Carnot Engine Model in a Chaplygin Gas

Sciknow Publications Ltd.
RJMS 2015, 2(2):42-47
DOI: 10.12966/rjms.05.02.2015
Research Journal of Modeling and Simulation
©Attribution 3.0 Unported (CC BY 3.0)
Carnot Engine Model in a Chaplygin Gas
Manuel Malaver*
Universidad Marítima del Caribe, Departamento de Ciencias Básicas, Catia la Mar, Venezuela
*Corresponding author (Email: mmf.umc@gmail.com)
Abstract –We extend the work of Panigrahi (2014) for Chaplygin gas, exotic matter used in some cosmological theories,
considering an equation of state of the form P  
B
n
with B  B V  3 and n is a constant. We have deduced an

0
expression for the thermal efficiency of Carnot engine that depends on the limits of maximum and minimal temperature imposed
on the cycle, as in the ideal gas and the photons gas.
Keywords –Chaplygin gas, Cosmological theories, Equation of state, Exotic matter, Carnot engine, Thermal efficiency
1. Introduction
An ideal gas is a gas composed of a group of randomly moving, non-interacting point particles. The ideal gas approximation is
useful because it obeys the ideal gas law and represents the vapor phases of fluids at high temperatures for which the heat engines
is constructed (Lee, 2001). A heat engine that can work with an ideal gas as working substance is the Carnot cycle. For the ideal
gas, Carnot cycle will be composed by two isothermal curves and adiabatic which will come given by the conditions
PV  const and PV   const , respectively (Dickerson, 1969; Nash, 1970) where  is an adiabatic exponent.
One of the great virtues of the Carnot cycle is its potential applicability to any working substance (Nash, 1970). In agreement
with Lee (2001) and Leff (2002), the Carnot cycle for a photon gas provides a very useful tool to illustrate the thermodynamics
laws and it is possible to use for introducing the concepts of creation and annihilation of photons in an introductory course of
physics. Bender et al.(2000), showed that the efficiency of a quantum Carnot cycle is the same as that of a classical Carnot cycle,
with the identification of the expectation value of the Hamiltonian as the temperature of the system. Unlike the ideal gas, the
pressure for a photon gas is a function only of the temperature and the internal energy function is dependent of volume (Leff,
2002).
Recent observational evidences suggest that the present universe is accelerating (Sushkov, 2005; Lobo, 2005; Lobo, 2006)
and the cosmological model of Chaplygin gas (CG) is one of the most reasonable explanations of recent phenomena. The form of
the CG equation of state is the following P   B where P is the pressure of the fluid,  is the energy density of the fluid and

B is a constant. The thermodynamical behavior of the CG model was studied by Panigrahi (2014) and it obtains that the third
law of thermodynamics is satisfied in this model and that the volume increases when temperature falls during adiabatic
expansions, which also is observed in a photon gas (Leff, 2002).
In this paper, an expression is deduced for the thermal efficiency of Carnot cycle for a Chaplygin gas from the thermal
equation of state for the pressure given for Panigrahi (2014). The thermodynamic variables as the heat Q, work W and internal
energy U are a function of the temperature and the volume. We have found that the efficiency of Carnot cycle for CG model only
will depend on the limits of maximum and minimal temperature imposed on the cycle, as in case of the ideal gas and the photons
gas. The article is organized as follows: in Section 2, the physical properties of Carnot cycle using an ideal gas are studied; in
Section 3, we show the deduction for the thermal efficiency of Carnot engine for a Chaplygin gas; in Section 4, we conclude.
2. Carnot Engine in an Ideal Gas
In this work, we have used the convention of Wark and Richards (2001) that defines the
W    PdV
work during a reversible process as
(1)
Research Journal of Modeling and Simulation (2015) 42-47
43
Following Dickerson (1969) and Nash (1970), in Fig. 1 we show the Carnot cycle for an ideal gas. In the first step of A to B,
that is the isothermal expansion, there is no change in the internal energy U in an ideal gas. This implies that
 W AB  Q AB  RTH ln
VB
VA
(2)
QAB is the absorbed heat in the first step, TH is the high temperature and R is the universal gas constant.
The second step of B to C is an adiabatic expansion. In this expansion QBC  0 and the change in internal energy is equal
to the work done
U BC  WBC  CV TL  TH 
(3)
where CV is the thermal capacity at constant volume and TL is the low temperature.
In the isothermal compression of C to D, the internal energy change is again zero and we obtain
 WCD  QCD  RTL ln
In the final adiabatic compression of D to A
VD
VC
(4)
QDA  0 and
U DA  WDA  CV TH  TL 
(5)
In a Carnot cycle for an ideal gas the net work done in the two adiabatic processes is zero and the adiabatic steps are related
by the equation
CV
The net work of the four steps is Wnet  WAB
T  R
VB
V
 A   L 
VC
VD  TH 
 WCD . Substituting (6) into eqs. (2) and (4) , we obtain
Wnet   RTH  TL ln
The heat absorbed at the high temperature is
VA
VB
(7)
QAB and the thermal efficiency of the entire cycle is
 Wnet
T

 1 L
Q AB
TH
Fig. 1. Carnot cycle for an ideal gas
3. Thermal Efficiency in a Chaplygin Gas
For a Chaplygin gas, the equations of state for the internal energy U (T ,V ) and pressure
(6)
P(T ,V ) (Panigrahi, 2014) are
(8)
44
Research Journal of Modeling and Simulation (2014) 42-47
1
T2
 2 B0  2 N 2 
U 
 V 1  2

 N 

n
1
2
(9)
1

T2  2
(10)


V
1


 2 

6n
is a constant, N 
and  is a universal constant with dimension of
3
B N
P   0 
 2 
where B0 is a positive universal constant ,
1



2
n
6
temperature.
Considering the Carnot cycle for a CG model, in the first step, the reversible isothermal expansion and replacing (10) in (1)
and integrating, the work done is
1
1


N
N
T2  2
1  H2  VB 2  V A 2



The expression for the differential of internal energy dU is given for
 U 
 U 
dU  
 dT  
 dV
 T V
 V T
 2 B0 
WI  

 N 
2

(11)
(12)
where C   U 
V
 T V
For an isothermal process
dT  0 and (12) it reduce to
 U 
dU  
 dV
 V T
(13)
and for Chaplygin gas,  U  takes the form
 V T
1
1
T2  2
 B N  2 n 
 U 

   0  V 6 1  H2 
 
 V T  2 

Substituting (14) in (13) and integrating we obtain for the change of internal energy
(14)
1
1
N
N
T2  2
 2 B0  2 
U I  
 1  H2  VB 2  V A 2 

  
 N  
In agreement with the first law of the thermodynamics, the absorbed heat in the first step is given by
 2 B0 
QI  

 N 
1
2
V N 2  V N 2 
A
 B

2
T
1  H2
2
H
2
T

(15)
(17)

In the second path, a reversible adiabatic expansion, (Wark & Richards, 2001) and the work done is
WII  U II 
TC
 U 
dT   
 dV
V  T
V2 
V3
C
V
TL
(18)
According Panigrahi (2014), the thermal capacity at constant volume for the CG model, can be written as
 2 B0 
CV  

 N 
1
2
Now, using eq. (14) and eq. (19) in eq. (18), we obtain

1 
2 
T2
 2B 
WII  U II   0  1  2
 N   


V



N
1
2
2
T

2
V N 2
 C
(19)
1

T2

1


2





3
2


N
N 
1
1
2 
2
 VB   V 

2

TH2
 1  TL
1


2
2








(20)
Research Journal of Modeling and Simulation (2015) 42-47
45
In the third path, the reversible isothermal compression, the work done is
WIII
and for the transferred heat
 2 B0 


 N 
1
2

T2
1  L2





1
2
V
N
D
2
 VC
N
2

(21)
QIII we obtain
QIII
 2 B0 


 N 
 2B 
WIV  U IV   0 
 N 
V N 2  V N 2 

C
TL2 
 D

2
TL2
1 2
2
(22)

 0 and using (14) and (19) in (18) In the fourth path, we have
For the final adiabatic compression, again QIV
1
1
2


2
1  T

2





1
2
V N 2
 A


N
N 
1
1
2 
2
 VD   V 

2

TL2
 1  TH
1


2
2








(23)
The net work of the four steps is
Wneto  WI  WII  WIII  WIV
(24)
and the net heat is
Qneto  QI  QIII
(25)
In agreement with the first law (Dickerson, 1969; Nash, 1970), for a cyclical process U  0 and the net work can be
written as


1 

2
2
2
N
N
N
N
TH
TL
 2 B0  
(26)
V 2  V 2  
V 2  V 2 
Wneto  Qneto  

A
B
C
D
2 
2 




N


T
T
 2 1  H2

 2 1  L2




The heat absorbed at TH is given by (17) and the efficiency is

N
TH2  N 2
 VC 2 
V
2  D

N
N
2 
2
2 
T
1  L2 V A  VB 
1
 Wnet
T
 1
QI
T
2
L
2
H
(27)

eq. (27) can be written as
 Wnet
T2

 1  L2
QI
TH
1
1
2
L
2
T

2
H
2
T

N

 N 2  VD 2

VC

1
N


2

 VC

N

 TL2  N 2  V A 2
1  2 VB
 1
N


 
2

 VB

 TH2
1  2


(28)
For a reversible adiabatic process in a Chaplygin gas (see Appendix)
V


1  2
T

2



1
 const
(29)
N
The eq. (29), we obtain
 VC

 VB
Substituting eq. (30) in eq. (28), we have



N
2

TH
TL2   2
TL TH2   2
(30)
46
Research Journal of Modeling and Simulation (2014) 42-47
 Wnet
T2

 1  L2
QI
TH
1
1
2
L
2
T

2
H
2
T

 N2


2
2  VD

TH TL  

1
 N2


 VC

N
2
V 2

 TL 
1  2 TL TH2   2  AN  1


2
  
 VB

 TH2
1  2
 
(31)
eq. (29) implies that
VB
V
 A
VC
VD

2
1  2
TH


2
1  2
TL




1



1
N
(32)
N
Thus, eq. (32) takes the form
VD V A

VC VB
(33)
With eq. (33), eq. (31) reduces to
  1
TL
TH
(34)
The efficiency of Chaplygin gas engine depends on the limits of maximum and minimal temperature of the cycle as in the
ideal gas and photon gas. If TH  TC ,  tends to 0.
4. Conclusions
We have deduced an expression for the thermal efficiency of a Chaplygin gas engine, which is a function only of the maximum
and minimal temperature of the cycle. The study of Chaplygin gas can enrich the courses of thermodynamics, which contributes
to a better compression of the thermal phenomena.
The thermodynamic equations that describe the behavior of the Chaplygin gas are tractable mathematically and offer a wide
comprehension of the accelerated universe expansion and of the basic ideas of the modern cosmology.
As in the ideal gas and the photon gas, for a Chaplygin engine to be 100% efficient, temperature of hot reservoir TH must be
infinite or zero for the cold reservoir TL and the second law says that a process cannot be 100% efficient in converting heat into
work.
References
Bender, C. M., Brody, D. C., & Meister, B. K. (2000). Quantum mechanical Carnot engine. Journal of Physics A: Mathematical and General, 33(24), 4427.
Dickerson, R. (1969). Molecular Thermodynamics, W. A., Benjamin, Inc, Menlo Park, California, ISBN: 0-8053-2363-5.
Lee, M. H. (2001). Carnot cycle for photon gas? American Journal of Physics, 69(8), 874-878.
Leff, H. S. (2002). Teaching the photon gas in introductory physics. American Journal of Physics, 70(8), 792-797.
Lobo, F. S. (2005). Stability of phantom wormholes. Physical Review D, 71(12), 124022.
Lobo, F. S. (2006). Stable dark energy stars. Classical and Quantum Gravity, 23(5), 1525 -1541.
Nash, L. (1970). Elements of Classical and Statistical Thermodynamics, Addison-Wesley Publishing Company, Inc, Menlo Park, California.
Panigrahi, D. (2014).Thermodynamical behaviour of the variable Chaplygin gas.http://arxiv.org/abs/1405.7667
Sushkov, S. (2005). Wormholes supported by a phantom energy. Physical Review D, 71(4), 043520.
Wark, K., & Richards, D. (2001). Termodinámica, McGraw-Hill Interamericana, Sexta Edición, ISBN: 84-481-2829-X.
Appendix
In this Appendix, we have deduced an expression for a reversible adiabatic process in a Chaplygin gas. In agreement with the
first law of the thermodynamics, when Q=0
A.1
dU  dW  Cv dT  PdV
Following Dickerson (1969), the internal energy
U (V , T ) we can express as
Research Journal of Modeling and Simulation (2015) 42-47
47
 U 
 U 
U (V , T )  
 dV  
 dT
 V T
 T V
A.2

 U  
 U 
  dV  
 dT  0
P  
 V T 
 T V

A.3
Substituting A. 2 in A.1, we have
With the well known thermodynamical relation (Dickerson, 1969; Nash, 1970)
 U 
 P 

  T
 P
 V T
 T V
A.4
A.3 can be written as
  P  
 U 
  dV  
 dT  0
T 
 T V
  T V 
A.5
Replacing eq. (10) and eq. (19) in A.5, we obtain
N dV

2 V
dT

T2
T 1  2





0
A.6
Integrating,
N V2 1 T22
ln
 ln
2
V1 2 T12
 T12   2
 2
2
 T2  

  0

A.7
Converting to the exponencial form,
 V2

 V1



N
2

2
1 2
T2


2
 1  2
T1







1
2
A.8
Then for a reversible adiabatic process with a Chaplygin gas we have
V


1  2
T

2



1
 const
N
A.9