50658252 Thermodynamics Ppt Notes

School of Mechanical SciencesIndian Institute of Technology, Bhubaneswar

E-mail: [email protected], Tel: 2306-273

P. Rath, Ph.D.Asst. Professor

mailto:[email protected]�

Course Structure► Mid-semester: 30%► End-semester: 50%► Quiz & Assignments: 15%► Attendance: 5%

Reference Books► Thermodynamics: An Engineering Approach – Yunus A.

Cengel and Michael A. Boles► Engineering Thermodynamics – P. K. Nag► Fundamentals of Classical Thermodynamics – Gordon J.

Van Wylen and Richard E. Sonntag

► Thermal Science and Engineering – D.S. Kumar

Essential Reading @ My Lecture Notes

Course Outline ► Fundamentals of Thermodynamics► Basic Concepts & Definitions► Zeroth Law of Thermodynamics

Thermodynamics derived from the greek words“therme” means heat and “dynamis” means power.

Thermodynamics is the science of energy

What is Thermodynamics?

► Thermodynamics is a science dealing with energy and its transformation

► It deals with equilibrium and feasibility of a process► Deals with relations between heat and work and the

properties of a system

Fundamentals of Thermodynamics

► 1st and 2nd laws of thermodynamics emergedsimultaneously in the 1850s.

► The above laws are derived out of the works of WilliamRankine, Rudolph Clausius and Lord Kelvin (formerlyWilliam Thomson).

► The term “thermodynamics” was first used in apublication by Lord Kelvin in 1849.

► The first textbook of thermodynamics was written in 1859by Wiiliam Rankine, a Professor at the University ofGlasgow.

Brief History

Fundamentals of Thermodynamics

Fundamentals of ThermodynamicsWhat Thermodynamics tells us?

► Is the proposed chemical reaction or a physical process possible?

► Does the reaction/process go to completion or does it proceed to a certain extent only beyond which it cannot be proceed?

► How much energy is required for the process to take place?

► What is the maximum efficiency of a heat engine or the maximum coefficient of performance of a refrigerator?

1. Zeroth law of thermodynamics deals with thermalequilibrium and provides a means of measuringtemperature.

2. The first law of thermodynamics tells about the conservationof energy and introduces the concept of internal energy.

3. The second law of thermodynamics dictates the limits ofconverting the internal energy into work and introduces theconcept of entropy. It also tells whether a particular processis feasible or not.

4. The third law of thermodynamics provides a datum for themeasurement of entropy.

Fundamentals of Thermodynamics (Cont’d)Principles of Thermodynamics

• It consists of four laws

1. Macroscopic Approach2. Microscopic Approach

Fundamentals of Thermodynamics (Cont’d)

Two approaches to study Thermodynamics

Macroscopic Approach

► The structure of matter is not considered.► Only a few variables are used to describe the state of matter.► The values of these variables can be measured.► Classical thermodynamics adopts the macroscopic approach.► It is based on continuum theory.

Microscopic Approach

► A knowledge of the structure of matter is essential.► A large number of variables are needed to describe

the state of matter. The values of these variables cannot be measured. Statistical thermodynamics adopts the microscopic approach.

Basic Concepts and DefinitionsSystem A definite quantity of matter bounded by some surface. The boundary surface may be real or imaginary. It may change in shape and size. Sometimes the system is also referred as control mass or cv. A system can exchange energy in the form of work and heat.

Fundamentals of Thermodynamics (Cont’d)

Surrounding The combination of matter and space external to the system

constitutes the surrounding.

3 types of Systems1. Open System (Control Volume): A properly selected region in

space that involves mass as well as energy flow across its boundary.Ex: Compressor, turbine, nozzle.

2. Closed System (Control Mass): It consists of a fixed amount ofmass in a selected region in space and no mass can cross itsboundary. Energy can flow across the boundary. Ex: Pressurecooker, refrigerator, cylinder fitted with a movable piston.

3. Isolated System: No mass and energy flow across the boundary of achosen space. Ex: Thermos flask, Universe.

Basic Concepts & Definitions (Cont’d)

► A property is any characteristic (which can bequantitatively evaluated) that can be used to describethe state of a system. Ex: P, V, T, etc.

Property

Essential features of a property

► It should have a definite unique value when the systemis in a particular state.

► The value of the property should not depend upon thepast history of the system.

► Property is a state function and not a path function.► Its differential is exact.


► Extensive Properties: Are those whose values depend on the size or extent of the system. Ex: Mass, Volume, Total Energy.

► Intensive Properties: Are those that are independent of the size of a system. Ex: T, P, ρ

Thermodynamics deals with relevant properties only.

Classification of property

1. Relevant Property: Associated with energy and itstransformation.

2. Irrelevant Property: Not associated with energy and itstransformation. Ex: Color, odor, taste

How to determine whether a property is intensive or extensive?


► Specific Property: Extensive property per unit mass

m V T P ρ

Extensive PropertyIntensive Property


m/2VTPρ

m/2VTPρ

EnergyBasic Concepts & Definitions (Cont’d)

► Ability to do work.

Modes of Energy► Macroscopic: Organized form of energy. Example:

Kinetic Energy (KE), Potential Energy (PE)► Microscopic: Disorganized form of energy. Example:

Internal Energy (U)

E = KE + PE + U

Total Energy (E): Sum of all macroscopic and microscopic modes of energy.

Steady StateIf the property of a system at any specified location are independent of time, then the system is said to be in a steady state.

Basic Concepts & Definitions (Cont’d)State► It is the condition of a system identified by its properties.► The number of properties required to fix the state of a

system is given by the state postulate.

The State PostulateThe state of a simple compressible system iscompletely specified by two independent, intensiveproperties.

Equilibrium► It is a concept associated with the absence of any

tendency for spontaneous changes when the system isisolated.

► In a state of equilibrium, the properties of system areuniform and only one value can be assigned to eachproperty.


Types of Equilibrium1. Thermal Equilibrium: Equality in temperature2. Mechanical Equilibrium: Equality in pressure3. Chemical Equilibrium: Equality in chemical potential4. Thermodynamic Equilibrium: 1 + 2 + 3

Basic Concepts & Definitions (Cont’d)Process► When a system changes from one equilibrium state to

another, the path of successive states through which thesystem passes is called a process.

Quasi Equilibrium Process► While a system passing from one state to the next, the

deviation from equilibrium is infinitesimal, a quasiequilibrium process occurs.

► It is otherwise called quasistatic process as the processproceeds very slowly under the influence of infinitesimaldriving forces (∆P, ∆T, etc).

► The system remains in infinitesimally close to anequilibrium state at all times.

Basic Concepts & Definitions (Cont’d)Nonequilibrium Process► If the system goes from one equilibrium state to another

through a series of nonequilibrium states, a nonequilibrium process occurs.

► It is represented by doted line.► Example: Combustion, free expansion of gas (sudden

expansion).

► Thermodynamics gives a broad definition of work.

Work and HeatWork

How to define “work” thermodynamically?

► If a battery connected to resistor circuit as shown below, does it doing any work?

Battery

Yes, it is doing some work. How?

If the resistor is replaced by a motor which is lifting a mass, a work is said to be done.

Work done by a system on its surrounding isdefined as an interaction whose sole effect,external to the system, could be reduced to theraising of a mass through a distance.

Work and Heat Cont’dThermodynamic definition of work

F = Generalized forceDl = Generalized displacement

dW = F · dl

Mechanical Work

Generalized force = Applied Force (F)Generalized displacement = Displacement (ds)

: where P is the absolute pressure.

Work and Heat Cont’dElectrical WorkGeneralized force = Applied Potential (V)Generalized displacement = Charge (q)

Magnetic Work

Generalized force = Applied Magnetic Field Strength (B)Generalized displacement = Magnetic Dipole Moment (m)

Moving Boundary Work

∫= dVPW

The area on a P-V diagram represents the work for a quasi-equilibrium process only.

Note

Non-equilibrium Work

Work and Heat Cont’d

► Work obtained from non-equilibrium process.► Work cannot be calculated using .► Example: Paddle wheel work, Free expansion

∫ dVP

Notes on Work► It is not a property of the system and its

differential is not exact.► It depends on the process path.► Work interactions depend upon the choice of

the system.

► Energy can cross the boundary of a closed system in theform of heat or work.

► The mode of energy transfer, which cannot beaccounted as work from a macroscopic point of view iscalled heat interaction.

► Energy transfer as heat occurs by virtue of temperaturedifference across the boundary of the system.

► It is not a property of the system and its differential is notexact.

Work and Heat Cont’dHEAT

Zeroth Law of Thermodynamics► Developed by R. H. Fowler in 1931.► This law is developed after the 1st and 2nd law of

thermodynamics.► It provides the basis for the measurement of

temperature of a systemStatementWhen two bodies are in thermal equilibrium with athird body, they are also in thermal equilibrium witheach other.

Temperature ScalesThermometryIt is defined as the act of measuring temperaturewith accuracy and precision.► Temperature measurement depends upon the

establishment of thermodynamic equilibrium between thesystem and the device used to measure the temperature.

► The sensing element of the device has certain physicalcharacteristics which change with temperature and thiseffect is taken as a measure of temperature.

Thermometric PropertyA property or physical characteristic which changes itsvalue as a function of temperature is called thermometricproperty.

A substance whose property or physical characteristicchanges as a function of temperature is known as thethermometric substance.

Temperature Scales Cont’dThermometric Substance

► A change in dimension. e.g., mercury-in-glassthermometer, gas thermometer, etc.

► A change in electrical resistance of metals andsemiconductors. e.g., resistance thermometers,thermistors, etc.

► A thermo-electric emf for two different metals and alloysjoined together. Ex. Thermocouples.

Physical Characteristics

► A change in the intensity and color of emitted radiations. Ex. Radiation thermometer.

► Fusion of materials when exposed to temperatures. Ex. Pyrometer.

Temperature Scales Cont’d

t ∼ x Relationship

BAxt += Linear relationshipThe constants A and B can be determined from any twofixed thermometric points, e.g., the ice point (freezingpoint) and the steam point (boiling point) of water.Note► A Fixed Point refers to an easily reproducible state of

an arbitrarily chosen standard system.

Let temperatures at ice and steam point in a scale isrepresented as ti and ts and the correspondingthermometric properties be xi and xs respectively, then


BAxt ii += BAxt ss +=

From above two equations

is

is

xxtt

A−−

= iis

isi x

xxtt

tB−−

−=

Hence,

is

iisi xx

xxtttt

−−

−+= )(

Centigrade and Fahrenheit ScalesTemperature Scales Cont’d

In Centigrade scale,

ti = 0 °C ts = 100 °C

Hence,

is

iC xx

xxt

−−

= 100

In Fahrenheit scale,

ti = 32 F ts = 212 F

Hence,


is

iF xx

xxt

−−

+= 18032

Note► The thermometric substance used is same while

measuring temperature in Centigrade and Fahrenheitscales.

Therefore,

18032

100−

= FC tt

( )3295

−= FC tt⇒

A temperature scale that is independent of theproperties of substance is known as theThermodynamic Temperature Scale.

Temperature Scales Cont’dThermodynamic Temperature Scale

► Thermodynamic temperature scale in the SI is the Kelvin scale (K).

► Thermodynamic temperature scale in the English system is the Rankine scale (R).

A temperature scale that turns out to be identical to the Kelvin scale is the ideal gas temperature scale.

1st Law Analysis

Process Path

dEWQ =−δδ

∫∫ = WQ δδ

Cyclic Process

P

V

1

2

Heat Transfer to the systemWork done by the system

Specific Heat (C)

1st Law Analysis

► Specific Heat at Constant Pressure: CP

► Specific Heat at Constant Volume: Cv

Energy required to raise the temperature of a unit mass of a substance by one degree

PP T

hC

∂∂

=V

v TuC

∂∂

=

► It is a measure of energy storage capabilities of various substances.

► It can be specified by two independent intensive properties.

For solids and liquids

1st Law Analysis

CCC vP ==

αβ 2vTCC vP =−

TPv

v

∂∂

−=1α

PTv

v

∂∂

−=1β

Properties of Pure Substance

A substance with fixed chemical composition

Pure Substance

► A pure substance is not necessarily consists of asingle chemical element or compound.

► A mixture of two or more phases of a puresubstance is also a pure substance as long asthe chemical composition of each phase is same.

PhaseProperties of Pure Substance Cont’d

A system that has distinct molecular arrangementwhich is homogeneous throughout is called aphase.

► If a system contains more than one phase, theyare separated by a phase boundary.

► Principal phases: Solid, liquid and gas.► There may be several phases within a principal

phase.

1st Law for Flow Process

From 1st Law of Thermodynamics

∫∫ ⋅+∂∂

=

CSCV

Sys AdedVetdt

dE vρρ

[ ]SysSys WQ

dtdE

−=

gzue ++=2v2

where

Work done by system

Heat transfer to the system

► Shaft Power ( )

1st Law Flow Process Cont’d

[ ] ∫∫ ⋅+∂∂

=−

CSCV

CV AdedVet

WQ vρρ

as ∆t → 0, the system and the CV coincides, hence

[ ] [ ]CVSys WQWQ −=−

othershPS WWWWW +++=

SWMachine such as a pump, a turbine, a fan or a compressor whose shaft protrudes through the control surface and the work transfer associated with all such devices is the shaft power.

Work done by CV

► Rate of work done by pressure forces on CV1st Law Flow Process Cont’d

( )∫ ⋅−=CS

P dAnPW ˆv

► Rate of work done by shear forces

∫ ⋅−=

CS

Sh dAW v τ

........++= magneticelectricalother WWW

► Rate of work done by other forces

n̂F

► The term Pv is the flow work, which is the workassociated with pushing a fluid into or out of acontrol volume per unit mass.

[ ] ∫∫ ⋅

++

∂∂

=−−−CSCV

CVotherShS AdPedVet

WWWQ vρ

ρρ

[ ] ( )∫∫ ⋅++∂∂

=−−−CSCV

CVotherShS AdPvedVet

WWWQ vρρ

[ ] ∫∫ ⋅

+++

∂∂

=−−−CSCV

CVotherShS AdgzhdVet

WWWQ v

2v2

ρρ


[ ] ∫ ⋅

+++

∂∂

=−−−CS

CVCVotherShS Adgzh

tEWWWQ

v2v2

ρ


[ ]CVCVotherShSii

iiee

ee EWWWQgzhmgzhm −−−−=

++−

++

2v

2v 22

ConditionInletiConditionExite

==

where the subscripts

► The state of matter at any location inside thecontrol volume does not change with time.

► The rate of energy transfer as heat and workacross the control surface are constant.

Steady Flow Process

∫∫ ⋅+∂∂

==

CSCV

Sys AddVtdt

dm v0 ρρ

RTT for mass conservation0 (Steady flow)

0v =−=⋅∫ ie

CS

mmAd ρ ie mm =⇒

► The steady state flow implies that there is noaccumulation of mass inside the control volume.That is the rate of inflow of mass is equal to therate of outflow of mass.

► The devices like turbines, compressors, pumps,etc. operates at steady state conditions exceptat the start up and shut down periods.

► An analysis of steady state flow processes isuseful in evaluating the performance of suchdevices and in the design of equipment.

Steady Flow Process Cont’d

Steady Flow: ApplicationsTurbine/Compressor

0=Q► Adiabatic Process, i.e.

mWgzhgzh S

ii

iee

e

−=

++−

++

2v

2v 22

► For turbine, is Positive► For Compressor, is Negative

SW

SW

Ideal Gas: FundamentalsProperties of Ideal Gases

An ideal (or perfect) gas has no intermolecular forces ofattraction or repulsion between the particles of gas andthe particles are in a state of continuous motion.

The collision of the molecules with one another and withthe walls of the container is perfectly elastic.

It does not change its phase during a thermodynamicprocess.

The volume occupied by the gas molecules is negligibleas compared to the volume of the gas.

It obeys a set of common rules governing change of itsproperties.

Real gases differ from ideal ones due to presence of theintermolecular forces and finite molecular volumes.

► The equation of state for ideal gas is given as

RTpv =

► The state of zero pressure of a real gas is called ideal state.

► Under special conditions (p → 0), even the real gases behave in a similar manner. Hence,

( ) 0→= pRTpv

Ideal Gas: Fundamentals Cont’d

► The identical behavior of a real gas at hightemperature and low pressure is called idealbehavior.

► The internal energy u and enthalpy h arefunctions of temperature alone. That is

)(1 Tfu = )(1 Tfh =

Ideal Gas: Fundamentals Cont’d

Boyle’s LawIf the state of a perfect gas changes at constanttemperature, then the volume of a given mass ofthe gas is inversely proportional to the absolutepressure.

p1

V ∝ pV = Const.

► Boyle’s law is essentially valid only at very low pressureand at moderately high temperature.

Charles’s LawIf the state of a perfect gas changes atconstant pressure, then the volume of agiven mass of perfect gas varies directly asabsolute temperature.

V ∝ T Const.=TV

► It was found by Gay-Lussac and Regnault that atconstant pressure, the change in volume of any perfectgas corresponding to a unit degree temperature changeis given by 1/273 of its volume at 0 °C.

According to Charles’s Law,

Charles’s Law Cont’dVo = Volume of the gas at 0 °C.

Vt = Volume of the gas at t °C.

)273( +== tAATVt

)2730( += AVo∴ ⇒273

oVA =

∴273

tVVV oot +=

oot V

tVV

=

−2731

Gay-Lussac Law

Avogadro’s LawUnder identical conditions of temperatureand pressure, equal volumes of all gaseshave same number of molecules.Gas – 1: M1, P, V, T Gas – 2: M2, P, V, T

According to Avogadro’s law, each gas will contain the same number of molecules, say n.

m1∝ n M1 = k n M1 m2∝ n M2 = k n M2

againm1 = ρ1 V m2 = ρ2 V

► The product of molecular mass and specific volume isconstant for all ideal gases under identical conditions ofpressure and temperature.

► The quantity vM is called molar volume. It represents thevolume of 1 kmol of ideal gas.

Avogadro’s Law

1

2

2

1

2

1

vv

MM

==ρρ

Hence,

2211 MvMv = Constant=Mv

► At standard conditions of t = 0 °C and p = 1.013bar, the volume of 1 kilo mole of all gases isequal to 22.4135 m3.

Avogadro’s Law

Molar volume = voM = 22.4135 m3/Kmol

► For molar volume of a gas, the characteristic gas equation can be written as

TRTRMpV umol ==

► Ru is called as the universal gas constant or the molargas constant.

Van der Waal’s Eqn. of State► Real gases differ from ideal ones due to

presence of the intermolecular forces andalso to the finite molecular volumes.

( ) TRbvvap uMM

=−

+ 2

vM = Molar volume

( ) TRbvMMvap u=−

+ 22

TRbMmV

MmV

ap u=

−

+2

2

2

where,

Mmn =

( ) TnRnbVVanp u=−

+ 2

2

Van der Waal’s Eqn. of State Cont’d

► Vander Waal’s equation for 1 mol of real gas is,

( ) TRbVVap u=−

+ 2

( ) 023 =−++− abaVVTRpbpV u

► At critical point all the three roots coincides.► At critical point , the isotherm has zero slope.



Isotherm Lines

At critical point,


0=

∂∂

CVp 02

2

=

∂∂

CVp

bVC 3=

uC bR

aT27

8=

227bapC =

Mixture of Gases

► Total mass of gases in the cylinder is given as

Gases a, b, c, …..Volume, V

Pressure, PTemperature, T

........+++= cba mmmm

........+++= cba nnnn

Gases a, b, c, …..Volume, V

Pressure, PTemperature, T

Mass Fraction

...........,,,mmx

mmx

mmx c

cb

ba

a ===

1........ =+++ cba xxx

Mole Fraction

...........,,,nny

nny

nny c

cb

ba

a ===

1........ =+++ cba yyy

Mixture of Gases Cont’d…

Partial Pressure

Gases a, b, c, …..V

PT

Gas aV

Gas bV

Gas cV

Pa PbPcT T T

TnRPV u=

TRnVP uaa = TRnVP ubb = TRnVP ucc =

Mixture of Gases Cont’d…

Partial pressure is defined as the pressure whicheach individual component of a gas mixture wouldexert if it alone occupied the volume of the mixtureat the same temperature.

Partial Pressure Cont’d…

( ) ( ) PVTnRTRnnnVPPP uucbacba ==+++=+++ ........

....+++= cba PPPP

The total pressure of a mixture of ideal gases isequal to the sum of the partial pressures of theindividual gas components of the mixture.

Dalton’s Law of Partial Pressure

Remarks on Dalton’s LawSpecific volume (ν)

► According to Dalton’s Law,

Gases a, b, c, …..V

T P

Gas aVa = V

T Pa

...==== cba VVVV

...==== ccbbaa mmmm υυυυ

Again,

........+++= cba mmmm

........+++=∴cc

c

bb

b

aa

a

vmm

vmm

vmm

mvm

Therefore,

........1111+++=

cba vvvv⇒

........+++= cba ρρρρ

Specific volume (ν) Cont’d…

Amagat-Leduc Law of Partial Volumes► The partial volume of a gas component represents the

volume that this gas component will occupy if itstemperature and pressure is kept equal to that of the gasmixture.

Amagat-Leduc Law Cont’d….► If P and T are the pressure and the absolute

temperature of the gas mixture, then

TnRPV u=

► For the component gases

,TRnPV uaa = ,TRnPV ubb = ,TRnPV ucc = …….

( ) ( ) PVTnRTRnnnVVVP uucbacba ==+++=+++∴ ........

VVVV cba =+++ ........⇒

1........ =+++VV

VV

VV cba

FractionVolume== aa r

VV

Amagat-Leduc Law Cont’d….

TnRPV u=

TRnVP uii =

: For the gas mixture

: For a constituent of the gas mixture

nn

PP ii =⇒

Now,

► The gas equation for the partial volumes of the gas can be written as

TRnPV uii =

nn

VV ii =∴

Hence,

VV

nn

PP iii ==

Partial Pressure Ratio

Mole Fraction

Volume Fraction

Amagat-Leduc Law Cont’d….

R of Gas MixtureFor each constituent of the gas mixture containedin a vessel of volume V and temperature T, we canwrite

,TRmVP aaa = ,TRmVP bbb = ,TRmVP ccc = …….

( ) ( )TRmRmRmVPPP ccbbaacba ........ +++=+++∴

Using Dalton’s law,

( )TRmRmRmVP ccbbaa ....+++=

For the gas mixture, we have( ) TmRTRmmmVP mmcba =+++= ....

Molecular Mass of Gas Mixture

R of Gas Mixture Cont’d…....+++=∴ ccbbaam RmRmRmmR

⇒ ∑∑ ==+++=i

iuiiccbbaam M

xRRxRxRxRxR ....

∑∑ ==+++==i

iu

i

ui

c

uc

b

ub

a

ua

m

um M

xRMRx

MRx

MRx

MRx

MRR ....

∑=+++=i

i

c

c

b

b

a

a

m Mx

Mx

Mx

Mx

M....1

⇒ In terms of Mass Fraction

Molecular Mass of Gas Mixture Cont’d….

Gibbs-Dalton Law

........+++= cba mmmm⇒ ........+++= ccbbaam MnMnMnnM

∴ ........+++= ccbbaam MyMyMyM

The internal energy, enthalpy and entropy of a gaseous mixtureare respectively equal to the sums of the internal energies, theenthalpies and the entropies which each component of the gasmixture would have, if each alone occupied the volume of themixture at the temperature of the mixture.

Gibbs-Dalton Law Cont’d…Internal Energy

∑=+++= iiccbbaa umumumummu ......

TcmTcmTcmTcmTcm ivicvcbvbavav ∑=+++= ,,,, ......

∑=+++= ivicvcbvbavav cxcxcxcxc ,,,, ......

∑=+++= iiccbbaa uxuxuxuxu ......

Specific Heats

∑=+++= iiccbbaa umumumummu ......

⇒

∴

⇒

According to Gibb-Dalton law, enthalpy of the mixture is given as

Gibbs-Dalton Law Cont’d…

∑=+++= iiccbbaa hmhmhmhmmh ......∴ TcmTcmTcmTcmTcm ipicpcbpbapap ∑=+++= ,,,, ......

⇒ ∑=+++= ipicpcbpbapap cxcxcxcxc ,,,, ......

Entropy

∑=+++= iiccbbaa smsmsmsmms ......⇒ ∑=+++= iiccbbaa sxsxsxsxs ......

Entropy Change in Mixing of GasesImagine a number of inert ideal gases separated from oneanother by suitable partitions, all the gases being at thesame temperature T and pressure P. The total initialentropy will be

∑=+++= kkccbbaai smsmsmsmS ......From property relation,

vdPdTcvdPdhTds p −=−=

The entropy of 1 kg of kth gas at T and P is,

kkpk CPRTcsk

+−= lnlnConst. of integration

∴ ( )∑ +−= kkpki CPRTcmSk

lnln

After the partitions are removed, the gases diffuse into oneanother at the same temperature (T) and pressure (P).According to Gibbs-Dalton law, the final entropy of themixture is the sum of the partial entropies, with each gasexerting its respective partial pressure. Thus

Entropy Change in Mixing of Gases

( )∑ +−= kkkpkf CPRTcmSk

lnln

Partial pressure of kth

gas in volume V at T

∴ ( ) ∑∑∑ −=−=−−=− kkkk

kkkkkif yRmPPRmPPRmSS lnlnlnln

⇒ ∑∑∑ −=−=−=−=∆ kkukkukkkif yynRynRyRmSSS lnlnln

50658252 Thermodynamics Ppt Notes

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