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Mathematical Analysis

The document discusses thermodynamics, focusing on the principles of heat exchange, work done in systems, and the laws governing thermal processes. It covers various thermodynamic processes such as adiabatic and isothermal processes, detailing how internal energy, pressure, volume, and temperature interact. Key equations and concepts are presented to illustrate the relationships between these variables in different thermodynamic scenarios.

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Devotuli Kar
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© © All Rights Reserved
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0% found this document useful (0 votes)
18 views23 pages

Mathematical Analysis

The document discusses thermodynamics, focusing on the principles of heat exchange, work done in systems, and the laws governing thermal processes. It covers various thermodynamic processes such as adiabatic and isothermal processes, detailing how internal energy, pressure, volume, and temperature interact. Key equations and concepts are presented to illustrate the relationships between these variables in different thermodynamic scenarios.

Uploaded by

Devotuli Kar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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T

Ihermodymamie
dias
ha bhaneh &1 phuy tek whu
bettween heot auud
Wi tte Hrlaitnkhip
WoHK Lalled Tetm0 &uraTu e

ypesrbyktaun
Oepu bqktasm, maH aud nengy Canbe
exchouNged beween Swale
ondSykem
(2)Lod y t : Oly enuay L a n be.
exhaagd bitwean- Swta

ND caML be iKchANAe

SyMem

Clodsyte 1oLakd_yhre
Dokymamie Shol
Tha dqinite Valueh b PrassuHe (P), Nolumel
CLnd Termpettatute(T) a qar at Ony hme
epeceids the ThertmodynamitStafa

Ctate Natu able


Whiem datemime tha
h vai able
unmo dynarmi behavidwL O O Lytem

a caldd stae varuiablut


and
y hi ytean ik a q a , ttlen P,V

To
VOuabler. NO matkeM h0 W a q a heA che
a pakteula equ ibuum Srat

lyper b1 theHmodynamuPhoteM 6

ULsttherrmal prote% T khe temporahwta


y t a m d o e WE
Choung e dwung
0

a prlo w * t e n
doer mtt ohahge uumq
t p HoCLM iN
LoUed *ArObau PHO Cess,
P
ochorie pHocese T, ti Volume dotk
h
UEhange dwumq a pOUM hhenm
PMOes iLalUed TyoUhoric poceMr.
4) Adiabahe procul 3 Tmo haat entem
OM aveN e Syttem ung daapMocess
t e eeDHto ess r Lallad AolkbahtepLe Less.

Isobau

PA
iGtunmal
Adiabahe

onK dAne by h e ga

t o n e by q a = F = PA
PA

dw- Fx da =
pA d
Total nOHKdbne=|dw d

NoHkdne by g a = tVe
PAdL
totkoleby
dv NoM Kdne tn qa -

VL
NonK dtneby qrtaph
7o Pv grLOLph , woHkddnis AHaa UmdeL
Vo lum aUr

T //
vo
W= tV

/
V
W:+L W -V
volummnd

NG-n-Lyeie and yoie pHoteSs

Syem (9ar) doen m &E MetwML to imihal

rade i.Lfrna glate yytem


ir di4terunt
HOm th iniHal tate.

Cg.o) ewTn toi4


inuhial shote
Da-L

WONK odHne P-
in.A4alc PHOCz

M N

NoMK 4oMpHouss ABL =


+Arua ABcNMA
NotK ort p CDA =-
A Hea tNMAD
Torad WorK Ata ABCNMA +(Aruaa
N MA D
AMua b Loop ADeDA
WoMK dOne im y cuc pHOLESS
1M- Cloak dise
and Anhi elbeKwis diteshio
n

Anh-ooeKwise= -V
&Fimd NoHKdbne pHaALWra,P = 2_V MOmm

Volume. Vto vN
Ca)v- )-v, e) y-vd) yv
Sol.

N- Pdw

2N dv

V
pHsswup-T*: tind
4
Valu
WoK d d n i o n No luhe,Vtov

)-n VI -h*k*E-
(d)Nonebz thuse
C)-v
T-Vnk7 TR
V N2 N
P-4
2t1 V,
V
-2t1 V,

nRvV
n'

nR CAws
V

PrusswuP= T*. Fimol nieltK Arne


0TL Tevp .T, to T

(amRT2 b)nk[T,- I,)-nK (T-T.)


(anRT,
( ) N.O.T
W P dv
PN RT

T nR
n R

r.nR dT
T
T
ON n )
nRdT
dv= KAT
T T
-nR[--{11)
oZeno Law df hetmo dymoamie
Wo booiek on yeen thak a in dheHmal
Lg ui Lib ttium wi th a i n d body ara 1m
hettmal cquuubtuium nilh eath' bhert.

mtMnal tnvg
Sul b IaKinehe and potnkia

Called t s 1mteermo u e q

FiLst La 4 Thstnodunamic
h h a enerLg uppued to a Sytem
i equal to nimutease m dh ndermal
nengy AU) b i Syster pluur H
WonRdeue by ha Syr

I WoHKdonby qak =+ve


( WoHK dbnR-0mga -Ve

(im) tleat addrd to HSyshem Ave

(iv) Heat ost by Sytem=-Ve

4U+ (v) AU = t l e , 1nCHAaR 1M

mhuHnal e n 4 t

Vi U, =-V2 outraNe
Da-

P
TIPplicalion binst Lar "lhenmboumamie
O Kelaion betwLLn Cp%Cy
CusiduHL mouy O1 a m idual ak. Suppo
n
+hat qas ir hmated od Lon
tant vo ume o
a t t h empeHot ww imHalefs by A

= C,AT
.OLum 1x L8nkrant, W= O

AJeto pinst Laww

7D CAT =4 0 + 0

7Cy 40

uppohe mow molek B a UuhLaled


at owtant pHussut So that fr

ncruoM by Samu amend 4T


emp.

Hu PA V
' PN nRT
.PAV nRAT
AU W

= 4U+ nkaAT

ALDO = mcpAT 47

4U+ nKAT
4T mAT

AU
-+ RAT AU
naT naT

Cy +R
pCytR

-CyR
BoiLing
n phouss Suppo a iguid O
T Contau
m aMs m V a p e w i s e at
Th
Prustwe p. TtN Volume in Ihe iquid
i Vy
Shake ir 1n Nap6-wL Sate
.

Vzaed
nthaexpas ion
Lek w ind h e WotKdbe.
0tetnal ueHL
Cund tthe Unong i n k e
VL
Pdv Pdv
Let, pecifre atur hut 6 Vapeurisationi

= mLy

AUt w
mLy 4U+ P(V,- Va)
AU =

my P-Y)
Melhinq prous
Nhen o Solid change into iquid shake,
th tennal eLHgy murLaseh. thitr l a n
be talulated juomfinst lau terromodyna-

Spcii latuk heat bffuion


S0, Hhat Hguidott melhn pHoteM
mL
mce dwunq mlHna pMo ces, thu haunge
im Youme Av) i Very m a l ansl a n
be Contid1ted a Zerto .
',NE PAV Px0 ="0
PT.O
Ae to fiHst Law,
AUt h

mL = AV+ O

AU = mL|

)IsstiH TnaM prLOtesS A TeHmodurmam


Proca W i h takerplau at com Faut

.
t. l y Pand y buntE T.

Le-wrhoi2KT Cwrront
PV MRT = =

P wranPV, - Pv]1

T
T
P
Haar keserWo ift

toHL iohe tmalpro uSs

T h yun du Sh&ud haw Conduchngwalb


iThL qa hdiuld be Compussed oH
al&wed to expand owl.
NOHK d0ne owlimq Sote Hmal pHLoLeS

NeKno0n tha
= nRI dv
V

RT
P nRT P =n
N
nRT.dv = T AogeV
N

= nRT Loge
VI

hkTelogn N

TLog.
l we ue tu on mula 4otL 9m than

w2.903RTLogo VI

qaN COmhan oM gm,


M
NN 2 303 VT Loq L
P.T:O
Fort Sattinnalprotes&

PVfV
T
V P
N 2-303 m RT Aog VI
=
2303 nRT og (P

NOTE FAiAotetma eypancio eP


Cemkau gatwitt hawe to-Supplied

Ci For iothanmal expausien (temp. Lenslaid)|


qa n amtut hia eg wal to OHK
d e n b y h a qa Wu haNe to Supplied
pplied

m t extLmmal SduM Leim 0HdeLto


main tain cons han temp. otal
eXpand S0, W t V

0twN

w
W tV hen = + V
( Fon ind het1mal ompHssi0N a,
anmbuut bj heal eaual t o WoK doas

COmp HSSdU W -N

AU+ hN

W WW- VeHeat 4low Dut)


m intthermal pro cuss , tenpetatwa

CAhangin intermal energa (AU0)


= AU + w

0+ w
= 2303 nRTADg
V

2.30 3 nkT Lo9


AdiabatpHote 9S

+oKele pl in Such a

a a v e dh iyAde
hoatr enten/ O

OurumgtheP HoLe &S 1 PV and ohange


Pt
Tsothema

Co

, adiabarie
ISulahnG >
Nall

toLAdiabahapHOLess,

hawe imSulahina
( i )T yundet ShBuld
Walu.

hould be Lompre &Sed


o
Thu g a
Very quiok.
albwed +o expand

Applicahen b finsd aur to adiabalie pH0teL


heat exdhange
We KmnN ha-
nttan
i n adiabahu DHOUSL.

AUE W
= AU W

A U -W

W= tve, Fom
AU= -Ve
0 11 duniakeA )
(epaiwiom .e.
AU=+Ve
-Ve dta N

(CorpreSsiom) (i.in Ciean)


P-1
E D{ nn adiabalt PHOtesS
Ne hawe ntted abbve that im an adi abai
PHOeM wnlike i bHenmal case, T wiu varu
A WekA a p A V. TheettHA adiabahte

PHOes S oHA idial qa i not diserubed


1m he terim dz PV=nRT alome becausc al
thrue skate voasuablkr P anAT) chounge
duouing dipMOtLIS.oind hen tthes
chengtr a r ralaked,
We ne ed a Second Lguahb1- .

PV,' =
Pv Herte-,P
P omskant|
PV XN = Llant
7
0msta
- COnrmE
T
Y-
Tx )
T, v, V- T, V
Fruom),
T RT Y LomshnF
P
T (nR) T
PY-I
Comerant
TxT C0wr han
P

p-

Y-
P = 0ran

WOHK b u dwung AdiabakiePHOusS


V Y

P. d =
|Kvdv KV

V
-Y+
FoH adi albaki PHOusS,

PV K Cwrtant -
P = k K K - 7
1-Y

-
.
K
1-T
Sy - -
= Y-I
-
KV, KV,X V

(P-P,V,)
W (P,P,V.)
W
-
RTRT
MM

W (T-T2
FoL 1 qttamL
L
M

PV nRT
PV M
RTK
M

NOTE PHHOLLSS PV=


Ldnrant=>p =ConVsfan
aolabah L PHo L s ,
Fot
P N = Lbmst

CUrra
P
>adi
P olecreaserfask aNV

imCrtearer im a m aiabak4

SS
i o t e H mol PHDLLS
6ttenmad v Aduabalt pROCe
T P,

IO
adi
Adi

V
V

9thAHnal Pr0tus AdiabaiL PHO


eonPXakwta_Hemaim No uat added on

Ctsreun (AT=0 ). Hamoved (A=0)

ITh hanaar ocur )The hanger 0lu


Suddeny
4iTue Syeon i )The syktem- i
+uMm ally conduwtra thetmalli nrulotad
+o h Surdtndimg. 4H0mHes uULDUndin

iv No chang in t e ) Tun ix hang


nteHnal eteMgu
Sytemi
Specte haatbecbm eh)Spetipiu hut become
Z O
Atanptmt 0nadiabki
TRNmal t n v , Cepe in CAULNe slope i mote
s s han the SUope 0FLAnL StOpe IS0RurmmalCwwa
A oliocbaic Cwwe.
RENERSI BLE IRREVERSIBLE PROCES

I.Kewem SblLpHoeeseAny PHOCLS Which Lan


bemadr to pHoteed in evense dination ho
ttar i Pamek thrnaugh exacty +h Same
tate ar im R di tett pHOLest ln- a l
aN pectr ir Callo a TveHSi b l pro tess.

P CondAt 8- 04 HwLHSibla
Proce ee
Th pHouss Should dake
Pla at eytruny ste
Yate.
)Th Syktem shauld
b i Hom Hiton

2r veHSibla pHotes
A PHouss that canntt- be maol +o pHo Ceed
i t h HWLHse dine chíon I CaA
irowsesibl pHo cesg. Tn a iroteveHsiLl
Los amd cammot e rteCoveMed back
EKomptl, i) DiKolvimg tLuga[Salt im Watu
th A iUCVenSiLteprroLess.
P-10

Limd eridu 81 4iHst lau o thutm0dymam ich


U T inst Lou doek mot imdieara te dirucion
trtair e b4 hiat.
teM anyhung
C)Te 4iust law doek ntt
utt comdithonr under wich 'hea
Canb LOnveHted into WotLK.

h
The 4inst Lau doe not telu Why
Cemnot be Convemted
Whole ozhcaE eLHay
9mto N0HK LOn Hni@- ul

Matwallyfotm a b hof
teat4tw
Cold body; "heat Nil neveH t
bodut0 a

a cold body +o a hot body


ithe o oH
ENTROPY
ik a measwt 07 t t ditoHdu
Entopy
A yhoteo.
P2-12
nthopPY ir a Measw &1 hdur mueh emergy
ON hea i
um-avauLalbleot Lom veHSi8imto
Ueul woHK when a Syrtem at Kebvin
ttimmpeHakwu. uoliqolh a HveHSible
PROtesS by abSotbimg ou ambunt ot heat
Its imcHLON 1HOPYL AS.

AS=S>

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