Thermodinamics Process

7/29/2019 Thermodinamics Process

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Equipo GContreras Viveros Alex----Isochoric processGutierrez Reyes Joel------Isobaric processResendiz Jimnez Omar---Isothermal processValtierra Jimnez --------Adiabatic processRojas Herrera Tania-------Polytropic process

Thermodynamic process

A thermodynamic process may be defined as the energetic evolution of athermodynamic system proceeding from an initial state to a final state. Paths

through the space of thermodynamic variables are often specified by holdingcertain thermodynamic variables constant. It is useful to group these processesinto pairs, in which each variable held constant is one member of a conjugatepair.Isochoric Process

An isochoric process, also called an isovolumetric process, is a process duringwhich volume remains constant. The name is derived from the Greek isos,"equal", and khora, "place."If an ideal gas is used in an isochoric process, and the quantity of gas staysconstant, then the increase in energy is proportional to an increase intemperature and pressure. Take for example a gas heated in a rigid container:the pressure and temperature of the gas will increase, but the volume willremain the same.In the ideal Otto cycle we found an example of an isochoric process when weassume an instantaneous burning of the gasoline-air mixture in an internalcombustion engine car. There is an increase in the temperature and thepressure of the gas inside the cylinder while the volume remains the same.Equations

If the volume stays constant: (V = 0), this implies that the process does nopressure-volumework, since such work is defined by:

W= PV

where P is pressure (no minus sign; this is work done bythe system).By applying the first law of thermodynamics, we can deduce that Uthe changein the system's internal energy, is:

U= Q
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for an isochoric process: all the heat being transferred to the system is added tothe system's internal energy, U. If the quantity of gas stays constant, then thisincrease in energy is proportional to an increase in temperature,

Q = mCVT

where CV is molar specific heat for constant volume.On a pressure volume diagram, an isochoric process appears as a straightvertical line. Its thermodynamic conjugate, an isobaric process would appear asa straight horizontal line.

Isochoric Process in the Pressure volume diagram. In this diagram, pressureincreases, but volume remains constant.

Isobaric Process

An isobaric process is a thermodynamic process in which the pressure staysconstant: p = 0 The term derives from the Greek isos, "equal," and barus,"heavy." The heat transferred to the system does work but also changes theinternal energy of the system:
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The yellow area represents the work done

According to the first law of thermodynamics, where W is work done by thesystem, U is internal energy, and Q is heat. Pressure-volume work (by thesystem) is defined as: ( means change over the whole process, it doesn'tmean differential)

but since pressure is constant, this means that

.

Applying the ideal gas law, this becomes

assuming that the quantity of gas stays constant (e.g. no phase change duringa chemical reaction). Since it is generally true that

then substituting the last two equations into the first equation produces:

.

The quantity in parentheses is equivalent to the molar specific heat for constantpressure:

cp = cV+ R
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and if the gas involved in the isobaric process is monatomic then and

.

An isobaric process is shown on a P-V diagram as a straight horizontal line,connecting the initial and final thermostatic states. If the process moves towardsthe right, then it is an expansion. If the process moves towards the left, then it isa compression.If the volume compresses (delta V = final volume - initial volume < 0), then W 0), then W > 0.That is, during isobaric expansion the gas does positive work, or equivalently,the environment does negative work. Restated, the gas does positive work onthe environment.Enthalpy. An isochoric process is described by the equation Q = U. It wouldbe convenient to have a similar equation for isobaric processes. Substituting thesecond equation into the first yields

The quantity U + p V is a state function so that it can be given a name. It iscalled enthalpy, and is denoted as H. Therefore an isobaric process can bemore succinctly described as

.

Variable density viewpoint. A given quantity (mass M) of gas in a changingvolume produces a change in density . In this context the ideal gas law iswrittenR(T) = M Pwhere T is thermodynamic temperature above absolute zero. When R and Mare taken as constant, then pressure P can stay constant as the density-tempertature quadrant (,T) undergoes a squeeze mapping.

Isothermal Process

Isothermal process or isothermic process to the change of reversibletemperature in a thermodynamic system is denominated, being this change ofconstant temperature in all the system. For a substance that carries out achange of state static and isothermally, the transferred heat can be calculatedof advisable way according to the second law of the thermodynamics.

Q12 = T dS = T(S2- S1) = mT(s2-s1)..eq. 1

Combining the previous equation with the first law of the thermodynamics, thatis obtained for a closed system carries out a static change of state isothermally.
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Wideal = (U1-T1S1) - (U2-T2S2)

On the other hand, of the definition of the function of Helmholtz or function ofwork.

A= U-TS

where: A it is work, U it is internal energy, T is temperature and S is entropy.Reason why eq 1 can be written like:

Wideal = (A1-A2)T = -(A2-A1)T

One concludes then that the diminution in the function of Helmholtz of a systemrepresents the maximum work that can develop that one during an isothermalprocess by means of some appropriate device. In the same way, the diminutionin the function of Helmholtz of a system that is in an suitable device to absorbwork, the minimum work necessary to carry out the process isothermally. This isthe reason for which the function of Helmholtz also considers a potentialfunction him.

An isothermal curve is a line that on a diagram represents the successivevalues of the diverse variables of a system in an isothermal process.The isotherms of an ideal gas in a diagram P-V, call diagram of Clapeyron, areequilateral hyperbolas, whose equation is PV = constant.

Example of an isothermal problem.

Two kilograms of gaseous nitrogen confined in a cylinder that lodges a piston,

static carry out a change of state from 300 k and 101,325 kPa to 300 a final


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state to 300 k and 20.000 kPa. Heat transference can be between nitrogen anda heat deposit that is 300k.A) It determines the work transferred by nitrogenB) It determines the total heat transferred between nitrogen and the deposit. .Because it is a static and isothermal work.

A) 1W2 = m[(u1-u2)T = -T(s1-s2)].eq 1

Using data of the table of Nitrogen it is obtained:.

u1 =h1 p1v1= (311.163-101.325 x .8786) kJ/ Kg= 222.138 kJ/Kg

S1 = 6.8418 kJ/Kg Ku2 = h2 p2 v2= (279.010 20,000 x 0.004704) kJ/Kg= 184.93 kJ/Kgs2= 5.1630 kJ/KgReplacing the numerical values in the equation.1W2 = 2[ ( 222.138-184.93 ) 300(6.8418 5.1630) ] kJ= -932,864 kJ. The negative sign means that the work takes place in the gas.Since the process static and isothermal, it is had of the second law1Q2 = m T (s2 s1)= 2 x 300 (5.1630 6.8418) kJ

= -1007.28 kJ.The negative sign means that the heat is extracted of the gas.

BIBLIOGRAPHI

Ingeniera termodinmica. Fundamentos y aplicaciones. Francis. F. Huang,Primera edicin, Mxico 1997. Editorial continental

Adiabatic Process

n thermodynamics, an adiabatic process or an isocaloric process is athermodynamic process in which no heat is transferred to or from theworking fluid. The term "adiabatic" literally means impassable, coming from

the Greek roots - ("not"), - ("through"), and ("to pass"); thisetymology corresponds here to an absence of heat transfer. Conversely, aprocess that involves heat transfer (addition or loss of heat to the surroundings)is generally called adiabatic.

IFor example, an adiabatic boundary is a boundary that is impermeable to heattransfer and the system is said to be adiabatically (or thermally) insulated; an

insulated wall approximates an adiabatic boundary. Another example is theadiabatic flame temperature, which is the temperature that would be achieved
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by a flame in the absence of heat loss to the surroundings. An adiabaticprocess that is reversible is also called an isentropic process. Additionally, anadiabatic process that is irreversible and extracts no work is in an isenthalpicprocess, such as viscous drag, progressing towards a nonnegative change inentropy.

One opposite extremeallowing heat transfer with the surroundings, causingthe temperature to remain constantis known as an isothermal process. Sincetemperature is thermodynamically conjugate to entropy, the isothermal processis conjugate to the adiabatic process for reversible transformations.

A transformation of a thermodynamic system can be considered adiabatic whenit is quick enough that no significant heat is transferred between the system andthe outside. At the opposite extreme, a transformation of a thermodynamicsystem can be considered isothermal if it is slow enough so that the system'stemperature remains constant by heat exchange with the outside.

Adiabatic heating occurs when the pressure of a gas is increased from workdone on it by its surroundings, ie a piston. Diesel engines rely on adiabaticheating during their compression stroke to elevate the temperature sufficientlyto ignite the fuel. Similarly,jet engines rely upon adiabatic heating to create thecorrect compression of the air to enable fuel to be injected and ignition to thenoccur

Adiabatic cooling occurs when the pressure of a substance is decreased as itdoes work on its surroundings. Adiabatic cooling does not have to involve afluid. One technique used to reach very low temperatures (thousandths andeven millionths of a degree above absolute zero) is adiabatic demagnetisation,where the change in magnetic field on a magnetic material is used to provideadiabatic cooling. Adiabatic cooling also occurs in the Earth's atmosphere withorographic lifting and lee waves, and this can form pileus or lenticular clouds ifthe air is cooled below the dew point.

Ideal gas (reversible case only)

For a simple substance, during an adiabatic process in which the volumeincreases, the internal energy of the working substance must decrease

The mathematical equation for an ideal fluid undergoing a reversible (i.e., noentropy generation) adiabatic process is

where P is pressure, V is volume, and

CP being the specific heat for constant pressure and CV being the specific heatfor constant volume. is the number of degrees of freedom divided by 2 (3/2 formonatomic gas, 5/2 for diatomic gas). For a monatomic ideal gas, = 5 / 3, andfor a diatomic gas (such as nitrogen and oxygen, the main components of air) = 7 / 5. Note that the above formula is only applicable to classical ideal gases

and not Bose-Einstein or Fermi gases.
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For reversible adiabatic processes, it is also true that

where T is an absolute temperature.

Bibliography:

http://en.wikipedia.org/wiki/Adiabatic_process

Polytropic Process

When a gas undergoes a reversible process in which there is heat transfer, theprocess frequently takes place in such a manner that a plot of the Log P(pressure) vs. Log V (volume) is a straightline. Or stated in equation form PVn =a constant.

This type of process is called apolytropic process.

An example of a polytropic process is the expansion of the combustion gassesin the cylinder of a water-cooled reciprocating engine.

2- Example:Compression or Expansion of a Gas in a Real System such as a

Turbine


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Many processes can be approximated by the law:

where,P= Pressure,v= Volume,n= an index depending on the process type.

Polytropic processes are internally reversible. Some examples are vapors andperfect gases in many non-flow processes, such as: n=0, results in P=constant i.e. isobaric process. n=infinity, results in v=constant i.e. isometric process. n=1, results in P v=constant, which is an isothermal process for a perfect

gas.

n= , which is a reversible adiabatic process for a perfect gas.

Some Polytropic processes are shown in figure below:

The initial state of working fluid is shown by point 0 on the P-V diagram. Thepolytropic state changes are:

0 to 1= constant pressure heating, 0 to 2= constant volume heating, 0 to 3= reversible adiabatic compression, 0 to 4= isothermal compression, 0 to 5= constant pressure cooling, 0 to 6= constant volume cooling, 0 to 7= reversible adiabatic expansion,
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0 to 8= isothermal expansion.

polytropic process is a thermodynamic process that obeys the relation:

PVn = C,

where P is the pressure, V is volume, n is any real number (the polytropicindex), and C is a constant. This equation can be used to accuratelycharacterize processes of certain systems, notably the compression orexpansion of a gas, but in some cases, possibly liquids and solids.

Under standard conditions, most gases can be accurately characterized by the

ideal gas law. This construct allows for the pressure-volume relationship to bedefined for essentially all ideal thermodynamic cycles, such as the well-knownCarnot cycle. (Note however that there may be instances where a polytropicprocess occurs in a non-ideal gas.)

For certain indices n, the process will be synonymous with other processes:

if n = 0, then PV0=P=const and it is an isobaric process (constantpressure)

if n = 1, then for an ideal gas PV=NkT=const and it is an isothermal

process (constant temperature) if n = = cp/cV, then for an ideal gas it is an adiabatic process (no heattransferred)

Note that , since (see:adiabatic index)

if n = , then it is an isochoric process (constant volume)

When the index n occurs between any two of the former values (0,1,gamma, orinfinity), it means that the polytropic curve will lie between the curves of the twocorresponding indices.

The equation is a valid characterization of a thermodynamic process assumingthat:

The process is quasistatic The values of the heat capacities,cp and cV, are almost constant when 'n'

is not zero or infinity. (In reality, cp and cV are a function of temperature,but are nearly linear within small changes of temperature).
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Polytropic fluids

Polytropic fluids are idealized fluid models that are used often in astrophysics. Apolytropic fluid is a type of barotropic fluid for which the equation of state iswritten as:

P = K(1 + 1 / n)

where P is the pressure, K is a constant, is the density, and n is a quantitycalled the polytropic index.

This is also commonly written in the form:

P = K

where in this case, = (1 + 1 / n) (Note that need not be the adiabatic index

(the ratio of specific heats), and in fact often it is not. This is sometimes a causefor confusion.)

Gamma

In the case of an isentropic ideal gas, is the ratio of specific heats, known asthe adiabatic index.

An isothermal ideal gas is also a polytropic gas. Here, the polytropic index isequal to one, and differs from the adiabatic index .

In order to discriminate between the two gammas, the polytropic gamma issometimes capitalized, .

To confuse matters further, some authors refer to as the polytropic index,rather than n. Note that
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Bibliography:

www.taftan.com/thermodynamics/POLYTROP.HTM

http://www.answers.com/topic/polytropic-process

http://www.engineersedge.com/thermodynamics/polytropic_process.htm

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/ImportantThermalProcess.html

interesting article:

During the 19th century many advances in many different fields of physics tookplace, creating a new atmosphere conducive to the continuing investigationsinto the natural world. One such field, that was investigated rigorously was thatof the stars. Through out the history of mankind, and many years before, it was

evident that the sun had not changed its behavior much. This is a fact that ledto the original idea that one could consider a star a gas sphere, that essentiallymust remain in hydrostatic equilibrium with its own gravitation. Any situationcontrary to this would lead to accelerations either inward or outward, whichwould cause many short term changes, that have not been observed in normalstars over the years.i

Although this is an amazing achievement in ones consideration of thestars, a full understanding of the complex processes was still not at hand. Thisdescription, though informative, still does not address the question of why starsradiate, or what a stars energy source may be. Half of this question wasanswered with the introduction of quantum theory of blackbody radiation by Max

Plank in 1900.ii After the introduction of the blackbody spectrum, it wasdetermined that a star was essentially radiating according to the rules of a black
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body, with the notable exception of the Fraunhofer absorption lines, which werediscovered by Fraunhofer in 1819, and emissions lines, which originate from thestellar atmosphere and surrounding gas. This, however, still did not explain thestars energy source. Once it was suggested that a star may live from itsgravitational potential energy, but this is not the case, as such a star would have

only a fraction of the lifetime of a real star. This being the case a newassumption was made, that the star had some sort of undefined internal energysource. With this in mind Karl Schwarzschild began his work on radiativetransfer of energy in stellar atmospheres around 1906. iii This established theaddition of a new dynamic to stellar structure, whose importance is equal to thatof the first assumption based on hydrostatic equilibrium. It was essentiallydetermined that the structure of a star was supported by internal pressure aswell as radiation pressure against its gravitational pull.

With this theoretical basis in place another important milestone wasreached at about the same time. Hertzsprung and Russell recognized that allstars are not of the same mass, temperature, and luminosity and therefore have

a number of differing properties dependant upon these quantities. In measuringthe luminosity and effective temperature of each star they found that stars of thesame class occupy distinguishable places in relation to these quantities. Withthe creation of the Hertzsprung-Russell diagram many of the classes, such asthe main sequence stars and red giants, that are quite familiar today firstbecame apparent. Soon after this connection between star class and effectivetemperature was established, Eddington established a mass-luminosityrelationship for the main sequence stars and further researched the internalmechanisms of the stars. In the 1930s the Dirac Fermi statistics for adegenerate electron gas were published. It was quickly realized that this toowas applicable to the stellar interiors of certain star classes, specifically whitedwarfs of high mass. S. Chandrasekar was instrumental in this determination,moreover he was also able to find the critical maximum mass for a white dwarfstar, known today as the Chandrasekar mass.

It is at this point in history where the Polytrope representation of starsfirst was developed. It is a method that today still lends valuable methods andinsights to the internal structure of stars. It has also proven to be most versatilein the examination of a variety of situations, including the analysis of isothermalcores, convective stellar interiors, and fully degenerate stellar configurations.Even the case of an ideal gas can be related to a polytrope of index n = 3/2.The justification for such a theory is that, as the name implies, it is extremely

versatile. As will be shown later such a class of models allows for the derivationand prediction of many stellar properties, which continue to be of significantinterest to the astrophysics community. The derivation of polytropic star modelsaccording to R. Kippenhahn and W. Weigert, as well as S. Chandrasekar andWilliam K. Rose is outlined in the following section.

The polytropic theory of stars essentially follows out of thermodynamicconsiderations, that deal with the issue of energy transport, through the transferof material between different levels of the star. We simply begin with thePoisson equation and the condition for hydrostatic equilibrium:

P

r

M G

r

M

rr

r

r

=

=

2

24


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Eq. 1 & 2

where P is the pressure, M(r) is the mass of a star at a certain radius r, and isthe density, at a distance r from the center of a spherical star. iv Combination ofthese equations yields the following equation, which as should be noted, is anequivalent form of the Poisson Equation.

Eq. 3

From these equations one can then obtain the Lane-Emden equation throughthe simple supposition that the density is simply related to the density, whileremaining independent of the temperature. We already know that in the case ofa degenerate electron gas that the pressure and density are ~ P3/5. v

Assuming that such a relation exists for other states of the star we are led toconsider a relation of the following form:

Eq.3

where K and n are constants, at this point it is important to note that n is thepolytropic index. Using this as a basis to classify different interior states withinthe star we can also conclude that a non-relativistic degenerate electron gas isa Polytrope of n = 3/2. Based upon these assumptions we can insert thisrelation into our first equation for the hydrostatic equilibrium condition and fromthis rewrite equation to:

Eq.5

Where the additional alteration to the expression for density has been insertedwith representing the central density of the star and that of a relateddimensionless quantity that are both related to through the following relation.

Eq.6

142

2

r

d

dr

r dP

drG

=

P K n=

1

1

K n

G r

d

dr

rd

dr

n n( )+

=

1

4

11 12

2

= n


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Additionally, if place this result into the Poisson equation, we obtain adifferential equation for the mass, with a dependance upon the polytropic indexn. Though the differential equation is seemingly difficult to solve, this problemcan be partially alleviated by the introduction of an additional dimensionlessvariable , given by the following:

Eq. 7

Inserting these relations into our previous relations we obtain the famous formof the Lane-Emden equation, given below:

Eq.8

At this point it is also important to introduce the boundary conditions, which arebased upon the following boundary conditions for hydrostatic equilibrium, andnormalization considerations of the newly introduced quantities and . Whatfollows for r = 0 is

Eq.9

Eq. 10

Taking these simple relations into consideration, it is also evident that one canproduce additional conditions, based upon a modified form of the Lane-EmdenEquation given by:

Eq. 11

Here it is apparent that as approaches 0 the first term of the equationapproaches . As a result an additional condition must be introduced in orderto maintain the conditions of Eq. 9 and 10 simultaneously:

r a

an K

Gn

=

=+

( )1

4

11

1

2

12

2

r

d

d

d

d

n

=

= =( )0 1

r = =0 0

2 2

2

d

d

d

d

n+ =

d

d

=

=0

0


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Eq. 12

Once the boundary conditions have been determined it is an easy matter toobtain a number of solutions for the Lane-Emden equation. In addition tovarious numerical methods, which will be explained later, this equation actually

has 3 known analytical solutions for polytropes of index n = 0, 1, and 5, givenbelow:

Eq. 13, 14 & 15where the subscripts represent the index number n for a specified solution andthe subscripts represent its value for = 0.

These results are useful in a few respects and deal with some actualstate equation for stars, however they are more important for inferring generalforms of the Lane-Emden solutions. Below are the plots of vs. for theaforementioned solutions, generated using Mathmatica.

Polytrope n = 1

Please note that all the following graphs are modeled after the following: x axis gives values and the y axis give values. Here we can see that thefunction basically follows the same form as that for an index n = 0, with a fewminor differences, however the Polytrope of index n = 0 also terminates at afinite radius just as is observed in the relation for a Polytrope of index n = 1.The other main difference that we observe in this case is that the terminationpoint of the star is markedly larger, at about a value of 3,15 as opposed to thetermination value of about 2,48 for the polytrope n = 0 as seen below.

Polytrope Index n = 0

0

2

1

1 1

5

2

1

61 6

1

11

3

= + =

= =

=+

=

,

sin( ),

,


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Here we can see that the values for start at one in accordance with ourboundary conditions and then eventually reduces to = 0. This is essentiallyan indicator that the stars material ceases to exist outside of this area as the

density drops to 0 at this point. The next graph shows many similarities to thisone.

Polytrope Index n = 5

Though these two solutions for n = 1 and n = 0 share many characteristics, thesolution for the Polytrope of index n = 5, contains some radically different andunexpected characteristics. In this case the behavior of the function is markedlydifferent than that of its predecessors. Here the density of the star initiallydecreases rapidly as radius increases but slows rapidly once a value ofaround three is reached. At this point the decrease slows continually. Thoughit may not be apparent on the graphic provided, the function never reaches 0. Itis therefore evident that a polytropic star of index n = 5 has an infinite radius,and in reality cannot exist, however the case itself deserves further study.Despite this fact such a model provides important theoretical perspectiveconcerning the theory, as one may view this as the border between polytropicthat are physically feasible. It is also of interest to note, as will be shown later,that such a stellar model has, in spite of the infinite radius, a finite mass.Additionally, other stellar models, that are created in a layered fashion, whereeach layer consists of a Polytrope of a different index, may also utilize thisfunction for a portion of the star, in which case a finite radius would be possible.

In addition to these relations there are also a number of otherconclusions that one can draw from the polytropic model of stars. For relations


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of this type, there exists a relation between the polytropic index, mass of a starand the radius. It is perhaps evident in the discussion of the analytic solutionsof the polytropic index that one could possibly infer a relation between thepolytropic index of the star and the radius that one would calculate from thatstar. In the attempt to find a relation the most immediate result is obtained from

the simple equations of stellar state. Integration of dM/dr on the limit from 0 tothe radius of the star, yields the following expression.

Eq. 16

This equation, is in itself not possible to integrate as we have an r dependencein the density . However, this problem can be easily alieveated through theuse of the polytropic variables. By replacing and r by , where lambda is thecentral density, and a the process of integration can not only be carried outmore easily, but a relation dependent upon central density the maximum Radiusand the other polytropic variables obtained. Integration after such a substitionyeilds the follwoing:

Eq.17

Though this relation is extremely useful in the polytropic model of stars thereare also a number of additional relations, that can augment usefulness of thisrelation as well. The first of these is the relationship between the central densityand the average density of the star. It is easily seen, that for the case of a starthe average density is simply given by the division of the total mass through thetotal volume. For the assumption of a spherical star, which yeilds a volume of4/3r^3, and using the relation for the total mass. We find simply that the ratioof the average density to the central density is given by:

Eq.18

Where is the density and is the average density.Another case, though not nearly as important as the case presented

above, is found in the relationship for the radius of a star. Beginning with thestandard polytropic for the radius (Eq.6) relation and inserting the value for awe obtain:

M r drtot

R

= 4 20

M Rd

dtot=

=

41

3

11

< >= = =

M

V

d

d

tot3

11

RK n

G

n

n=+

( )1

4

12

1

21


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Eq. 20

It is now apparent that the case of n = 1 will, as in the case for the total mass ofthe star at n = 3 which will be covered shortly, yield an expression that isindependent of the central density of the star. This relation is extremely

interesting, as it reveals that there is the potential for fundamentally differentclasses of stars to exist, whose masses differ, but not the radii. Though thismay seem counterintuitive it is important to note that as the mass in a real starincreases, the equations of state may change, which limits the actual range ofthe masses for such a star in practice. This, however, does not diminish fromthe theoretically interesting aspects of this model.

There is additionally another case, which is also independent of thecentral density as well. For the case of a Polytrope of index n = 5, an analyticsolution is obtainable, as was shown above. Insertion of this relation for intothe equation for total mass yields a relation that in itself appears rathermundane, however we may also insert the value for the maximum radius, which

has already been determined to be infinite. By evaluation of the resulting limit:

Eq. 21

It is readily apparent that the total mass of this star with an infinite radius yieldsthe surprising result of a finite mass. Though the significance of this case is inpractice not important, as there are not any stars that exist with an infinite

radius, it is a case of theoretical interest, as it essentially represents a bordercase in polytropic star models, as models with n > 5 diverge. This case,however, is not nearly groundbreaking or interesting as that of the first relationor the one to be discussed next, it is merely a theoretical model that provides aninteresting point of consideration, though composite models may contain suchpolytropes and make use of such a relation.

In addition to the relations that we discussed here there remains anadditional relation between the mass and the radius that remains to bediscussed. In this case we start with the equations for the maximum radius ofthe star as well as the total mass for a star of polytropic index n. Betweenthese two equations, we can eliminate the central density and obtain a

relationship that is independent of this quantity. Through this we obtain the

result given below:

Eq. 21, 22, & 22

M ad

da

K

Gtot

=

=

=

4 4 3

3

1

3

1

22

5

1

N GM R K

Nn w

wd

d

n

n

n

n

n

n

n

n

n

n

n

n

+

=

=

=+

=

1 3

0

1

1

0 1

1

1

1

1

4

1


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This is one of the most important results of the polytropic theory of stars, as itnot only provides, for a given radius a mass or vice versa, but also provides, inits simplified form, some of the framework that is used in order to obtainnumerical results for the other intrinsically important values for a star. Thesevariables and notation are usually used rather universally in the literature

regarding polytropic star models. An additional point that is of paramountimportance is the fact that a polytrope of index n = 3 will yield a massexpression independent of the radius. As earlier discussed the case where then = 3 corresponds to the polytropic star representation of a relativisticdegenerate electron gas. This is essentially the model for the Chandrasekarmass for stars (given as =1.54 x Suns mass), and this can indeed be obtainedby inserting value obtained from the polytropic models in combination with thestate equation of a relativistic degenerate gas. Though this may not seemimportant, as the Chandrasekar border mass is a well known value, it isimportant to point out that this derivation for the mass is a result as it is incomplete agreement with a value that was originally calculated through

conventional methods through the use of statistical mechanics. This argumentnot only lends credence and further support to the border mass concept, but italso lends credibility to the polytropic theory of stars, while demonstrating theeffectiveness of the method. This is further bolstered by the fact that there areno known white dwarfs that exceed this mass. This is a powerful statement withregard to the theory of stellar states, that has been widely accepted for anumber of years.

This being said there remains one final relation for the central pressurethat is easily derivable from the expression for the central density. In order toobtain this relation we begin with the expression for the central density given by,

Eq. 25

As this relation is dependant upon K and the central density it is apparent thatwe can replace both of these relations by the values determined for the Mass Radius relation as that determined for the value of the average density inrelation to that of the central density. The average density can then be replacedby its dependence on the mass and the volume and these relations finally yield:

Eq. 26

Though it is often not mentioned it is interesting to note that this relation canalso be used as a good check on the accuracy of ones stellar model. It isapparent that as the central pressure changes that the equation of state may

P K P K

n

n

c

n

n= =

+ +

1 1

P N GM R wGM

R

w

nd

d

c n

n

n

n

n

n

nn

n

= =

=

+

+

=

1 3 1 2

4

2

1

4 1

1

( )


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change with it as well. For example if it is known that the electron gas isdegenerate we have two cases for the degeneracy. The first of these is therelativistic case and the second of these is the non relativistic case. Each ofthese possesses its own equation of state that is different the first. For therelativistic case we have a polytrope of n = 3 and for the non - relativistic case

we have a polytrope of n = 3/2. In any case if we have a star of a known mass,radius, and general chemical composition one could easily determine thepolytropic index. For the case of n = 3 this is of extreme importance as we areable to determine the border mass. This gives a maximum value of the centraldensity before the collapse of a star into a neutron star, and the value for thenon-relativistic case give values that correspond to approaching values oflargest central pressure. At the point where the model exceeds the pressure ofthe Chandrasekar mass we know that the model most certainly is that of aneutron star or just invalid. In any case it provides a nice guide to the models ofthe stars that one would be concerned with and also provides a good basis forthe inclusion of models or exclusion of models, based on the validity of the

equations of state at different values for the pressure.Having demonstrated the importance of the polytropic star relations, it is

now important to consider cases, in which one is not able to use the exactsolutions of the polytropic relations. Examples of such systems would be that ofa non-relativistic electron gas, where, as mentioned earlier, the polytropic indexis n = 3/2. Even the all important case of the fully degenerate electron gas (n =3) is not covered by the exact solutions. There are a number of ways that thisproblem could be attacked. The first of these is to use a program such asMathmatica in order to obtain solutions to the Lane Emden equations.However under the assumption that one does not have access to such aprogram there are also a number of other ways that the problem at hand can beattacked. The first of these methods is to attempt a power series solution forthe Lane-Emden equation:

Eq. 11

One can then proceed to formulate a power series solution to the Lane

Emden equation. Which is of the general form:

Eq. 31

where c subscript n represent arbitrary constants.

By matching the coefficients and inserting the appropriate border conditions, theresulting solution is given by

2 2

2

d

d

d

d

n+ =

( ) ...= + 1 16 1202 4

n

( ) = cn n


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Eq. 32

One can see immediately that this result is consistent with the analytical

solution of n = 0, as the additional terms for the series solution are n dependant.These solutions, though not the only available for use in determination ofphysical quantities, are generally satisfactory for representing differentpolytropic indices.

A method that is readily available is a variation of the Euler method, forsecond order differential equations. In considering this method on creates anarbitrary step, suitable to the task at hand, and from initial values of theequation extrapolates a solution. In this case one considers the initial value ofd/d as well as . At different itterations one can now extrapolate values fromthe functions slope. This yields a recursive relation for this method of:

Eq. 33

where h is the step size and the primes represent derivatives of the function .Using the initial values and the Lane-Emden equation, one can then findapproximate values of the function and for the next step. Reinserting thesevalues yields a new value of allowing one to extrapolate another value.Repetition of this process allows one to crate an approximate data curve,thereby describing a polytrope of a certain type.

Although this method may be of use it is quite evident that there remain anumber of other techniques that may be superior, which result in numericalresults that one can use in order to determine the necessary factors that may beplaced in the various relations, which were introduced in the preceding section,in order to make predictions for different stellar properties. The first, and alsothe most used throughout the literature (See Schwarzschild, Chandrasekar,Kippenhahn and Weigert) is the Runge-Kutta Method. In order to determine thenumerical solutions of these equations, one must first begin with initial values asin the preceding section. The first of these is given by our normalizationcondition, and the second of these is acquired through observation of the Lane-Emden equation, in the expanded form, given by equation 13. As the

normalization conditions requires that (0)=1 it is apparent that the second termof the equation will diverge if the value of d(0)/d is not equal to zero. It istherefore apparent that a second condition exists, which makes thedetermination of solutions through numerical methods possible. The generalprocedure for the Runge-Kutta method is as follows. As the Lane-Emdenequation is a second order differential, one must use an expanded form.According to the methods outlined in a number of mathematical texts thevarious values of , d/d, for certain values of can be determined through astep mehtod. The size of the steps h are taken to be constant and provide aniterated view of the function ().vi

' ' ' '

' '

i i i

i i i

h

h

+

+

= +

= +1

1


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Eq. 31

Where the separate coefficients k are given by,

Eq. 32

Calculation of these parameters is in this case not limited to the exact polytropicsolutions, but can be tabulated for any polytropic index n, note that n of theRunge-Kutta method is not n the polytropic index. As an example a list of anumber of solutions is included to below.

Where each of these variables represent the variables defined in the primaryrelations for polytropic relations.vii

Now in order to convince the reader of the validity of such relations, interms of actual theory data agreement, the model of a fully degenerate whitedwarf is considered. As indicated earlier such a model is that of a polytropewith index n = 3. It is also of use to remind the reader that the mass relationintroduced earlier produces the Chandrasekar mass, which defines themaximum mass of an exhausted star, before which it would simply collapsefurther until it would reach the state of a neutron star. Inserting the appropriatevalues into the mass radius relations, given by Equations 21, 22, and 23, wefind the radius of the star to be on the order of 28km - 40km.4 This radius isnecessarily much smaller than that of a typical star, but larger than that of aneutron star, on the order of ~10km. This being a crude, however satisfying,example of a polytropic star models practical use, is only one of manyconceivable uses for such a theory.

Throughout this survey on the Polytropic theory of stars, it has beendemonstrated that in many respects these models provide a versatile andaccurate method for the determination of many stellar parameters. Though the

n ? ?(d?/d?) ?(central)/?(average) o_w_n N_n w_n - [(n+1) ? [ d?/d? ] ] -1

0 2.4494 4.8988 1 0.333 - 0.11936 0.5

1 3.14159 3.14159 3.28987 - 0.63662 0.26227 0.51.5 1.65375 5.99071 5.99071 132.3843 0.42422 0.7714 0.53849

3 6.89485 2.01824 54.1825 2.01824 0.36394 11.05666 0.85432

4 14.97755 1.79723 622.408 0.729202 0.4772 247.558 1.66606

5 Infinite 1.73205 Infinite 0 Infinite Infinite Infinite

( )

n n n n n n

n n n

hk k k k

h

+

+

= + + + +

= +

1 1 2 3 4

1

62 2' '

'

' ' ' ( , , ' )

' '

( . , . ' , ' . )

( . , . ' , ' . )

( , ' , ' . )

n n n n

n n

n n n n n n

n n n n n n

n n n n n n

f

k

k f h h k

k f h h k

k f h h k

=

=

= + + +

= + + +

= + + +

1

2 1

3 2

2 3

5 5 5

5 5 5

5


24/31

Lane-Emden equation itself only has three known solutions, this difficulty caneasily be overcome with the introduction of a number of numerical solutions,that can be found through a numerous variety of methods, for whichapproximate solutions can be found to varying degrees of accuracy. This beingthe case the polytropic theory of Stars has proven to be a versatile and able

theory worthy of its name.


25/31

iMartin Schwarzschild. Structure and Evolution of Stars.Princeton University Press, Princeton 1958ii

H.H. Voigt. Abri der Astronomie (5. berarbeitete Auflage).Bibliografisches Institut und F.A. Brockhaus A.G., Mannheim 1991.

iii

William K. Rose. Advanced Stellar Astrophysics.Cambridge University Press, Cambridge (UK) 1998.iv

R. Kippenhahn, A. Weigert. Stellar Structure and Evolution.Springer-Verlag, Heidelberg und Berlin 1990.v

S. Chandrasekar. An Introduction the Study of Stellar Structure.Chicago University Press, Chicago 1931.vi

William E. Boyce, Richard C. DiPrima. Elementary Differential Equations and BoundaryValue Problems. John Wiley and Sons, New York 1997vii

Branden T. Allen. Various Lectures given at Gttingen, Germany on thepolytropic theory of Stars.

PROBLEMS OF TERMODINAMICS PROCESS

A mass of 0.6 [g] of air are in a cylinder with a piston 3 pressure [bar], a temperature of

176 [oC] and an initial volume of 260 [cm3]. For each point calculate the final state, thework, the heat involved in the process and the type of process that question.a. Adiabatic expands until the initial pressure is 1 [bar] according to the PVK where k =constant = 1.4.b. The pressure is reduced to one tenth of the initial pressure unchanged in volume.

c. Is compressed to one quarter of its original volume assuming that change is inaccordance with the relationship PV1.3 = constant.d. The pressure is increased to twice the initial value according to the PV = constantrelationship.e. Is compressed at constant pressure and temperature is tripled.f. The volume expands 5 times the initial value according to the PV = constant relationship

where P +3 = 5 V.


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Thermodinamics Process