 · 1 + · (from (1 + x)n ≈ 1 + n · x for x 1)
p0 κ
κ−1
= p0 + p1 ·
κ
κ−1
=⇒ p3 ≈p1 · (from p3 = p0 + p3 )
κ
' $
p1
=⇒ κ ≈ (3.4.21)
p1 − p3
h1
⇒ κ≈ (from pi = · g · hi ). (3.4.22)
h1 − h3
& %
Experimentation
Hints:
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Ar
N2
CO2 h
Figure 3.4.2.: Experimental setup and p(V ) diagram for determination of the isentropic
exponent according to Clément and Desormes.
• For the tutor: Please make sure to turn down the pressure regulator setting before
opening the main valve.
ATTENTION – Do not get confused: For a pressure regulator the pressure increases
when the knob is turned clockwise. This behaviour is unfamiliar because it is the
exact opposite of the behaviour of most other valves (e. g. water tap)!
• For students: Adjust the gas flow solely(!) using the valves in the hoses. The pressure
regulator settings may only be changed by your tutor.
• Do not apply too high gas pressure (increase slowly). Otherwise at the setup accor-
ding to Rüchardt and Flammersfeld the oscillating metal cylinder may “shoot out”
the top of the glass tube at rather high speed! At the setup according to Clément
and Desormes the water may splash out of the liquid column manometer.
Measurement according to Rüchardt and Flammersfeld:
1. Note down the ambient air temperatur (in the laboratory there are several thermo-
meters available at the different experimental setups) and the ambient air pressure
(in the hallway you find a mercury barometer as well as an electronic weather sta-
tion with pressure display – you may also have an absolute pressure sensor in your
smartphone).
2. Choose on of the gases Ar, N2 or CO2 and connect the outlet of the pressure regulator
with the gas inlet of the glass bulb.
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3. Flush the bulb for ≈ 5 min with the selected gas.
4. Carefully adjust a small overpressure using the pressure regulator (≈ 20 mbar =
20 hPa), until the metal cylinder performs harmonic oscillations around the hole in
the glass tube wall. The hole should be in the middle of the movement.
5. Measure the time necessary for 300 oscillations (i. e. 600 passages of the photoelectric
switch).
6. Repeat the measurement (items 2 – 5) for the other two gases.
Hint: Make sure to flush the bulb again for ≈ 5 min after changing to another gas.
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Measurement according to Clément und Desormes:
7. If necessary fill some water into the liquid column manometer.
Hint: Be careful not to “contaminate” the “bottle” with water, it should be only in
the manometer. Otherwise water vapour will distort the measurement. If you should
notice water in the bottle (from the previous group?) ask your tutor how to remove
it.
8. Choose on of the gases Ar, N2 or CO2 and connect the outlet of the pressure regulator
with the gas inlet of the glass bottle.
9. Flush the bulb for ≈ 5 min with the selected gas.
10. Close the gas inlet and outlet at the bottle. Carefully adjust a small overpressure
using the inlet valve (height of the water column ≈ 10 cm) and close the inlet valve
again.
11. Wait a few minutes until the gas has reached room temperature and then note down
the height h1 of the water column.11
12. Briefly open the gas outlet to completely remove the overpressure and close the
valve immediately afterwards.
13. Wait again for a few minutes until the gas has reached room temperature.
14. Note down the overpressure using the liquid column manometer (height h3 of the
water column).
15. Repet the measurement two more times with the same gas.
16. Repeat the measurement (items 8 – 15) with the other two gases.
IMPORTANT: Close the main valves of all compressed gas cylinders once you have finis-
hed the measurements. Do not change the pressure regulator settings.
Data analysis Measurement according to Rüchardt and Flammersfeld:
1. Calculate the adiabatic exponent for all three gases Ar, N2 und CO2 using equation
(3.4.16).
Measurement according to Clément und Desormes:
2. Calculate the adiabatic exponent for all three gases Ar, N2 und CO2 using equation
(3.4.22).
Hint: Calculate one value for κ and its associated uncertainty for each repetition of
the experiment. Subsequently calculate the weighted mean value.
11
It is best to note both raw numbers at the upper and lower end of the column and calculate the
difference later.
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General:
3. Quantitatively compare your results for both measurement principles (imems 1 and
2) with each other and with the values in table 3.4.2 by performing suitable signifi-
cance tests.
Questions and exercises
1. Explain the connection between the isentropic exponent κ and the number f of
degrees of freedom in a gas.
2. Show, that the three Poisson equations for isentropic processes can be converted
into each other using the ideal gas law.
3. What is the connection between the “Föhn”, which very often results in sunny
weather at the lake of Constance, and adiabatic processes?
Hint: Use the terms “moist adiabatic” and “dry-adiabatic”.12 Those are not fully
sufficient to explain the foehn wind phenomenon, but still play an important role
during most oft the “foehn” weather conditions and amplify the effect (see e. g.
[Sei93]).
Supplementary information
General remark: This section is currently only available in German.
Dispersion der Schallgeschwindigkeit
Die Dispersion der Schallgeschwindigkeit in bestimmten mehratomigen Gasen, also ihre
Abhängigkeit von der Frequenz der Schallwelle, lässt sich erklären, wenn man die Prozesse
bei der Schallausbreitung etwas näher betrachtet. Die Schallwelle erzeugt im Gas durch
adiabatische Kompression und Expansion periodische Temperaturschwankungen. Dabei
verteilt sich die Wärmeenergie sowohl auf die äußeren“ Freiheitsgrade (Translation und
”
Rotation), als auch auf die inneren“ Freiheitsgrade (kinetische und potentielle Energie
”
der Eigenschwingungen der Moleküle). Letztere brauchen aber zu ihrer Anregung eine
gewisse Zeit und werden daher immer weniger angeregt, je höher die Schallfrequenz ist.
Die effektive“ Zahl der Freiheitsgrade nimmt dann also ab, die Wärmekapazitäten cV
”
und cp = cV + R sinken, und somit steigt der Isentropenexponent κ = f +2 f
= ccVp . Für die
Schallgeschwindigkeit in einem Gas gilt
"
p
c= ·κ (3.4.23)
12
engl.moist adiabatic = dt.feuchtadiabatisch, engl.dry-adiabatic = dt.trockenadiabatisch
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mit
c = Schallgeschwindigkeit,
p = mittlerer Druck,
= mittlere Dichte,
κ = Isentropenexponent = cp /cV .
In Übereinstimmung mit dieser Überlegung findet man e. g. bei CO2 ein Ansteigen der
Schallgeschwindigkeit mit der Frequenz. Dieser Effekt wurde bereits im Jahr 1881 von H.
A. Lorentz vorausgesagt, aber erst im Jahr 1925 durch Pierce experimentell gefunden.
Der Wert der Schallgeschwindigkeit in Luft konnte historisch überhaupt erst durch die
isentropische Zustandsgleichung verstanden werden, wie man bei Betrachtung von equa-
tion 3.4.23 leicht einsehen kann. Newton ging von der Gasgleichung p ∼ V −1 aus und
folgerte daraus wegen dV /dp = −V /p eine Kompressibilität von 1/p. Damit müsste die
Schallgeschwindigkeit c = p/ sein, was aber einen zu niedrigen Wert von ≈ 280 m/s
ergibt. Der genauere Wert von ≈ 330 m/s war damals durchaus schon bekannt, konn-
te aber erst durch Laplace erklärt werden, der die kurz zuvor von Poisson aufgestellte
isentropische Zustandsgleichung p ∼ V −γ auf die Berechnung der Schallgeschwindigkeit
anwandte und so eine Kompressibilität von 1/(κ p), sowie die Schallgeschwindigkeit nach
equation 3.4.23 fand.
Bei konstanter absoluter Temperatur T ist p ∼ , also hängt c nicht vom Gasdruck ab
(im Gebirge ist die Schallgeschwindigkeit genauso groß wie auf Meereshöhe). Je wärmer
√
aber die Luft ist, desto geringer ist bei gegebenem p, denn ∼ p/T . Es folgt c ∼ T .
In warmer Luft läuft der Schall also schneller (e. g. im Sommer schneller als im Winter).
References
Temperature dependence of the contribution of rotations and vibrations to heat capaci-
ty: [Fli93] (very exact, quantitatively correct even for rotations), [Atk01] (slightly more
illustrative, not as exact), [TM04] ( Das Versagen des Gleichverteilungssatzes“, p. 573 ff.,
”
well comprehensible)
Original puclications: [Rüc29, Fla72]. The author of this text did not succeed until now
to read the even older publications [Ass52, Mül83, HD64] cited herein [HD64] seems to
be a typing error.
If you want to get more involved with the topic of degrees of freedom, you may have a look
at the detailed presentation found in various textbooks on theoretical physics. In [Gol91]
all of chapter X (pages 352–384) is dedicated to the treatment of “small oscillations”.
However the detailed calculation is not given there, instead there are references to other
publications like [Her].
The contribution of rotations to the specific heat capacity of gases is also treated in [Rai92]
in section 1.2.3 ( Innere Energie, Spezifische Wärmekapazität“) and in [Fli99].
”
Concerning the explanation of the foehn winds: [Sei93].
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Literaturverzeichnis
[Ass52] Assmann. Pogg. Ann., 85:1, 1852.
[Atk01] Atkins, Peter W.: Physikalische Chemie. Wiley-VCH, Weinheim, 3. Auflage,
2001.
[BM99] Banwell, Colin N. und Elaine M. McCash: Molekülspektroskopie: ein
Grundkurs. R. Oldenbourg Verlag, München, 1999.
[Fla72] Flammersfeld. Zeitschrift für Naturforschung, 27a:540, 1972.
[Fli93] Fließbach, Torsten: Statistische Physik. Bibliographisches Institut & F.
A. Brockhaus AG, Mannheim, 1993.
[Fli99] Fließbach, Torsten: Lehrbuch zur Theoretischen Physik, Band IV: Stati-
stische Physik. Spektrum Akademischer Verlag, Heidelberg, Berlin, 3. Auflage,
1999.
[Gol91] Goldstein, Herbert: Klassische Mechanik. AULA-Verlag GmbH, Wiesba-
den, 11. Auflage, 1991.
[HD64] Hafner, E. M. and J. G. Duthie. American Journal of Physics???, 1964???
[Her] Herzberg, G.: Infrared and Raman Spectra of Polyatomic Molecules.
[HRW03] Halliday, David, Robert Resnick und Jearl Walker: Physik. WILEY-
VCH GmbH & Co. KGaA, Weinheim/Germany, 1. Auflage, 2003. Authorized
translation from the English language 6. edition.
[Kuc88] Kuchling, Horst: Taschenbuch der Physik. Verlag Harri Deutsch, Thun und
Frankfurt am Main, 1988.
[Mül83] Müller, P. A. W. Ann., 18:94, 1883.
[Rai92] Raith, Wilhelm (Herausgeber): Bergmann-Schaefer – Lehrbuch der Expe-
rimentalphysik, Band V: Vielteilchen-Systeme. Walter de Gruyter, Berlin, 1.
Auflage, 1992.
[Rüc29] Rüchardt. Physikalische Zeitschrift, 30:58, 1929.
[Sei93] Seibert, Petra: Der Föhn in den Alpen – Ein aktueller Überblick. Geogra-
phische Rundschau, 45(2):116–123, 1993.
[TM04] Tipler, Paul A. und Gene Mosca: Physik für Wissenschaftler und Inge-
nieure. Elsevier GmbH, München, 2. Auflage, 2004. (2. Auflage der deutschen
Übersetzung 2004 basierend auf 5. Auflage der amerikanischen Originalausgabe
2003).
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