Thermodynamics Prelim

8/6/2019 Thermodynamics Prelim

1/12

[THERMODYNAMICS]


2/12

1. Ideal Gas

An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles.The ideal gas concept is useful because it obeys theideal gas law, a simplified equation of state, and isamenable to analysis under statistical mechanics.

At normal conditions such as standard temperature and pressure, most real gases behave qualitativelylike an ideal gas. Many gases such as air, nitrogen,oxygen, hydrogen, noble gases, and some heaviergases like carbon dioxide can be treated like ideal gases within reasonable tolerances.[1] Generally,deviation from an ideal gas tends to decrease with higher temperature and lower density (i.e.lower pressure),

[1]as the work performed by intermolecular forces becomes less significant compared

with the particles' kinetic energy, and the size of the molecules becomes less significant compared to theempty space between them.The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forcesand molecular size become important. It also fails for most heavy gases, such as water vapor ormany refrigerants.

[1]At some point of low temperature and high pressure, real gases undergo a phase

transition, such as to a liquid or a solid. The model of an ideal gas, however, does not describe or allowphase transitions. These must be modeled by more complexequations of state.The ideal gas model has been explored in both the Newtonian dynamics (as in "kinetic theory") andin quantum mechanics (as a "gas in a box"). The Ideal Gas model has also been used to model thebehavior of electrons in a metal (in the Drude model and the free electron model), and it is one of the

most important models in statistical mechanics.

The thermodynamic properties of an ideal gas can be described by two equations: The equation ofstate of a classical ideal gas is the ideal gas law

and the internal energy at constant volume of an ideal gas given by:

where P is the pressure V is the volume n is the amount of substance of the gas (in moles) R is the gas constant (8.314 JK

1mol

-1)

T is the absolute temperature

U is the internal energy is the dimensionless specific heat capacity at constant volume, 3/2 for monatomic gas,

5/2 for diatomic gas and 3 for more complex molecules.The amount of gas in JK

1is nR= NkB where

N is the number of gas particles kB is the Boltzmann constant (1.38110

23JK

1).

The probability distribution of particles by velocity or energy is given bythe Boltzmann distribution.The ideal gas law is an extension of experimentally discovered gas laws. Real fluidsat low density and high temperature approximate the behavior of a classical idealgas. However, at lowertemperatures or a higher density, a real fluid deviatesstrongly from the behavior of an ideal gas, particularly as it condenses from a gasinto a liquid or solid. The deviation is expressed as acompressibility factor.

Boyles LawBoyles law states that at constant temperature for a fixed mass, the absolute pressure and the volume ofa gas are inversely proportional. The law can also be stated in a slightly different manner, that the productof absolute pressure and volume is always constant.Most gases behave like ideal gases at moderate pressures and temperatures. The technology of the 17thcentury could not produce high pressures or low temperatures. Hence, the law was not likely to havedeviations at the time of publication. As improvements in technology permitted higher pressures andlower temperatures, deviations from the ideal gas behavior would become noticeable, and the


3/12

relationship between pressure and volume can only be accurately described employing realgas theory. The deviation is expressed as the compressibility factor.Robert Boyle (and Edme Mariotte) derived the law solely on experimental grounds. The law can also bederived theoretically based on the presumed existence of atoms and molecules and assumptions aboutmotion and perfectly elastic collisions (see kinetic theory of gases). These assumptions were met withenormous resistance in the positivist scientific community at the time however, as they were seen aspurely theoretical constructs for which there was not the slightest observational evidence.Daniel Bernoulli in 1738 derived Boyle's law using Newton's laws of motion with application on amolecular level. It remained ignored until around 1845, when John Waterston published a paper buildingthe main precepts of kinetic theory; this was rejected by the Royal Society of England. Later worksof James Prescott Joule, Rudolf Clausius and in particular Ludwig Boltzmann firmly established thekinetictheory of gases and brought attention to both the theories of Bernoulli and Waterston.The debate between proponents of Energetics and Atomism led Boltzmann to write a book in 1898, whichendured criticism up to his suicide in 1906. Albert Einstein in 1905 showed how kinetic theory applies tothe Brownian motion of a fluid-suspended particle, which was confirmed in 1908 by Jean Perrin.[edit]EquationThe mathematical equation for Boyle's law is:

where:p denotes the pressure of the system.Vdenotes the volume of the gas.k is a constant value representative of the pressure and volume of the system.

So long as temperature remains constant the same amount of energy given to thesystem persists throughout its operation and therefore, theoretically, the valueofkwill remain constant. However, due to the derivation of pressure asperpendicular applied force and the probabilistic likelihood of collisions with otherparticles through collision theory, the application of force to a surface may not beinfinitely constant for such values ofk, but will have a limit when differentiating suchvalues over a.Forcing the volume Vof the fixed quantity of gas to increase, keeping the gas at theinitially measured temperature, the pressurep must decrease proportionally.Conversely, reducing the volume of the gas increases the pressure.Boyle's law is used to predict the result of introducing a change, in volume and

pressure only, to the initial state of a fixed quantity of gas. The before and aftervolumes and pressures of the fixed amount of gas, where the before and aftertemperatures are the same (heating or cooling will be required to meet thiscondition), are related by the equation:

Boyle's law, Charles's law, and Gay-Lussac's law form the combined gas law.The three gas laws in combination with Avogadro's law can be generalized bythe ideal gas law.

Charless LawBoyles law states that at constant temperature for a fixed mass, the absolute pressure and the volume ofa gas are inversely proportional. The law can also be stated in a slightly different manner, that the product

of absolute pressure and volume is always constant.Most gases behave like ideal gases at moderate pressures and temperatures. The technology of the 17thcentury could not produce high pressures or low temperatures. Hence, the law was not likely to havedeviations at the time of publication. As improvements in technology permitted higher pressures andlower temperatures, deviations from the ideal gas behavior would become noticeable, and therelationship between pressure and volume can only be accurately described employing realgas theory. The deviation is expressed as the compressibility factor.Robert Boyle (and Edme Mariotte) derived the law solely on experimental grounds. The law can also bederived theoretically based on the presumed existence of atoms and molecules and assumptions about


4/12

motion and perfectly elastic collisions (see kinetic theory of gases). These assumptions were met withenormous resistance in the positivist scientific community at the time however, as they were seen aspurely theoretical constructs for which there was not the slightest observational evidence.Daniel Bernoulli in 1738 derived Boyle's law using Newton's laws of motion with application on amolecular level. It remained ignored until around 1845, when John Waterston published a paper buildingthe main precepts of kinetic theory; this was rejected by the Royal Society of England. Later worksof James Prescott Joule, Rudolf Clausius and in particular Ludwig Boltzmann firmly established thekinetictheory of gases and brought attention to both the theories of Bernoulli and Waterston.The debate between proponents of Energetics and Atomism led Boltzmann to write a book in 1898, whichendured criticism up to his suicide in 1906. Albert Einstein in 1905 showed how kinetic theory applies tothe Brownian motion of a fluid-suspended particle, which was confirmed in 1908 by Jean Perrin.EquationThe mathematical equation for Boyle's law is:

where:p denotes the pressure of the system.Vdenotes the volume of the gas.k is a constant value representative of the pressure and volume of the system.

So long as temperature remains constant the same amount of energy given to thesystem persists throughout its operation and therefore, theoretically, the valueofkwill remain constant. However, due to the derivation of pressure asperpendicular applied force and the probabilistic likelihood of collisions with otherparticles through collision theory, the application of force to a surface may not beinfinitely constant for such values ofk, but will have a limit when differentiating suchvalues over a.Forcing the volume Vof the fixed quantity of gas to increase, keeping the gas at theinitially measured temperature, the pressurep must decrease proportionally.Conversely, reducing the volume of the gas increases the pressure.Boyle's law is used to predict the result of introducing a change, in volume andpressure only, to the initial state of a fixed quantity of gas. The before and aftervolumes and pressures of the fixed amount of gas, where the before and aftertemperatures are the same (heating or cooling will be required to meet thiscondition), are related by the equation:

Boyle's law, Charles's law, and Gay-Lussac's law form the combined gas law.The three gas laws in combination with Avogadro's law can be generalized bythe ideal gas law.

Gas ConstantThe gas constant (also known as the molar, universal, orideal gas constant, denoted by thesymbol RorR) is a physical constant which is featured in a large number of fundamental equations in thephysical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmannconstant, but expressed in units of energy (i.e. the pressure-volume product) per kelvin permole (ratherthan energy per kelvin perparticle).Its value is

The two digits in parentheses are the uncertainty (standard deviation) in the last two digits of thevalue. The relative uncertainty is 1.810

6.

The gas constant occurs in the ideal gas law, as follows:

where P is the absolute pressure, V is the volume of gas, n is the number of moles of gas,and T is thermodynamic temperature. The gas constant has the same units as molar entropy.


5/12

Relationship with the Boltzmann constant

The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant byworking in pure particle count, N, rather than number of moles, n, since

where NA is the Avogadro constant. For example, the ideal gas law in terms ofBoltzmann's constant is

Measurement

As of 2006, the most precisemeasurement ofRis obtained by measuring the speed

of sound ca(p, T) in argon at the temperature Tof the triple point of water (used todefine the kelvin) at different pressuresp, and extrapolating to the zero-pressurelimit ca(0, T). The value ofR is then obtained from the relation

where: 0 is the heat capacity ratio (5/3 for monatomic gases such as argon); T is the temperature, TTPW = 273.16 K by definition of the kelvin; Ar(Ar) is the relative atomic mass of argon and Mu = 10

3kg mol

1.

Specific gas constant

The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant,divided by the molar mass (M) of the gas/mixture.

Just as the ideal gas constant can be related to the Boltzmann constant, so can the specific gas constantby dividing the Boltzmann constant by the molecular mass of the gas.

Another important relationship comes from thermodynamics. This relates the specific gas constant to thespecific heats for a calorically perfect gas and a thermally perfect gas.

where cp is the specific heat for a constant pressure and cv is the specific heat for a constant volume.It is common, especially in engineering applications, to represent the specific gas constant by thesymbol R. In such cases, the universal gas constant is usually given a different symbol such as R to

Rspecificfor dryair

Units

286.9 Jkg1

K1

53.3533 ftlbflb1

R1

1716.59 ftlbfslug1R1

Based on a mean molarmassfor dry air of 28.9645 g/mol.


6/12

distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whetherthe universal or specific gas constant is being referred to.U.S. Standard Atmosphere

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R* as:

The USSA1976 does recognize, however, that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. This disparity is not a significant departure fromaccuracy, and USSA1976 uses this value ofR*for all the calculations of the standard atmosphere. Whenusing the ISO value ofR, the calculated pressure increases by only 0.62 pascal at 11 kilometers (theequivalent of a difference of only 0.174 meter or 6.8 inches) and an increase of 0.292 Pa at 20 km (theequivalent of a difference of only 0.338 m or 13.2 in).

Specific HeatThe heat capacity at constant volume ofnR= 1 JK

1of any gas, including an ideal gas is:

This is the dimensionless heat capacity at constant volume, which is generally a function of

temperature. For moderate temperatures, the constant for a monoatomic gas is while

for a diatomic gas it is . It is seen that macroscopic measurements on heat capacityprovide information on the microscopic structure of the molecules.The heat capacity at constant pressure of 1 JK

1ideal gas is:

where H= U+pV is the enthalpy of the gas.

2. Processesof Ideal Gas3.

Isometric Process

An isochoric process, also called a constant-volume process, an isovolumetric process, oran isometric process, is a thermodynamic process during which the volume of the closedsystemundergoing such a process remains constant. In nontechnical terms, an isochoric process isexemplified by the heating or the cooling of the contents of a sealed non-deformable container: Thethermodynamic process is the addition or removal of heat; the isolation of the contents of the containerestablishes the closed system; and the inability of the container to deform imposes the constant-volumecondition.

An isochoric thermodynamic process is characterized by constant volume, i.e. V= 0. The process doesno pressure-volume work, since such work is defined by

W= PV,where P is pressure. The sign convention is such that positive work is performed by the system on

the environment.For a reversible process, the first law of thermodynamics gives the change in the system's internalenergy:dU= dQ dWReplacing work with a change in volume givesdU= dQ PdVSince the process is isochoric, dV= 0, the previous equation now givesdU= dQUsing the definition of specific heat capacity at constant volume,Cv = dU / dT,


7/12

dQ = nCvdTIntegrating both sides yields

Where Cis the specific heat capacity at constant volume, a is initial temperature and b isfinal temperature. We conclude with:

Isochoric Process in the Pressure volume diagram. In this diagram, pressure increases, but volumeremains constant.On a pressure volume diagram, an isochoric process appears as a straight vertical line. Itsthermodynamic conjugate, an isobaric process would appear as a straight horizontal line.Ideal gasIf an ideal gas is used in an isochoric process, and the quantity of gas stays constant, then the increasein energy is proportional to an increase intemperature and pressure. Take for example a gas heated in arigid container: the pressure and temperature of the gas will increase, but the volume will remain thesame.Ideal Otto cycle

The ideal Otto cycle is an example of an isochoric process when it is assumed that the burning ofthe gasoline-air mixture in an internal combustion engine car is instantaneous. There is an increase in thetemperature and the pressure of the gas inside the cylinder while the volume remains the same.Etymology

The noun isochorand the adjective isochoric are derived from the Greek words (isos)meaning "equal", and (chora) meaning "space."

Isobaric Process

An isobaric process is a thermodynamic process in which the pressure stays constant. The term derivesfrom the Greek isos, (equal), and barus, (heavy). The heat transferred to the system does work but alsochanges the internal energy of the system.

According to the first law of thermodynamics, whereW is work done by the system, U is internalenergy, and Q is heat. Pressure-volume work by the closed system is defined as:

where means change over the whole process, whereas d denotes a differential. Sincepressure is constant, this means that

.Applying the ideal gas law, this becomes

assuming that the quantity of gas stays constant, e.g., there is no phasechange during a chemical reaction. According to the equipartition theorem, thechange in internal energy is related to the temperature of the system by

,where cV is specific heat at a constant volume.Substituting the last two equations into the first equation produces:

,where cP is specific heat at a constant pressure.


8/12

Specific heat capacity

To find the molar specific heat capacity of the gas involved, the following equations apply for any generalgas that is calorically perfect. The property is either called the adiabatic index or the heat capacity ratio.Some published sources might use k instead of .Molar isochoric specific heat:

.Molar isobaric specific heat:

.

The values for are = 1.4 for diatomic gasses like air and its major components, and formonatomic gasses like the noble gasses. The formulas for specific heats would reduce in these specialcases:Monatomic:

andDiatomic:

andAn isobaric process is shown on a P-V diagram as a straight horizontal line, connecting the initial andfinal thermostatic states. If the process moves towards the right, then it is an expansion. If the processmoves towards the left, then it is a compression.Sign convention for work

The motivation for the specific sign conventions of thermodynamics comes from early development ofheat engines. When designing a heat engine, the goal is to have the system produce and deliver workoutput. The source of energy in a heat engine, is a heat input.If the volume compresses (delta V = final volume - initial volume < 0), then W < 0. That is, during isobaric

compression the gas does negative work, or the environment does positive work. Restated, theenvironment does positive work on the gas.If the volume expands (delta V = final volume - initial volume > 0), then W > 0. That is, during isobaricexpansion the gas does positive work, or equivalently, the environment does negative work. Restated, thegas does positive work on the environment.If heat is added to the system, then Q > 0. That is, during isobaric expansion/heating, positive heat isadded to the gas, or equivalently, the environment receives negative heat. Restated, the gas receivespositive heat from the environment.If the system rejects heat, then Q < 0. That is, during isobaric compression/cooling, negative heat isadded to the gas, or equivalently, the environment receives positive heat. Restated, the environmentreceives positive heat from the gas.Defining enthalpy

An isochoric process is described by the equationQ = U. It would be convenient to have a similar

equation for isobaric processes. Substituting the second equation into the first yields

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

.Enthalpy and isobaric specific heat capacity are very useful mathematical constructs, since whenanalyzing a process in an open system, the situation of zero work occurs when the fluid flows at constantpressure. In an open system, enthalpy is the quantity which is useful to use to keep track of energycontent of the fluid.


9/12

Variable density viewpoint

A given quantity (mass m) of gas in a changing volume produces a change in density . In this contextthe ideal gas law is written

R(T) = M Pwhere T is thermodynamic temperature. When R and M are taken as constant, then pressure Pcan stayconstant as the density-temperature quadrant (,T) undergoes a squeeze mapping.

Isothermal Process

An isothermal process is a change of a system, in which the temperature remains constant: T= 0.This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and thechange occurs slowly enough to allow the system to continually adjust to the temperature of the reservoirthrough heat exchange. In contrast, an adiabatic process is where a system exchanges no heat withits surroundings (Q = 0). In other words, in an isothermal process, the value T= 0 but Q 0, while in anadiabatic process, T 0 but Q = 0.

Details for an ideal gas

For the special case of a gas to which Boyle's law applies, the product pV is a constant if the gas is keptat isothermal conditions. The value of the constant isnRT, where n is the number of moles of gas presentand R is the ideal gas constant. In other words, the ideal gas lawpV= nRTapplies. This means that

holds. The family of curves generated by this equation is shown in the graph presented here. Eachcurve is called an isotherm. Such graphs are termedindicator diagrams and were first usedby James Watt and others to monitor the efficiency of engines. The temperature corresponding toeach curve in the figure increases from the lower left to the upper right.Calculation of work

The yellow area represents "work" for this isothermal changeIn thermodynamics, the work involved when a gas changes from state A to state B is simply

For an isothermal, reversible process, this integral equals the area under the relevantpressure-volume isotherm, and is indicated in yellow in the figure (at the bottom right-hand ofthe page) for an ideal gas. Again, p = nRT / V applies and with T being a constant (as this is an

isothermal process), we have:

By convention work is defined as the work the system does on the surroundings. It is also worth notingthat, for many systems, if the temperature is held constant then the internal energy of the system also isconstant, and so U= 0. From First Law of Thermodynamics, Q = U+ W , so it follows that Q = Wforthis same isothermal process.

Applications

Isothermal processes can occur in any kind of system, including highly-structured machines, andeven living cells. Various parts of the cycles of some heat engines are carried out isothermally and maybe approximated by a Carnot cycle. Phase changes, such as melting or evaporation, are also isothermalprocesses.In Isothermal non flow Process, the work done by compressing the perfect gas (Pure Substance) is a

negative work, as work is done on the system, as result of compression, the volume will decrease, andtemperature will reduce. To mentain the temperature at constant value ( as process is isothermal) heatenergy has to be supplied to the system. The amount of energy supplied to the system is equal to thework done (by compressing the perfect gas). Thence Q = W. Heat supplied to the system will be positive.In equation of work, the term nRT can be replaced by PV of any state. The product of pressure andvolume is infact, 'Moving Boundary Work'; the systems boundaries are compressed. For Expansion thesame theory is applied.

As per Joule's Law, Internal energy is the function of absolute temperature. In isothermal process thetemperature is constant. Hence, the internal energy is constant. And net internal energy is ZERO.


10/12

Isentropic Process

In thermodynamics, an isentropic process orisoentropic process (= "equal"(Greek); entropy = "disorder"(Greek)) is one in which for purposes of engineering analysis andcalculation, one may assume that the process takes place from initiation to completion without anincrease or decrease in the entropy of the system, i.e., the entropy of the system remains constant. It canbe proved that any reversible adiabatic process is an isentropic processBackground

Second law of thermodynamics states that,

where Q is the amount of energy the system gains by heating, T is the temperature of the system,and dS is the change in entropy. The equal sign will hold for a reversible process. For a reversibleisentropic process, there is no transfer of heat energy and therefore the process is also adiabatic.For an irreversible process, the entropy will increase. Hence removal of heat from the system(cooling) is necessary to maintain a constant internal entropy for an irreversible process in order tomake it isentropic. Thus an irreversible isentropic process is not adiabatic.For reversible processes, an isentropic transformation is carried out by thermally "insulating" thesystem from its surroundings. Temperature is the thermodynamic conjugate variable to entropy,thus the conjugate process would be an isothermal process in which the system is thermally"connected" to a constant-temperature heat bath.Isentropic flow

An isentropic flow is a flow that is both adiabatic and reversible. That is, no heat is added to theflow, and no energy transformations occur due to friction ordissipative effects. For an isentropic flowof a perfect gas, several relations can be derived to define the pressure, density and temperaturealong a streamline.Note that energy can be exchanged with the flow in an isentropic transformation, as long as itdoesn't happen as heat exchange. An example of such an exchange would be an isentropicexpansion or compression, with entail work done on or by the flow.[edit]Derivationof the isentropic relationsFor a closed system, the total change in energy of a system is the sum of the work done and theheat added,

The work done on a system by changing the volume is,

wherep is the pressure and V is the volume. The change in enthalpy () is given by,

Since a reversible process is adiabatic (i.e. no heat transfer occurs),

so . This leads to two important observations,

, and

or

=> The heat capacity ratio can be written as,

For an ideal gas is constant. Hence on integrating theabove equation, assuming a perfect gas, we get

i.e.


11/12

Using the equation of state for an ideal

gas, ,

also, for constant Cp = Cv+ R(per mole),

and

Thus for isentropic processes with an ideal gas,

or

Table of isentropic relations for an ideal gas

Derived from:

Where:= Pressure

= Volume

= Ratio of specific heats =

= Temperature= Mass

= Gas constant for the specific gas =

= Universal gas constant

= Molecular weight of the specific gas= Density

= Specific heat at constant pressure


12/12

= Specific heat at constant volume

Polytropic ProcessA polytropic process is a thermodynamic process that obeys the relation:

wherep is the pressure, V is volume, n, the polytropic index, is any real number, and Cis a

constant. This equation can be used to accurately characterize processes of certain systems,notably the compression or expansion of a gas and in some cases liquids and solids.

Applicability

The equation is a valid characterization of a thermodynamic process assuming that the processis quasistatic and the values of the heat capacities, Cp andCV, are almost constant when n is not zero orinfinity. (In reality, Cp and CVare actually functions of temperature, but are nearly constant within smallchanges of temperature).Under standard conditions, most gases can be accurately characterized by the ideal gas law. Thisconstruct allows for the pressure-volume relationship to be defined for essentially all ideal thermodynamiccycles, such as the well-known Carnot cycle. Note however that there may also be instances where apolytropic process occurs in a non-ideal gas.[edit]Relationship to ideal processes

For certain values of the polytropic index, the process will be synonymous with other common processes.

Some examples of the effects of varying index values are given in the table.

Variationof polytropic index n

Polytropicindex

Relation Effects

n < 0 An explosion occurs

n = 0pV =p(constant)

Equivalent to an isobaric process (constant pressure)

n = 1pV= NkT(constant)

Equivalent to an isothermal process (constant temperature)

1 < n < A quasi-adiabatic process such as in an internal combustion engine during

expansion, or in vapor compression refrigeration during compression

n =

= is the adiabatic index, yielding an adiabatic process (no heat transferred)

Equivalent to an isochoric process (constant volume)

When the index n is between any two of the former values (0, 1, gamma, or infinity), it means that thepolytropic curvewill bounded by the curves of the two corresponding indices.

Note that 1 < < 2, since .Notation

In the case of an isentropic ideal gas, is the ratio of specific heats, known as the adiabatic index or asadiabatic exponent.

An isothermal ideal gas is also a polytropic gas. Here, the polytropic index is equal to one, and differsfrom the adiabatic index .In order to discriminate between the two gammas, the polytropic gamma is sometimes capitalized, .To confuse matters further, some authors refer to as the polytropic index, rather than n. Note that

Thermodynamics Prelim

Documents

Transcript of Thermodynamics Prelim