01-Thermodynamic Process (Theory)

Notes by 

Notes by  : : Pradeep Pradeep KshetrapalKshetrapal

genius

genius

φ  φ  P P V  V  dx  dx  F=PA F=PA

2

2 Thermodynamic P Thermodynamic Processesrocesses

13.1 Introduction 13.1 Introduction..

(1)

(1) Thermodynamics :Thermodynamics : It is a ranch o! science "hich deals "ith e#change o! heat energyIt is a ranch o! science "hich deals "ith e#change o! heat energy et"een odies and con$ersion o!

et"een odies and con$ersion o! the heat energy into mechanical energy and $ice%$ersa.the heat energy into mechanical energy and $ice%$ersa.

((&&)) TheThermrmodyodynanamimic c syssystem tem :: ' collection o! an e#tremely large numer o! atoms or' collection o! an e#tremely large numer o! atoms or mole

moleculcules es conconnened d "it"ith h in in cercertaitain n ououndandaries such ries such thathat t it it has a has a cercertaitain n $al$alue ue o! o! prpressessururee $o

$olulume me anand d tetempmpererataturure e is is cacalllled ed a a ttheherrmomodydynanamimic c ssysysttemem. . 'n'nyyththining g ooututssidide e ththee thermodynamic system to "hich energy or matter is e#changed is called i

thermodynamic system to "hich energy or matter is e#changed is called i ts surroundings.ts surroundings. Example

Example : *as : *as encloenclosed in sed in a cylinder tted "ith a a cylinder tted "ith a pistopiston !orms the n !orms the therthermodynmodynamic systemamic system ut the atmospheric air around the cylinder

ut the atmospheric air around the cylinder mo$ale piston urnermo$ale piston urner etcetc. are all . are all the surroundings.the surroundings.  Thermodynamic system may e

 Thermodynamic system may e o! three typeso! three types

(i) +pen system : It e#change oth energy and matter "ith the

(i) +pen system : It e#change oth energy and matter "ith the surrounding.surrounding. (ii) Closed system : It e#change only energy (not matter) "ith the

(ii) Closed system : It e#change only energy (not matter) "ith the surroundings.surroundings. (iii) Isolated system : It e#change neither energy nor matter "

(iii) Isolated system : It e#change neither energy nor matter "ith the surrounding.ith the surrounding. (,)

(,) ThermodynaThermodynamic variables and equation of mic variables and equation of state :state : ' thermodynamic system can e' thermodynamic system can e descried y speci!ying its pressure $olume temperature internal energy and the numer o!  descried y speci!ying its pressure $olume temperature internal energy and the numer o!  mo

molesles. . ThThesese e papararamemeteters rs arare e cacallelled d ththerermomodydynanamimic c $a$ariarialleses. . ThThe e rerelalatition on eet"t"een een ththee thermodynamic $ariales (

thermodynamic $ariales (PP V V  T T ) o! the system is called e-uation o! state.) o! the system is called e-uation o! state. or

or  µ  µ  moles o! an ideal gas e-uation o! state is moles o! an ideal gas e-uation o! state is PV PV  / /  µ  µ RT RT  and !or 1 mole o! an it ideal gas is and !or 1 mole o! an it ideal gas is PV 

PV  / / RT RT  or

or µ  µ  moles o! a real gas e-uation o! state is moles o! a real gas e-uation o! state is V V  bb RT RT 

V  V  a a P P  µ  µ 

_ 

_ 

 

 

−−

 µ  µ 

==

µ µ 

 

 

 

 



 

 

 

 

++

(( )) & & & &

 and !or 1 mole

 and !or 1 mole o! a real gas it iso! a real gas it is

RT  RT  b b V  V  V  V  a a P P

_ 

_ 

−−

==

 

 

 

 



 

 

 

  ++

_&& (( ))

((00)) TherThermodymodynamnamic ic equiequilibrilibrium um :: hen the thermodynamic $ariales attain a steadyhen the thermodynamic $ariales attain a steady $alue

$alue i.e.i.e. they are independent o! time the system is said to e in the state o! thermodynamic they are independent o! time the system is said to e in the state o! thermodynamic e-uilirium. or a system to e in thermodynamic e-uilirium the !ollo"ing conditions must e e-uilirium. or a system to e in thermodynamic e-uilirium the !ollo"ing conditions must e !ullled.

!ullled.

(i) 2echa

(i) 2echanicnical al e-ue-uiliiliriurium m : : TheThere re is is no no ununalaalancenced d !or!orce ce etet"ee"een n the systthe system em and itsand its surroundings.

surroundings.

(ii) Thermal e-uilirium : There is a uni!orm temperature in all parts o! the system and is same as (ii) Thermal e-uilirium : There is a uni!orm temperature in all parts o! the system and is same as that o!

that o! surrousurrounding.nding.

(iii) Chemical e-uilirium : There i

(iii) Chemical e-uilirium : There is a uni!orm chemical composition through out the system ands a uni!orm chemical composition through out the system and the surrounding.

13.1 Introduction 13.1 Introduction..

(1)

(1) Thermodynamics :Thermodynamics : It is a ranch o! science "hich deals "ith e#change o! heat energyIt is a ranch o! science "hich deals "ith e#change o! heat energy et"een odies and con$ersion o!

et"een odies and con$ersion o! the heat energy into mechanical energy and $ice%$ersa.the heat energy into mechanical energy and $ice%$ersa.

((&&)) TheThermrmodyodynanamimic c syssystem tem :: ' collection o! an e#tremely large numer o! atoms or' collection o! an e#tremely large numer o! atoms or mole

moleculcules es conconnened d "it"ith h in in cercertaitain n ououndandaries such ries such thathat t it it has a has a cercertaitain n $al$alue ue o! o! prpressessururee $o

$olulume me anand d tetempmpererataturure e is is cacalllled ed a a ttheherrmomodydynanamimic c ssysysttemem. . 'n'nyyththining g ooututssidide e ththee thermodynamic system to "hich energy or matter is e#changed is called i

thermodynamic system to "hich energy or matter is e#changed is called i ts surroundings.ts surroundings. Example

Example : *as : *as encloenclosed in sed in a cylinder tted "ith a a cylinder tted "ith a pistopiston !orms the n !orms the therthermodynmodynamic systemamic system ut the atmospheric air around the cylinder

ut the atmospheric air around the cylinder mo$ale piston urnermo$ale piston urner etcetc. are all . are all the surroundings.the surroundings.  Thermodynamic system may e

 Thermodynamic system may e o! three typeso! three types

(i) +pen system : It e#change oth energy and matter "ith the

(i) +pen system : It e#change oth energy and matter "ith the surrounding.surrounding. (ii) Closed system : It e#change only energy (not matter) "ith the

(ii) Closed system : It e#change only energy (not matter) "ith the surroundings.surroundings. (iii) Isolated system : It e#change neither energy nor matter "

(iii) Isolated system : It e#change neither energy nor matter "ith the surrounding.ith the surrounding. (,)

(,) ThermodynaThermodynamic variables and equation of mic variables and equation of state :state : ' thermodynamic system can e' thermodynamic system can e descried y speci!ying its pressure $olume temperature internal energy and the numer o!  descried y speci!ying its pressure $olume temperature internal energy and the numer o!  mo

molesles. . ThThesese e papararamemeteters rs arare e cacallelled d ththerermomodydynanamimic c $a$ariarialleses. . ThThe e rerelalatition on eet"t"een een ththee thermodynamic $ariales (

thermodynamic $ariales (PP V V  T T ) o! the system is called e-uation o! state.) o! the system is called e-uation o! state. or

or  µ  µ  moles o! an ideal gas e-uation o! state is moles o! an ideal gas e-uation o! state is PV PV  / /  µ  µ RT RT  and !or 1 mole o! an it ideal gas is and !or 1 mole o! an it ideal gas is PV 

PV  / / RT RT  or

or µ  µ  moles o! a real gas e-uation o! state is moles o! a real gas e-uation o! state is V V  bb RT RT 

V  V  a a P P  µ  µ 

_ 

_ 

 

 

−−

 µ  µ 

==

µ µ 

 

 

 

 



 

 

 

 

++

(( )) & & & &

 and !or 1 mole

 and !or 1 mole o! a real gas it iso! a real gas it is

RT  RT  b b V  V  V  V  a a P P

_ 

_ 

−−

==

 

 

 

 



 

 

 

  ++

_&& (( ))

((00)) TherThermodymodynamnamic ic equiequilibrilibrium um :: hen the thermodynamic $ariales attain a steadyhen the thermodynamic $ariales attain a steady $alue

$alue i.e.i.e. they are independent o! time the system is said to e in the state o! thermodynamic they are independent o! time the system is said to e in the state o! thermodynamic e-uilirium. or a system to e in thermodynamic e-uilirium the !ollo"ing conditions must e e-uilirium. or a system to e in thermodynamic e-uilirium the !ollo"ing conditions must e !ullled.

!ullled.

(i) 2echa

(i) 2echanicnical al e-ue-uiliiliriurium m : : TheThere re is is no no ununalaalancenced d !or!orce ce etet"ee"een n the systthe system em and itsand its surroundings.

surroundings.

(ii) Thermal e-uilirium : There is a uni!orm temperature in all parts o! the system and is same as (ii) Thermal e-uilirium : There is a uni!orm temperature in all parts o! the system and is same as that o!

that o! surrousurrounding.nding.

(iii) Chemical e-uilirium : There i

(iii) Chemical e-uilirium : There is a uni!orm chemical composition through out the system ands a uni!orm chemical composition through out the system and the surrounding.

 Thermodynam

 Thermodynamic Processesic Processes 33 (3)

(3) ThermodynaThermodynamic process mic process :: The process o! change o! state o! a system in$ol$es change The process o! change o! state o! a system in$ol$es change o! thermodynamic $ariales such as pressure

o! thermodynamic $ariales such as pressure PP $olume $olume V V  and temperature and temperature T T  o! the system. The o! the system. The process is 4no"n as

process is 4no"n as thermodynamic process. Some important processes arethermodynamic process. Some important processes are (i)

(i) IsotIsothermahermal l proprocess cess (ii) (ii) 'dia'diaatic atic proprocess cess (iii) (iii) IsoIsoaric aric procprocess ess (i$) (i$) IsocIsochorichoric (iso$olumic process)

(iso$olumic process) ($) Cy

($) Cyclic anclic and non%cd non%cyclic pyclic procrocess ess ($i) 5($i) 5e$erse$ersile and ile and irrirre$ersie$ersile prle processocess  13.2 Zeroth Law of Thermodynamics

 13.2 Zeroth Law of Thermodynamics..

I! systems

I! systems A A and and BB are each in thermal e-uilirium "ith a third system are each in thermal e-uilirium "ith a third system CC thenthen A A and and BB are in are in thermal e-uilirium "ith

thermal e-uilirium "ith each othereach other..

(1) The 6eroth la" leads to the concept o! temperature. 'll odies in thermal e-uilirium (1) The 6eroth la" leads to the concept o! temperature. 'll odies in thermal e-uilirium must ha$e a common property "hich has the same $alue !or all o! them. This property is called must ha$e a common property "hich has the same $alue !or all o! them. This property is called the temperature.

the temperature.

(&) The 6eroth la" came to light long a!ter the rst and seconds la"s o! thermodynamics (&) The 6eroth la" came to light long a!ter the rst and seconds la"s o! thermodynamics had een disco$ered and numered. 7ecause the concept o! temperature is !undamental to had een disco$ered and numered. 7ecause the concept o! temperature is !undamental to those t"o la"s the la" that estalishes temperature as a $alid concept should ha$e the lo"est those t"o la"s the la" that estalishes temperature as a $alid concept should ha$e the lo"est numer. Hence it is called 6eroth la".

numer. Hence it is called 6eroth la".

 13.3 uantities Involved in !irst Law of Thermodynamics  13.3 uantities Involved in !irst Law of Thermodynamics..

(1)

(1) "eat #"eat # QQ$ $ :: It is the energy that is trans!erred et"een a system and its en$ironmentIt is the energy that is trans!erred et"een a system and its en$ironment ecause o! the temperature di8erence et"een them. Heat al"ays 9o" !rom a ody at higher ecause o! the temperature di8erence et"een them. Heat al"ays 9o" !rom a ody at higher temperature to a ody at lo"er temperature till their temperatures ecome e-ual.

temperature to a ody at lo"er temperature till their temperatures ecome e-ual. Important points

Important points (i) Heat is a !orm

(i) Heat is a !orm o! energy so it is a o! energy so it is a scalar -uantity "ith dimensionscalar -uantity "ith dimension ;;MLML&&T T −−&&::.. (ii) <nit :

(ii) <nit : Joule Joule (S.I.)(S.I.) CaloieCaloie (practical (practical unit) unit) and and 11 caloiecaloie / 0.& / 0.& Joule Joule (iii) Heat is a

(iii) Heat is a path dependent -uantitypath dependent -uantity e.!.e.!. Heat re-uired to change the temperature o! a gi$enHeat re-uired to change the temperature o! a gi$en gas at a constant pressure is di8erent !rom that re-uired to change the temperature o! same gas gas at a constant pressure is di8erent !rom that re-uired to change the temperature o! same gas through same amount at constant $olume.

through same amount at constant $olume. (i$) or solids and

(i$) or solids and li-uids :li-uids : ∆∆_{"" / / mLmL ;or ;or change change in in state state andand} ∆∆_{"" / / mcmc}∆∆_{TT ;or change in;or change in}

temperature temperature

or gases "hen heat is asored and temperature changes : or gases "hen heat is asored and temperature changes :

T  T  C C " " _V V 

==

_V V 

∆

∆

∆

∆

)) µ µ 

(( ;or constant ;or constant $olume $olume andand ((

∆

∆

""))_PP

==

µ µ CC_PP

∆

∆

T T  ;or ;or constantconstant pressure

pressure (&)

(&) %or& #%or& # W W $ :$ : oor4 can e r4 can e dened as the energy that is trans!erred !rom one ody dened as the energy that is trans!erred !rom one ody to theto the other o"ing to a !orce that acts et"een them

other o"ing to a !orce that acts et"een them I!

I! PP  e the pressure o! the gas in the cylinder then !orce  e the pressure o! the gas in the cylinder then !orce e#erted y the gas on the piston o! the

e#erted y the gas on the piston o! the cylindercylinder F F  / / PAPA In a

In a small displsmall displacement o! piston thracement o! piston throughough dx dx  "or4 done y "or4 done y the gas the gas dV  dV  P P dx  dx  P PAA dx  dx  F  F  d#  d# == .. == == ∴

∴ _{Total amount o! "or4 done Total amount o! "or4 done} V V  (( _{$ $  ii})) Vi Vi PPdV dV  PP V V  V V  d#  d#  #  # 

=

=

=

=

$ $ 

=

=

−

−

∆

∆

∫ ∫ 

∫ ∫ 

dx  dx  F=PA F=PA

Important points

(i) =i4e heat "or4 is also a path dependent scalar physical -uantity "ith dimension

: ;ML&T −&

(ii) rom

∆

# 

=

P

∆

V 

=

P(V _$ 

−

V _i)

∆_{#  / positi$e i!} V $ 

>

V i i.e. system e#pands against some e#ternal !orce.

∆_{#  / negati$e i!} V $ 

<

V i i.e. system contracts ecause o! some e#ternal !orce e#erted

y the surrounding.

(iii) In P%V  diagram or indicator diagram the area under P%V  cur$e represents "or4 done. #  / area under P%V  diagram

It is positi$e i! $olume increases (!or e#pansion) It is negati$e i! $olume decreases (!or compression)

(i$) In a cyclic process "or4 done is e-ual to the area under the cycle. It is positi$e i! the cycle is cloc4"ise.

It is negati$e i! the cycle is anticloc4"ise.

($)

=

∫ 

$ 

i

V 

V  PdV 

# 

rom this e-uation it seems as i! "or4 done can e calculated only "hen P%V  e-uation is 4no"n and limits V   andi V $  are 4no"n to us. 7ut it is not so. e can calculate "or4 done i! "e

4no" the limits o! temperature.

or e#ample the temperature o! & moles o! an ideal gas is increased !rom T > to &T >

through a process

T 

P

=

α   and "e are interested in nding the "or4 done y the gas. Then

PV  / &RT  (ideal gas e-uation) ?..(i)

and

T 

P

=

α  ?..(ii)

@i$iding (i) y (ii) "e get

α  & &RT  V 

 =

 or dV  &RT dT  α  &

=

∴ _> & & & > > &RT  dT  &RT  T  dV  P #  T  T  V  V  $  i

=

 

 

 



 

 

 

 

 



 

 

=

=

∫ 

∫ 

α  α 

So "e ha$e !ound the "or4 done "ithout putting the limits o! $olume.

($i) I! mass less piston is attached to a spring o! !orce constant '  and a mass m is placed o$er the piston. I! the e#ternal pressure is P  and due to e#pansion o! gas the piston mo$es up>

through a distance x  then

 Total "or4 done y the gas # 

=

# 1

+

# &

+

# ,

P  A V  Positi$ e "or4 A#pansio n B _P  A V  Begati$ e "or4 Compressio n B  x  M M P P V  Positi$e "or4 P₁ P & B C (  A V ₁ V _& Cloc4"ise cyclic process P V  Begati$e "or4 P₁ P & B C (  A V ₁ V _& 'nticloc4"ise cyclic process

 Thermodynamic Processes( "here # 1 / or4 done against e#ternal pressure (P>)

# & / or4 done against spring !orce ('x )

# , / or4 done against gra$itational !orce (m!)

∴ _# 

=

_P>V 

+

_'x &

+

_m!

& 1

($ii) I! the gas e#pands in such a "ay that other side o! the piston is $acuum then "or4 done y the gas "ill e 6ero

's # 

=

P

∆

V 

=

> ;Here P / >

(,) Internal ener)y #U$ : Internal energy o! a system is the energy possessed y the system due to molecular motion and molecular conguration.

 The energy due to molecular motion is called internal 4inetic energy )'   and that due to

molecular conguration is called internal potential energy )P.

i.e. Total internal energy )

=

)' 

+

)P

(i) or an ideal gas as there is no molecular attraction ) _p =>

i.e. internal energy o! an ideal gas is totally 4inetic and is gi$en y ) )_'  µ RT  & ,

=

=

and change in internal energy

∆

)

=

µ R

∆

T  &

,

(ii) In case o! gases "hate$er e the process

T  R $  )

=

∆

∆

&  µ 

=

µ C_V 

∆

T  1 ) ( ) 1 (

−

−

=

∆

−

=

γ    µ  γ    µ  R T  RT $  T i 1

−

−

=

γ    µ   µ RT _$  RT _i 1 ) ( − − = γ   i i $  $ V  PV  P

(iii) Change in internal energy does not depends on the path o! the process. So it is called a point !unction i.e. it depends only on the initial and nal states o! the system i.e.

∆

)

=

)$ 

−

)i

(i$) Change in internal energy in a cyclic process is al"ays 6ero as !or cyclic process )$ 

=

)i

So ∆) =)_$  −)_i =>

Sample problems based on Q* U and  W 

Problem 1. ' thermodynamic system is ta4en through the cycle P"R*P process. The net "or4 done y the system is

+,rissa - 2//20 (a) &> J ()  &> J Dacuum *as R 1>> 'p " P * &>> 'p 1>> cc ,>> cc P V 

(c) 0>> J (d)  ,E0 J

*olutio& : () or4 done y the system / 'rea o! shaded portion on P%V  diagram

  J &> 1> ) 1> &>> ( 1> ) 1>> ,>> ( − F × − × , = = −

and direction o! process is anticloc4"ise so "or4 done "ill e negati$e i.e. ∆_{# / }

&> J.

Problem 2. 'n ideal gas is ta4en around ABCA as sho"n in the ao$e P%V  diagram. The "or4 done during a cycle is

+T #n)).45ed.$ 2//10 (a) &PV  () PV  (c) 1G&PV  (d) ero

*olutio& : (a) or4 done / 'rea enclosed y triangle  ABC

PV  P P V  V  BC  AC (, ) (, ) & & 1 & 1

=

−

×

−

×

=

×

=

Problem 3.  The P%V  diagram sho"s se$en cur$ed paths (connected y $ertical paths) that can e !ollo"ed y a gas. hich t"o o! them should e parts o! a closed cycle i! the net "or4 done y the gas is to e at its ma#imum $alue

+657 #n)).$ 2///0 (a) ac

() c! (c) a$  (d) cd

*olutio& : (c) 'rea enclosed et"een a  and $  is ma#imum. So "or4 done in closed cycles !ollo"s a and $  is ma#imum.

Problem '. I! C₊ 

=

0.FcalGmole'  then increase in internal energy "hen temperature o! & moles o! this gas is increased !rom ,0> '  to ,0& ' 

+89T 1;0

(a) &E.J> cal () 1.J0 cal (c) 1,.> cal (d) .& cal

*olutio& : () Increase in internal energy

cal T 

C

)

=

. ₊ .

∆

=

&

×

0.F

×

(,0&

−

,0>)

=

&

×

0.F

×

&

=

1.J0

∆

µ 

Problem (. 'n ideal gas o! mass m in a state  A  goes to another state B  $ia three di8erent processes as sho"n in gure. I! "1"&  and ", denote the heat asored y the

gas along the three paths then +59 9T 120

(a) "1

<

"&

<

", () "1

<

"&

=

", B C (P, ,V ) P E ( V   A (P,V ) (,P, ,V ) V  P b d $  c e ! a V  1 & , B  A P

 Thermodynamic Processes; (c) "1

=

"&

>

",

(d) "1

>

"&

>

",

*olutio& : (a) 'rea enclosed y cur$e 1  'rea enclosed y cur$e &  'rea enclosed y cur$e ,

∴ "₁

<

"_&

<

"_{, ;'s} ∆_{) is same !or all cur$es}

Problem .  The relation et"een the internal energy ) and adiaatic constant γ   is

(a) = ₋₁ γ   PV  ) _() 1 − = γ   γ   PV  ) _(c) γ   PV  ) = _(d) PV  )

=

γ  

*olutio& : (a) Change in internal energy /

∆

)

=

µ c₊ 

∆

T  ⇒ )_&

−

)₁

=

µ c₊ (T _&

−

T ₁) =et initially T ₁

=

> _so )₁

 =

> and nally T _&

=

T  _and )_&

=

)

+  + T  T  c c )

=

 µ 

=

µ 

×

1 1 = − − × = γ   γ   PV  R R PV    ;'s PV = µ RT  ∴ R PV  T 

=

 µ  and 1 − = γ   R c_{+  }  13.' -oule<s Law.

hene$er heat is con$erted into mechanical "or4 or mechanical "or4 is con$erted into heat then the ratio o! "or4 done to heat produced al"ays remains constant.

i.e. # ∝ _" or   J

" # 

=

 This is LouleMs la" and J is called mechanical e-ui$alent o! heat. Important points

(1) rom #  / J" i! " / 1 then J / # . Hence the amount o! "or4 done necessary to produce unit amount o! heat is dened as the mechanical e-ui$alent o! heat.

(&) J is neither a constant nor a physical -uantity rather it is a con$ersion !actor "hich used to con$ert Joule or e! into caloie or -ilo caloies $ice%$ersa.

(,) Dalue o! J e -ilocaloi  J caloie e! caloie  J E , 1> & . 0 1> & . 0 & . 0

=

×

=

×

=

.

(0) hen "ater in a stream !alls !rom height  then its potential energy is con$erted into heat and temperature o! "ater rises slightly.

rom #  / J"

m! / J ms ∆_{t  ;"here m / 2ass s / Specic heat o! "ater}

∴_{5ise in temperature C}

  Js !. t 

=

°

∆

(3) The 4inetic energy o! a ullet red !rom a gun gets con$erted into heat on stri4ing the target. 7y this heat the temperature o! ullet increases y ∆_t .

rom #  / J" t  s m   J m+ &

=

∆

& 1

 ;"here m / 2ass +  / Delocity o! the ullet s / Specic heat o! the ullet

∴ _{5ise in temperature C}   Js +  t 

=

°

∆

& &

I! the temperature o! ullet rises upto the melting point o! the ullet and ullet melts then. rom #  / J" : ; & 1 & mL t  ms  J

m+ 

=

∆

+

  ;"here L / =atent heat o! ullet

∴ _{5ise in temperature} L s C  J +  t 

°





























−

=

∆

& &

(F) I! ice%loc4 !alls do"n through some height and melts partially then its potential energy gets con$erted into heat o! melting.

rom #  / J"

L m   J

m!.= N   _{;"here m / mass o! ice loc4 m}1  _{/ mass "hich}

melts

So -!

  JL m!. mN

=

I! ice%loc4 completely melts do"n then m! / J mL

∴ _{Height re-uired !or complete melting} mete !

  JL

.

=

Sample problems based on Joule's law

Problem ;. ater !alls !rom a height o! &1> m. 'ssuming "hole o! energy due to !all is con$erted into heat the rise in temperature o! "ater "ould e ( J / 0., JouleGcal)

+9b. 95T 2//20

(a) 0&OC () 0OC (c) >.0OC (d) 0.OC

*olutio& : (c) =oss in potential energy o! "ater / Increment in thermal energy o! "ater

⇒ m!.=  J ×ms∆t  ⇒ _.J

×

_&1>

=

  _0.,

×

_1>>>

∆

_t  ∴

_∆

_t 

₌

_>.0I

_°

_C

Problem =. ' loc4 o! mass 1>> !m slides on a rough hori6ontal sur!ace. I! the speed o! the loc4 decreases !rom 1> mGs to 3 mGs the thermal energy de$eloped in the process

is +79>6T 2//20

(a) ,.E3 J () ,E.3 J (c) >.,E3 J (d) >.E3 J

*olutio& : (a) Thermal energy de$eloped / =oss in 4inetic energy

 J +  +  m >.1 (1> 3 ) ,.E3 & 1 ) ( &

1 & & &

1 &

&

−

=

×

×

−

=

=

Problem .  The "eight o! a person is F> -!. I! he gets 1>3 _{caloies heat through !ood and the}

eciency o! his ody is &JQ then upto ho" much height he can clim

(appro#imately) +6!5 1;0

(a) 1>> m () &>> m (c) 0>> m (d) 1>>> m *olutio& : () Increment in potential energy o! mass / &JQ o! heat gained

⇒

 

 

 



 

 

_×

=

_1>3 1>> &J  J m!. ⇒

 

 

 



 

 

_×

×

=

×

×

_1>3 1>> &J & . 0 J .  F> . ⇒ .= &>>m

Problem 1/. Hailstone at >OC !alls !rom a height o! 1 -m on an insulating sur!ace con$erting "hole o! its 4inetic energy into heat. hat part o! it "ill melt (!=1>mGs&)

 Thermodynamic Processes (a) ,, 1 () J 1 (c) 1>0 ,, 1 ₋

×

(d) 'll o! it "ill melt *olutio& : (a) Anergy re-uired !or melting / =oss in potential energy ⇒  _J

×

_m

′

_L

=

_m!

⇒ 0.1J×(mN×J>×1>−,)= m×1>×1>>>⇒ ,, 1 N

=

m m   ;'s L / J> × _1>, caloie/-!

Problem 11. ' ullet mo$ing "ith a uni!orm $elocity +  stops suddenly a!ter hitting the target and the "hole mass melts e m specic heat * initial temperature &3OC melting point 0E3OC and the latent heat L. Then

(a)  J m+  ms mL & ) &3 0E3 ( &

+

−

=

()  J m+  mL ms & ) &3 0E3 ( &

=

+

−

(c)  J m+  mL ms & ) &3 0E3 (

−

+

=

(d)  J m+  mL ms & ) &3 0E3 ( &

=

−

−

*olutio& : () K.A. o! ullet / Heat re-uired to raise the temperature o! ullet !rom &3OC to 0E3OC R heat re-uired to melt the ullet

⇒ _{; (0E3 &3) :} & 1 _& mL ms  J m+ 

=

−

+

⇒  J m+  mL ms & ) &3 0E3 ( &

=

+

−

.

Problem 12. ' lead ullet at &EOC ust melts "hen stopped y an ostacle. 'ssuming that &3Q o! heat is asored y the ostacle then the $elocity o! the ullet at the time o! stri4ing (2.P. o! lead / ,&EO C specic heat o! lead / >.>, calG!mOC latent heat o! !usion o! lead / F calG!m and J / 0.& JGcal) +IIT?- 1=10

(a) 01> mGsec () 1&,> mGsec (c) ,>E.3 mGsec (d) Bone o! these

*olutio& : (a) <sing e#pression otained in prolem (11) "e get : &E ,&E ( ; & 1 Q E3 m+ &

 

=

 J ms

−

+

mL

 

 



 

 

Sustituting s= >.,×1>, calG-!oC _and L = F×1>,calG-! _{"e get} + = 01>mGs

Problem 13. ' drilling machine o! po"er P "atts is used to drill a hole in Cu loc4 o! mass m

-!. I! the specic heat o! Cu is 1 1

3 − −

°C

-!

  J  and 0>Q o! po"er lost due to heating

o! machine the rise in temperature o! the loc4 in T sec ("ill e in OC) (a) ms PT  F . > () msT  P F . > (c) ms PT  0 . > (d) msT  P 0 . >

*olutio& : (a) 's "e 4no"

) (  Time ) ( or4 ) ( Po"er T  #  P

=

∴ _{#  / P} × _T 

's 0>Q energy is lost due to heating o! machine so only F>Q energy "ill increase the temperature o! the loc4

∴ _{F>Q o! #  / m} × _s × ∆_t ⇒ ms PT  ms #  t 

=

>.F

=

>.F

∆

.

 13.( !irst Law of Thermodynamics.

'ccording to it heat gi$en to a system (∆_{") is e-ual to the sum o! increase in its internal}

energy (∆_{)) and the "or4 done (}∆_{# ) y the system against the surroundings.}

#  )

"

=

∆

+

∆

∆

Important points

(1) It ma4es no distinction et"een "or4 and heat as according to it the internal energy (and hence temperature) o! a system may e increased either y adding heat to it or doing "or4 on it or oth.

(&) ∆_" and ∆_{#  are the path !unctions ut} ∆_{) is the point !unction.}

(,) In the ao$e e-uation all three -uantities ∆_" ∆_{)  and} ∆_{#  must e e#pressed either in}

 Joule or in caloie.

(0) Lust as 6eroth la" o! thermodynamics introduces the concept o! temperature the rst la" introduces the concept o! internal energy.

(3) Sign con$entions

∆_" Positi$e hen heat is supplied to a system

Begati$e hen heat is dra"n !rom the system

∆_#  Positi$e hen "or4 done y the gas (e#pansion)

Begati$e hen "or4 done on the gas (compression)

∆₎

Positi$e hen temperature increases internal energy increases

Begati$e hen temperature decreases internal energy decreases

(F) hen a thermos ottle is $igorously sha4en :

Bo heat is trans!erred to the co8ee ∆_{" / > ;'s thermos 9as4 is insulated !rom}

the surrounding

or4 is done on the co8ee against $iscous !orce ∆_{#  / ()}

Internal energy o! the co8ee increases ∆_{) / (R)}

and temperature o! the co8ee also increases ∆_{T  / (R)}

(E) =imitation : irst la" o! thermodynamics does not indicate the direction o! heat trans!er. It does not tell anything aout the conditions under "hich heat can e trans!ormed into "or4 and also it does not indicate as to "hy the "hole o! heat energy cannot e con$erted into mechanical "or4 continuously.

Sample problems based on First law of thermodynamics

Problem 1'. I! 13> J o! heat is added to a system and the "or4 done y the system is 11>  J then change in internal energy "ill e

+657 #n)).$ 1@ A"7 2///0

(a) &F> J () 13> J (c) 11> J (d) 0> J

 Thermodynamic Processes11 Problem 1(. 11> J o! heat is added to a gaseous system "hose internal energy change is 0>

 J then the amount o! e#ternal "or4 done is

+A> 95T 13@ 6!5 1@ -I958 2///0

(a) 13> J () E> J (c) 11> J (d) 0> J

*olutio& : () ∆" = ∆) +∆#  ⇒ _11>

=

 _0>

+

∆

_#  ⇒ ∆# = E>  J

Problem 1. hen an ideal diatomic gas is heated at constant pressure the !raction o! the heat energy supplied "hich increases the internal energy o! the gas is

+IIT?- 1/@ 89T 2///0 (a) 3 & () 3 , (c) E , (d) E 3 *olutio& : (d) E 3 3 G E 1 1 ) 1 G( ) 1 G(

=

=

=

−

−

=

∆

∆

=

∆

∆

γ   γ   γ   γ   R R T  C T  C " ) P + 

Problem 1;. 'n electric !an is s"itched on in a closed room. The air in the room is +59 9T 10

(a) Cooled () Heated

(c) 2aintains its temperature

(d) Heated or cooled depending on the atmospheric pressure

*olutio& : () hen an electric !an is s"itched on in a closed room con$entional current o! air 9o"s. Hence due to $iscous !orce mechanical energy is con$erted into heat and some heat is also produced due to thermal e8ect o! electric current in motor o! !an. Problem 1=. ' gas is compressed at a constant pressure o! _{3> G} &

m

N  _{!rom a $olume o!} 1>m,

to a $olume o! _0m,

. Anergy o! 1>> J  is then added to the gas y heating. Its

internal energy is +5B8 1'0

(a) Increased y 0>> J () Increased y &>> J (c) Increased y 1>> J (d) @ecreased y &>> J

*olutio& : (a) ∆" = ∆) +∆#  ⇒ ∆" = ∆) + ∆dV ⇒ _1>>

=

∆

₎

+

_3>(0

−

_1>) ⇒ _1>>

=

∆

₎

−

_,>> ∴

  J

) = 0>>

∆

Problem 1. ' thermodynamic process is sho"n in the gure. The pressures and $olumes corresponding to some points in the gure are : P _A

=

,

×

1>0PaP_B

=

J

×

1>0Pa and

, , , , _{ 3 1>} 1> & m V  m V  _A

=

×

− ₍

=

×

−

In process AB F>> J o! heat is added to the system and in process BC &>> J o! heat is added to the system. The change in internal energy o! the system in process  AC

"ould e +A> 95T 120

(a) 3F> J () J>> J (c) F>> J (d) F0> J V  P (  A B C 0

*olutio& : (a) 7y adoining graph #  _AB

=

> _and # _BC

=

J

×

1>0;3

−

&:

×

1>−,

=

&0> J

∴ #  _AC

=

#  _AB

+

# _BC

=

>

+

&0>

=

&0> J

Bo"

∆

" _AC

=

∆

" _AB

+

∆

"_BC

=

F>>

+

&>>

=

J>> J

rom rst la" o! thermodynamics

∆

" AC

=

∆

) AC

+

∆

#  AC ⇒ J>>

=

∆

)AC

+

&0> ⇒ .

3F> J ) _AC

=

∆

Problem 2/. I! R / uni$ersal gas constant the amount o! heat needed to raise the temperature o! & mole o! an ideal monoatomic gas !rom &E, '  to ,E, '  "hen no

"or4 is done +59 9T 1/0

(a) 1>> R () 13> R (c) ,>> R (d) 3>> R

*olutio& : (c) ∆" = ∆) ;,E, &E,:

1 , 3 & 1

 

 

_ 

∆

=

×

₋

−

 

 

 

−

=

∆

=

C₊  T  R T  R γ    µ   µ  / ,>> R ;'s !or monoatomic gas , 3

=

γ     13. Isothermal 9rocess.

hen a thermodynamic system undergoes a physical change in such a "ay that its temperature remains constant then the change is 4no"n as isothermal changes.

In this process P and V  change ut T  / constant i.e. change in temperature ∆_{T  / >}

(1) ssential condition for isothermal process

(i) The "alls o! the container must e per!ectly conducting to allo" !ree e#change o! heat et"een the gas and its surrounding.

(ii) The process o! compression or e#pansion should e so slo" so as to pro$ide time !or the e#change o! heat.

Since these t"o conditions are not !ully realised in practice there!ore no process is per!ectly isothermal.

(&) quation of state : rom ideal gas e-uation PV  /  µ RT 

I! temperature remains constant then PV / constant i.e. in all isothermal process 7oyleMs la" is oeyed.

Hence e-uation o! state is PV  / constant. (,) Cample of isothermal process

(i) 2elting process ;Ice melts at constant temperature >O C

(ii) 7oiling process ;"ater oils at constant temperature 1>>O C. (0) Indicator dia)ram *as Conductin g "alls T _, T _& T ₁ T ₁T _& T _, P V  P V  θ  P V  or4

 Thermodynamic Processes13

(i) Cur$es otained on PV  graph are called isotherms and they are hyperolic in nature. (ii) Slope o! isothermal cur$e : 7y di8erentiating PV  / constant. e get

> = +V dP dV  P ⇒ P dV = −V dP ⇒ V  P dV  dP

−

=

∴ V  P dV  dP

−

=

=

θ  tan

(iii) 'rea et"een the isotherm and $olume a#is represents the "or4 done in isothermal process.

I! $olume increases ∆_{#  / R 'rea under cur$e and i! $olume decreases} ∆_{#  /  'rea under}

cur$e

(3) >peciDc heat : Specic heat o! gas during isothermal change is innite.

's

=

∞

×

=

∆

=

> m " T  m " C ;'s ∆_{T  / >}

(F) Isothermal elasticity : or isothermal process PV  / constant

@i8erentiating oth sides PdV 

+

VdP

=

> ⇒ P dV = −V dP ⇒ E_θ 

V  dV  dP P

=

=

−

=

Strain Stress G

∴ _Eθ 

=

_{P i.e. isothermal elasticity is e-ual to pressure}

't B.T.P. isothermal elasticity o! gas / 'tmospheric pressure / 3 & G 1> >1 .

1

×

N m

(E) %or& done in isothermal process

∫ 

∫ 

=

=

$  i $  i V  V  V  V  V  dV  RT  dV  P #  µ  ;'s PV  /  µ RT 

 

 

 

 



 

 

=

 

 

 

 



 

 

=

i $  i $  e V  V  RT  V  V  RT 

#   µ  log &.,>,µ  log_1>

or

_ 

 

 

 





 

 

=

 

 

 

 





 

 

=

$  i $  i e P P RT  P P RT 

#   µ  log &.,>,µ  log_1>

(J) !LTE in isothermal process

#  )

"

=

∆

+

∆

∆

  ut

∆

)

∝

∆

T 

∴

∆

₎

=

_{> ;'s} ∆_{T  / >}

∴

∆

"

=

∆

#  _{i.e.  heat supplied in an isothermal change is used to do "or4 against}

e#ternal surrounding.

or i! the "or4 is done on the system than e-ual amount o! heat energy "ill e lierated y the system.

Sample problems based on Isothermal process

Problem 21. +ne mole o! 0& gas ha$ing a $olume e-ual to &&.0 lites  at >OC  and 1

atmospheric pressure in compressed isothermally so that its $olume reduces to 11.& lites. The "or4 done in this process is +59 9T 13@ AF9 2//30

(a) 1FE&.3 J () 1E&J J (c)  1E&J J (d)  13E&.3 J *olutio& : (d) or4 done in an adiaatic process

 J V  V  RT  #  _e i $  e   J., &E, ( >.F) 13E& 0 . && & . 11 log &E, , . J 1 log

_ 

=

×

×

−

≈

−

 

 



 

 

×

×

×

=

 

 

 

 



 

 

=

 µ 

Problem 22. 'n ideal gas A and a real gas B ha$e their $olumes increased !rom V  to &V  under isothermal conditions. The increase in internal energy

+A> 95T 13@ -I958 2//1* 2//20

(a) ill e same in oth A and B () ill e 6ero in oth the gases (c) +! B "ill e more than that o! A (d) +! A "ill e more than that o! B

*olutio& : (c) In real gases an additional "or4 is also done in e#pansion due to intermolecular attraction.

Problem 23. hich o! the !ollo"ing graphs correctly represents the $ariation o! 

V  dP dV G )G ( − =

β   _{"ith P !or an ideal gas at constant temperature}

+IIT?- #>creenin)$ 2//20

(a) () (c) (d)

*olutio& : (a) or an isothermal process PV  / constant ⇒ _PdV 

+

 _VdP

=

_> ⇒

P dP dV  V  1 1

=

 

 

 



 

 

−

So P 1

=

β  ∴ _{graph "ill e rectangular hyperola.}

Problem 2'. Consider the !ollo"ing statements

6ssertion ( A): The internal energy o! an ideal gas does not change during an isothermal process

8eason #R$ : The decrease in $olume o! a gas is compensated y a corresponding increase in pressure "hen its temperature is held constant.

+! these statements +>86 1'0

(a) 7oth A and R are true and R is a correct e#planation o! A () 7oth A and R are true ut R is not a correct e#planation o! A (c) A is true ut R is !alse

(d) 7oth A and R are !alse (e) A is !alse ut R is true

*olutio& : ()'s ∆₎ ∝ ∆_{T  so !or isothermal process internal energy "ill not change and also}

V  P

∝

1

(7oyleNs la")

Problem 2(. Ho" much energy is asored y 1> -! molecule o! an ideal gas i! it e#pands !rom an initial pressure o! J atm to 0 atm at a constant temperature o! &EOC

+8oor&ee 120

(a) E

1> E&J .

1

×

 J () 1E.&J

×

1>E J (c) 1.E&J

×

1>I J (d) 1E.&J

×

1>I J

*olutio& : (a) or4 done in an isothermal process

P β  β  P β  P β  P

 Thermodynamic Processes1(  J P P RT  #  _e $  i e , 1>0 J., ,>> >.F, 1.E&J 1>E 0 J log ,>> , . J ) 1> 1> ( log

_ 

=

×

×

×

=

×

 

 



 

 

×

×

×

×

=

 

 

 

 





 

 

=

µ 

Problem 2. 3 moles o! an ideal gas undergoes an isothermal process at 3>> '  in "hich its $olume is douled. The "or4 done y the gas system is

(a) ,3>> J () 100>> J (c) 1EJ>> J (d) 3&>> J *olutio& : () log 3 J., 3>> log & 3 J., 3>> >.FI 100>>J.

V  V  V  V  RT  #  _e i $  e

 

=

×

×

×

≈

 

 



 

 

×

×

×

=

   

 

 

 

=

∆

µ 

Problem 2;. or4 done y a system under isothermal change !rom a $olume V 1  to V & !or a

gas "hich oeys Dander aalNs e-uation &RT 

V  & P & V  =               + − ) & ( β  α  (a)

   

 

 

  −

+

   

 

 

 

−

−

& 1 & 1 & 1 & log V  V  V  V  & & V  & V  &RT  _e α  β  β  ()

   

 

 

  −

+

   

 

 

 

−

−

& 1 & 1 & 1 & 1> log V  V  V  V  & V  V  &RT  α  αβ  αβ  (c)

_  

 

 



 

  −

+

  

 

 



 

 

−

−

& 1 & 1 & 1 & log V  V  V  V  & & V  & V  &RT  _e β  α  α  (d)

_  

 

 



 

 

−

+

  

 

 



 

 

−

−

& 1 & 1 & & 1 log V  V  V  V  & & V  & V  &RT  _e α  β  β 

*olutio& : (a) 7y Dander aalMs e-uation _&

& V  & & V  &RT  P α  β 

−

−

=

or4 done

=

∫ 

=

∫ 

₋

−

∫ 

&

1 & 1 & 1 & & V  V  V  V  V  V  _V  dV  & & V  dV  &RT  PdV  #  α  β 

[

]

_ 

 

 

 



 

 

−

+

−

−

=









+

−

=

& 1 & 1 & 1 & & 1 _log ) ( log & 1 & 1 _V V  V  V  & & V  & V  &RT  V  & & V  &RT  _e V  V  V  V  e α  β  β  α  β   13.; 6diabatic 9rocess.

hen a thermodynamic system undergoes a change in such a "ay that no e#change o! heat ta4es place et"een it and the surroundings the process is 4no"n

as adiaatic process.

In this process P V  and T  changes ut ∆_{" / >.}

(1) ssential conditions for adiabatic process

(i) There should not e any e#change o! heat et"een the system and its surroundings. 'll "alls o! the container and the piston must e per!ectly insulating.

(ii) The system should e compressed or allo"ed to e#pand suddenly so that there is no time !or the e#change o! heat et"een the system and its surroundings.

Since these t"o conditions are not !ully realised in practice so no process is per!ectly adiaatic.

(&) Cample of some adiabatic process

(i) Sudden compression or e#pansion o! a gas in a container "ith per!ectly non%conducting "alls.

(ii) Sudden ursting o! the tue o! icycle tyre.

(iii) Propagation o! sound "a$es in air and other gases. (i$) A#pansion o! steam in the cylinder o! steam engine.

*as

Insulating "alls

(,) !LTE in adiabatic process :

∆

"

=

∆

)

+

∆

# 

ut !or adiaatic process

∆

"

=

> ∴

∆

₎

+

∆

_# 

=

_>

I! ∆_{#  / positi$e then} ∆_{) / negati$e so temperature decreases i.e.  adiaatic e#pansion}

produce cooling.

I! ∆_{#  / negati$e then} ∆_{) / positi$e so temperature increases i.e.  adiaatic compression}

produce heating.

(0) quation of state : 's in case o! adiaatic change rst la" o! thermodynamics reduces to >

=

∆

+

∆

) #   i.e. d)

+

d# 

=

> ?.. (i)

7ut as !or an ideal gas d)

=

 µ C_V dT  and d# 

=

PdV 

A-uation (i) ecomes  µ C₊ dT 

+

PdV 

=

> ?.. (ii)

7ut !or a gas as PV  /  µ RT  PdV +V dP= µ RdT  ?.. (iii)

So eliminating dT  et"een e-uation (ii) and (iii) (

+

)

+

PdV 

=

> R dP V  dV  P C₊   µ   µ  or > ) 1 ( ) (

=

+

−

+

dV  P dP V  dV  P γ  



_









−

=

) 1 ( as γ   R C₊  or γ  PdV +V dP= > i.e.

+

=

> P dP V  dV  γ  

hich on integration gi$es C P V  _e e

+

log

=

log γ    i.e. log(PV γ  )

=

C or γ  

₌

_consta PV  ?.. (i$)

A-uation (i$) is called e-uation o! state !or adiaatic change and can also e re%"ritten as

constan 1

₌

− γ   TV  ;as P / ( µ RT GV ) ?.. (i$) and _γ  −1

=

consta γ   P T 









₌

P RT  V  µ  as ?.. ($i) (3) Indicator dia)ram

(i) Cur$e otained on PV graph are called adiaatic cur$e. (ii) Slope o! adiaatic cur$e : rom PV γ  

=

consta

7y di8erentiating "e get dPV γ   + P V γ  −1dV = >

γ  

 

 

 



 

 

−

=

−

=

− V  P V  PV  dV  dP γ   γ   γ   γ   1

∴_{Slope o! adiaatic cur$e}

 

 

 



 

 

−

=

V  P γ   φ  tan

7ut "e also 4no" that slope o! isothermal cur$e

V  P

−

=

φ  tan So 1 ) G ( ) G ( cur$e isothermal o! Slope cur$e adiaatic o! Slope

>

=

=

−

−

=

+   p C C V  P V  P γ   γ   φ  P V 

 Thermodynamic Processes1; (F) >peciDc heat : Specic heat o! a gas during adiaatic change is 6ero

's >

=

>

∆

=

∆

=

T  m T  m " C ;'s " / >

(E) 6diabatic elasticity : or adiaatic process γ  

₌

_consta

PV 

@i8erentiating oth sides d PV γ   + P V γ  −1dV = >

γ   φ  γ   E V  dV  dP P

=

=

−

=

Strain Stress G P E_φ  =γ  

i.e. adiaatic elasticity is γ   times that o! pressure ut "e 4no" isothermal elasticity E_θ 

=

P

So γ   γ   θ  φ 

=

=

=

P P E E elasticity Isothermal elasticity 'diaatic

i.e. the ratio o! t"o elasticities o! gases is e-ual to the ratio o! t"o specic heats. (J) %or& done in adiabatic process

∫ 

=

$  i V  V  PdV  # 

=

∫ 

$  i V  V  _V  dV  '  γ  













 

 

 



 

  =

_γ   V  '  P 's or

















−

−

=

_{−1 −1} : 1 ; 1 γ   γ   γ   _$  V _i '  V  ' 













+

−

=

∫ 

D ( − +1) 's 1 % γ   γ   γ  _dV  V  or ) 1 ( : ; γ  

−

−

=

P$ V $  PiV i : 's ; ' = PV γ   = P$ V $ γ   = PiV iγ   or _{(1 )};T $  T i: R

−

−

=

γ    µ  ;'s P$ V $ 

=

µ RT $  and P_iV _i

=

µ RT _i So ) 1 ( ) ( ) 1 ( : ;

−

−

=

−

−

=

γ    µ  γ   $  i $  $  i iV  P V  RT  T  P

() !ree eCpansion : ree e#pansion is adiaatic process in "hich no "or4 is per!ormed on or y the system. Consider t"o $essels placed in a system "hich is enclosed "ith thermal insulation (asestos%co$ered). +ne $essel contains a gas and the other is e$acuated. The t"o $essels are connected y a stopcoc4. hen suddenly the stopcoc4 is opened the gas rushes into the e$acuated $essel and e#pands !reely. The process is adiaatic as the $essels are placed in thermal insulating system (d" / >) moreo$er the "alls o! the $essel are rigid and hence no e#ternal "or4 is per!ormed (d#  / >).

Bo" according to the rst la" o! thermodynamics d) / >

I! )  andi )$  e the initial and nal internal energies o! the gas

then )$  −)i => ;'s )$ 

=

)i

 Thus the nal and initial energies are e-ual in !ree e#pansion. (1>) >pecial cases of adiabatic process

consta

=

γ   PV  ∴ _γ   V  P

∝

1  consta 1−γ  

₌

γ   PT  ∴

∝

γ  −1 γ   T  P   and TV γ  −1

=

consta ∴ 1 1 −

∝

_γ   V  T  Type of )as _γ   V  P

∝

1 ₋₁

∝

γ   γ   T  P 1 1 −

∝

_γ   V  T  2onoatomic γ   / 3G, , G 3 1 V  P

 ∝

P

 ∝

T 3G& 1_&G, V  T 

 ∝

@iatomic γ   / EG3 3 G E 1 V  P

 ∝

P

 ∝

T EG& 3 G & 1 V  T 

 ∝

Polyatomic γ   / 0G, , G 0 1 V  P

 ∝

P

 ∝

T 0 ₁1_G, V  T 

 ∝

(11) omparison between isothermal and adiabatic process (i) ompression : I! a gas is compressed isothermally and adiaatically !rom $olume V i to V $  then !rom the slope o! the graph

it is clear that graph 1 represents adiaatic process "here as graph & represent isothermal process.

or4 done # adiaatic  # isothermal

inal pressure Padiaatic  Pisothermal

inal temperature T adiaatic  T isothermal

(ii) Cpansion : I! a gas e#pands isothermally and adiaatically !rom $olume V i to V $   then

!rom the slope o! the graph it is clear that graph 1 represent isothermal process graph & represent adiaatic process.

or4 done # isothermal  # adiaatic

inal pressure Pisothermal  Padiaatic

inal temperature T isothermal  T adiaatic

Sample problems based on Adiabatic process

Problem 2=. @uring an adiaatic process the pressure o! a gas is !ound to e proportional to the cue o! its asolute temperature. The ratio C G p C+  !or the gas is

+6I 2//30 (a) & , () , 0 (c) & (d) , 3

*olutio& : (a) *i$en _P

 _∝

_T ,_{. 7ut !or adiaatic process}

1 −

∝

γ   γ   T  P . So γ  −1=, γ   ⇒ & ,

=

γ   ⇒ & ,

=

+   p C C P V _i 1 & V  $  V  P V _$  1 & V _i V 

 Thermodynamic Processes1 Problem 2. 'n ideal gas at &EOC is compressed adiaatically to

&E J

 o! its original $olume. I! 

, 3

=

γ    then the rise in temperature is

+95T 1='@ A> 95T 1@ T 2//3@ 79>6T 2//30

(a) 03> '  () ,E3 '  (c) &&3 '  (d) 0>3 ' 

*olutio& : () or an adiaatic process _TV γ  −1

₌

 constant

∴ 1 1 & & 1 −













=

γ   V  V  T  T  _⇒ '  V  V  T  T  FE3 J &E ,>> J &E ,>> , G & 1 , 3 1 & 1 1 &



=









=









=











=

− − γ   ⇒ '  T = FE3−,>>= ,E3 ∆

Problem 3/. I! γ   / &.3 and $olume is e-ual to 1GJ times to the initial $olume then pressure P ′

is e-ual to (initial pressure / P)

+89T 2//30 (a)

_P

′

=

_P

()

_P

′

=

_&

_P

(c) _P

_′

₌

_P

_×

_(&)13G& _{(d) P}

′

₌

_EP

*olutio& : (c) or an adiaatic process _PV γ  

₌

  constant _⇒

γ  













=

& 1 1 & V  V  P P _⇒

′

₌

_J3G& P P ⇒ & G 13 ) & (

×

=

′

P P

Problem 31. In adiaatic e#pansion o! a gas +59 9T 2//30

(a) Its pressure increases () Its temperature

!alls

(c) Its density decreases (d) Its thermal energy

increases

*olutio& : () ∆" = ∆) +∆# _{ In an adiaatic process} ∆_{" / >} ∴

−

∆

₎

=

∆

_# 

In e#pansion ∆_{#  / positi$e} ∴ ∆_{) / negati$e. Hence internal energy i.e.}

temperature decreases.

Problem 32. P%V  plots !or t"o gases during adiaatic process are sho"n in the gure. Plots 1 and & should correspond respecti$ely to

+IIT?- #>creenin)$ 2//10 (a) 1e and 0&

() 0& and 1e

(c) 1e and A  (d) 0& and N&

*olutio& : ()Slope o! adiaatic cur$e ∝ _γ   ∝

gas o! the 'tomicity

1

. So γ   is in$ersely proportional to atomicity o! the gas.

;'s γ   = 1.FF!or monoatomic gas γ   / 1.0 !or diatomic gas and γ   / 1.,, !or triatomic

non%linear gas.

rom the graph it is clear that slope o! the cur$e 1 is less so this should e adiaatic cur$e !or diatomic gas (i.e. 0&).

Similarly slope o! the cur$e & is more so it should e adiaatic cur$e !or monoatomic gas (i.e. 1e).

P

V   1

Problem 33. ' monoatomic ideal gas initially at temperature T 1 is enclosed in a cylinder

tted "ith a !rictionless piston. The gas is allo"ed to e#pand adiaatically to a temperature T & y releasing the piston suddenly. I! L1 and L& are the lengths o! 

the gas column e!ore and a!ter e#pansion respecti$ely then T 1G T & is gi$en y

+IIT?- #>creenin)$ 2///0 (a) , G & & 1

 

 

 

 



 

 

L L () & 1 L L (c) 1 & L L (d) , G & 1 &

 

 

 

 



 

 

L L

*olutio& : (d) or an adiaatic process & & 1 1 1 1 − − ₌ γ   γ   _T V  V  T  ⇒ , G & 1 & 1 , G 3 1 & 1 1 & & 1













=













=













=

− − L L  A L  A L V  V  T  T  γ   . Problem 3'. our cur$es A B C and ( are dra"n in the adoining gure !or a gi$en amount o! 

gas. The cur$es "hich represent adiaatic and isothermal changes are +95T 1=@ 79>6T 10

(a) C and ( respecti$ely () ( and C respecti$ely (c) A and B respecti$ely (d) B and A respecti$ely

*olutio& : (c) 's "e 4no" that slope o! isothermal and adiaatic cur$es are al"ays negati$e and slope o! adiaatic cur$e is al"ays greater than that o! isothermal cur$e so is the gi$en graph cur$e  A  and cur$e B represents adiaatic and isothermal changes respecti$ely.

Problem 3(. ' thermally insulated container is di$ided into t"o parts y a screen. In one part the pressure and temperature are P and T  !or an ideal gas lled. In the second part it is $acuum. I! no" a small hole is created in the screen then the temperature o! 

the gas "ill +89T 10

(a) @ecrease () Increase (c) 5emain same (d) Bone o! these

*olutio& : (c) In second part there is a $acuum i.e. P / >. So "or4 done in e#pansion / P∆_V  /

>

Problem 3.  T"o samples A and B o! a gas initially at the same pressure and temperature are compressed !rom $olume V  to V G& ( A isothermally and B adiaatically). The nal

pressure o! A is +59 9T 1* @ 59 95T 1;* 0

(a) *reater than the nal pressure o! B () A-ual to the nal pressure o! B (c) =ess than the nal pressure o! B (d) T"ice the nal pressure o! B *olutio& : (c) or isothermal process

& N & 1 V  P V 

P

=

⇒ _P&N

=

_{&P1 ?..(i)}

or adiaatic process

γ   γ   _           = & & 1 V  P V 

P ⇒ _P&

₌

_{& P}γ   _{1 ?..(ii)}

Since γ  >1_{. There!ore P&}

>

_P&N

V  P B  A C (

 Thermodynamic Processes21 Problem 3;. ' gas has pressure P and $olume V . It is no" compressed adiaatically to

,& 1

times the original $olume. I! (,&)1.0

=

1&J the nal pressure is

(a) ,& P () 1&J P (c)

1&J P

(d) ,&

P

*olutio& : () or an adiaatic process γ   γ  

& & 1 1V  P V  P

=

⇒ γ  

   

 

 

 

=

& 1 1 & V  V  P P 1.0 ,& G  _         = V  V  1&J ) ,& ( 1.0

=

=

∴ _{inal pressure / 1&J P.}

 13.= Isobaric 9rocess.

hen a thermodynamic system undergoes a physical change in such a "ay that its pressure remains constant then the change is 4no"n as isoaric process.

In this process V  and T  changes ut P remains constant. Hence CharleMs la" is oeyed in this process.

(1) quation of state : rom ideal gas e-uation PV  /  µ RT  I! pressure remains constant V ∝ _{T  or}

consta & & 1 1

₌

₌

T  V  T  V 

(&) Indicator dia)ram : *raph I represent isoaric e#pansion graph II represent isoaric compression.

Slope o! indicator diagram

=

> dV  dP

(,) >peciDc heat : Specic heat o! gas during isoaric process C_P $ 

_ 

R

 

 



 

  +

=

1 & (0) Aul& modulus of elasticity : >

G

=

∆

−

∆

=

V  V  P '  ;'s ∆_{P / >}

(3) %or& done in isobaric process : V  ; _{$  i}:

V  V  V  PdV  P dV  P V  V  #  $  i $  i

−

=

=

=

∆

∫ 

∫ 

;'s P / constant ∴

∆

# 

=

P(V _$ 

−

V _i)

=

 µ R;T _$ 

−

T _i:

=

µ R

∆

T 

(F) !LTE in isobaric process :

∆

)

=

µ C_V 

 ∆

T  R

∆

T 

−

=

) 1 (γ    µ    _and

∆

# 

=

µ R

∆

T  rom =T@ ∆" = ∆)+ ∆#  ∴ " R T  R T  R T  R T  _  R∆T           − = − ∆ =       + − ∆ = ∆ + ∆ − = ∆ 1 1 1 1 1 ) 1 ( γ   γ    µ  γ   γ    µ  γ    µ   µ  γ    µ  T  C "

=

_P

 ∆

∆

µ 

(E) Camples of isobaric process

(i) Con$ersion o! "ater into $apour phase (oiling process) :

hen "ater gets con$erted into $apour phase then its $olume increases. Hence some part o! asored heat is used up to increase the $olume against e#ternal pressure and remaining

P V  P₁ P_&  A I B C II (