Cp/Cv
Matthieu Schaller et Xavier Buffat
[email protected]
[email protected]
12 avril 2008
Table des matières
1 Introduction
2
2 Theory
2.1 Specific heat . . . . . . . . . . . . . . .
2.2 The specific heat at constant volume .
2.3 The specific heat at constant pressure
2.4 The Cp /Cv ratio . . . . . . . . . . . .
2.5 Degrees of freedom . . . . . . . . . . .
2.6 Ruchardt’s experiment . . . . . . . . .
2.7 Hafner and Duthie’s experiment . . . .
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2
2
2
3
3
4
4
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3 Results and discussion
3.1 Ruchardt’s method . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Hafner-Duthie’s method . . . . . . . . . . . . . . . . . . . . .
7
7
7
4 Conclusion
9
5 Apendix
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10
1
1
INTRODUCTION
1
2
Introduction
In thermodynamic physics, perfect gases play an important role because
they are the only model that corresponds closelly to the theory. All sorts of
thermodynamic machines use gases to produce mechanical energy by compressing and expanding gases. Some processes in these machines are called
adiabatic when no heat transfer is needed. For these processes an important
C
coefficient is the ratio Cvp often called γ.
In this experiment, we are going to mesure this coefficient for three differnet
gases, air, argon and CO2 .
2
Theory
2.1
Specific heat
When you heat up a certain quantity of gaz, there are three possbile
results that you can observe.
1. An elevation of the pressure if you keep the volume constant
2. An elevation of the volume if you keep the pressure constans
3. A modification of both volume and pressure if you let the system move
freely
This constation leads us to the definition of two coefficients, Cv , the
secific heat at constant volume and Cp , the specific heat at constant pressure.
2.2
The specific heat at constant volume
A natural definition of this coefficient is :
∂E Cv =
∂T v
(2.1)
The total energy of a perfect gaz if defined as :
cRT
(2.2)
M
where c is a coefficient depending on the nature of the gaz, M the atomic
mass of moles of the gaz and T , the temperature of the gaz. If we insert the
equation 2.2 in the equation 2.1, we get :
E=
Cv = c
R
M
(2.3)
2
THEORY
2.3
3
The specific heat at constant pressure
If you consider a mass m of gaz in a volume v at pressure p, we have
before heating, the relation :
m
RT
(2.4)
M
, where M is the atomic mass of the gaz. After heating (at constant pressure),
we have :
m
R(T + ∆T )
(2.5)
p(v + ∆v) =
M
By substrating these two equations, we get :
pv =
m
R∆T
M
The heating of the gaz is made of two parts :
p∆v =
(2.6)
1. An elevation ∆T of its temperature Q1 = Cv m∆T
2. an augmentation ∆v of its volume in order to maintain the pressure
at the same value. Q2 = p∆v.
The total amount of heat that the gas absorb is :
∆Q = ∆Q1 + ∆Q2 = Cv m∆T +
Q = (Cv +
m
R∆T
M
(2.7)
R
)m∆T
M
(2.8)
But by definition, Q = Cp m∆T . This leads us to the value of Cp :
Cp = Cv +
2.4
R
M
(2.9)
The Cp /Cv ratio
The value of the ratio Cp /Cv is often called γ. Its value is calculable
using the relations we fond before :
γ=
R
Cv + M
Cp
=
=
Cv
Cv
cR
M
+
cR
M
R
M
=1+
1
c
This is the value we are going to measure in this experiment.
(2.10)
2
THEORY
2.5
4
Degrees of freedom
If we want to determinate theoretically the value of γ, we need to know
the value of c.This value is defined as :
1
c= w
2
(2.11)
where w is the number of degree of freedom of a single gaz molecule. For a
mono-atomic gaz, this number is equals to w = 3 because each molecule can
move freely in all the 3 dimensions of space.
The cas of a diatomic molecule is a little bit more complex. Such a molecule can be represented as couple of spheres connected by a solid bar. To
represent the position of a molecule, you need three coordinates for th position of the first sphere and two angles to specify the position of the second
sphere. The total of independant variables is then w = 5.
In our experiment, we are going to use CO2 which is a more complicated
molecule. Both of the chemcial bounds in this molecule are rigid. This means
that the molecule can be represented as a solid. This means that you need 6
independant coordinates (three for the position and the tree Euler-angles),
w = 6.
Now that we have these informations, we can put them together with
the equation 2.10, and calculate the values of γ for different gaz :
Gaz
Argon
Air
CO2
Type
monoatomic
considered as diatomic
complex
Degrees of Freedom
3
5
6
c
3
2
5
2
6
2
γ
1.67
1.40
1.33
Fig. 1 – γ for different gaz
We are going to use two different methods to determinate experimentally
these coefficients. The experiments are explained in the next sections.
2.6
Ruchardt’s experiment
The goal of this experiment is to mesure the amortized oscillation period
of a small ball falling in a vertical cylinder which diameter is a little bit bigger
as the ball one. The cylinder is placed on a big container filled with the gaz
we want to probe. The left part of figure 2 shows such a system.
Let p be the air pressure in the container, p0 the atmospheric pressure
over the ball, m its mass, V the volume of the container, S the section of
the tube and g the standard gravity.
The equilibrium condition of the ball at a height z0 in the tube can be
written as :
2
THEORY
5
Fig. 2 – Both of the experiments
mg
(2.12)
S
The ball’s equation of motion aroud this equilibrium position is then :
p = p0 +
mz̈ = Sdp
(2.13)
The ball moves quick enough to admit that the variations of pressure
are adiabatic. These means that pv γ = cst. By derivation of this equation
we find :
dv
Sz
= −γp
(2.14)
v
v
If we insert this relation in the equation 2.13 we obtain the differential
equation of the ball’s oscillations.
dp = −γp
γpS 2
z
v
From this trivial equation, we get the period of oscillation τ :
r
mv
τ = 2π
γpS 2
mz̈ = −
(2.15)
(2.16)
And from here, it’s easy to obtain the γ-coefficient :
Cp
4π 2 mv
=γ= 2 2
Cv
S pτ
(2.17)
2
THEORY
6
This the relation we are going to use after having measured the period
τ and all the other constant of the system.
2.7
Hafner and Duthie’s experiment
The goal of this experiment is to mesure the oscillation period of a small
cylinder in a vertical tube which diameter is a little bit bigger than the the
cylinder’s size. The cylinder is standing over a big gaz container. Such a
system is shown on the right part of figure 2.
Let m be the mass of the cylinder, d the diameter of the tube, v the
volume of gaz under the hole ,p0 the atmospheric pressure, p the gaz pressure
and g the standard gravity.
The equilibirum condition of the cylinder at a height of z0 (under the
hole) is given by :
4mg
πd2
The equation of motion of the ball around z0 is then :
p = p0 +
(2.18)
πd2
dp
(2.19)
4
Again, we can admit that the variations of pressure are adiabatic and
use the relation pv γ = cst., which leads us by derivation to :
mz̈ =
dv
πd2 z 1
= −γp
(2.20)
v
4 v
If we insert this relation in the equation of motion 2.19 we get :
dp = γp
π 2 d4
z=0
16v
from which we get the pulsation :
r
2π
πd2 γp
ω=
=
τ
4
mv
mz̈ + pγ
(2.21)
(2.22)
And this gives us a relation for γ :
Cp
64mv
=γ= 4 2
Cv
pd τ
This is te relation we are going to use in this experiment.
(2.23)
3
RESULTS AND DISCUSSION
3
7
Results and discussion
3.1
Ruchardt’s method
The caracterisitc of the system used are resumed in this tabular :
ball mass
tube diameter
Container volume
atmospheric pressure
16.578 ± 10−3 g
16.02 ± 10−2 mm
10−2 ± 10−3 m3
94962 ± 665 Pa
Using the measure shown in annex with equation , we can find the air’s γ
coefficient : γair = 1.36 ± 0.2.
Error is simply calculated by adding relative error of each factor used for the
calculation. This error is quit big principally because of the volume which is
extremely unprecise, by reducing the error by half, the error could also be
reduce by half. The other measure are quit precise and does not affect the
result too much. Although the measure of the time is quit random because
starts and stops are made manually with a simple chronometer, we can have
a relativliy good measure by making a lot of try, reducing the standard deviation which mean reducing the error.
The theoretical value of the γair = 1.4 is included in our error range and
is quit near from the value we have found. This mean no systematic error
are made during the experiment, therefore we can assue that this method is
a good way to determine the coefficient. In fact by simple improvement of
the measure of the system’s caracteristics, particularly the volume, we could
find a very good measure of γair . In addition, we could mechanize the time
measurment systeme to significatively improve the precision.
The principal default of this experiment is that it is using atmosphere gaz,
therefore we cannot measure the γ coefficient of another gaz using the same
protocole. We could for exemple fill the volume with another gaz at atmospheric presure, but this would require to make a full vaccum in the container
and put in a gaz at atmospheric pressure. But this experiment would be most
unprecise due outflows and complicated due to manipulation that needs to
be done. Fortunatly we can use Hafner-Duthie’s experiment to carry out
these measure.
3.2
Hafner-Duthie’s method
Argon The caracterisitcs of the system are resumed in the following tabular, atmospheric pressure has not change since first experiment.
piston mass
tube diameter
Container volume
7.277 ± 10−3 g
14.88 ± 10−2 mm
2 · 10−3 ± 0.2 · 10−3 m3
3
RESULTS AND DISCUSSION
8
Using equation , we find γAr = 1.21 ± 0.23, whereas Theoretical value is
1.67. This measure is unprecise and the theoretical value is not included in
the error range calculated.
C02 These are the system’s caracteristics, container volume is identical to
the one used with argon gaz.
piston mass
tube diameter
6.397 ± 10−3 g
14.45 ± 10−2 mm
Similarly, we find γCO2 = 1.06 ± 0.19, whereas theoretical value is 1.33.
Thus, this measure is as bad as the one carried out with the argon gaz.
The factors of error shown in section 3.1 also occur during this experiment,
because the protocole is very similar and nothing has been made to improve
them, moreover, less measurements have been made making the value of
the period more unprecise. However these imprecisions have less importance
in this experiment, because the measure are obviously far from theoretical
value. Indeed, in the two cases the measure have shown up to be way lower.
There is some explainations for this results.
Some gaz is flowing through the hole and through the tube. Even if the
ball is suppose to be keep it airtight, some flowout exists through it. Firstly,
these flowouts are not considered int the calculations and then are a source
of error, nevertheless these are not great fluctuations and should not have
such an effect on the results. Although, These two air entries are creating
small fluctuations of presure in this, consequently they induce fluctuation of
the oscillation’s period. Therefore, to get a precise measure of the period,
we have to measure it during a long time. There is the problem, the mean
period measured is about 0.4 s in the two cases which represent a frequency
of 2.5 Hz, which mean we have to follow the ball and count the oscillations
at a quit big rate during a long time. Although it is simple, it turns out to
be hard to realise precisely. Most likely, some oscillations have been systematicaly missed during the count making the measured time longuer, then
making the γ coefficient lower.
Then human factor causes the most trouble in this experiment, we could
considerably improve precision by using automatic methods to calculate the
oscillations period.
During the three experiment carried out, we used the perfect gaz model
to compare our results with theory, we should keep in mind that this model
is an approximation and thus theoretical value are probably a bit different
as what we calculate, particularly in the case of the CO2 . Indeed, the fact
that it is compose by three atom make its behavior slightly different from
the one predicted by the perfect gaz model.
4
CONCLUSION
9
In addition, we supposed that movement of the ball or pistons were adiabatic, which is not completely true in our case, heat might have been dissipated
through the side of the tube. Finally, we did not consider friction in all of
our calculation, although it is not futile.
These factors, which are quit small considered one by one, are important
when bring together and nedd some improvement to finally get a precise
experiment.
4
Conclusion
The experiment we carried out do not provide precise value of the ratio
Cp /Cv , although it has prooved to be a good way to get to it. Indeed, the
devices used are very basic and simple, some improvements, in particular
to avoid human factor by simply mecanise the measurement process, could
considerably improve the precision of the results and also avoid a systematic
error, therefore bringing better results.
The first experiment showed a perfect agreement between the pert gaz theory
and the reality, even though we are working with diatomic molecules.
5
5
APENDIX
10
Apendix
Measured time [s] τ [s] γair
11.63
1.16 1.25
11.25
1.13 1.34
10.88
1.09 1.43
11.1
1.11 1.38
11.28
1.13 1.33
11.22
1.12 1.35
10.97
1.1
1.41
11.02
1.1
1.4
11.19
1.12 1.35
11.05
1.11 1.39
11.1
1.11 1.38
Measurment with Ruchardt’s
method
Measured time [s] τ [s] γAr
40.73
0.41 1.2
41.96
0.42 1.13
38.26
0.38 1.36
42.01
0.42 1.13
Measurment with Hafner-Duthie’s
method using argon gaz
Measured time [s] τ [s] γCO2
41.3
0.41 1.15
42.25
0.42 1.1
44.59
0.45 0.99
44.17
0.44 1.01
Measurment with Hafner-Duthie’s
method using CO2