Download Byeong-Joo Lee

The First Law
Byeong-Joo Lee
POSTECH - MSE
[email protected]
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Various Forms of Work
0. Hydrostatic system
PdV
1. Surface film
SdA
2. Stretched wire
FdL
3. Reversible cell
εdZ
4. Dielectric slab
EdΠ
5. Paramagnetic rod
μoHdM
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Is Heat an Energy?
▷ Count Rumford (1798): heat produced during boring of cannon was roughly
(Benjamin Thompson) proportional to the work performed during the boring
▷ Humphrey Davy (1799): End of Caloric Theory
← Melting of two blocks of ice by rubbing them in vacuum
▷ Mayer, Helmholtz 등 에너지 보존 법칙의 가능성을 언급
▷ James Joule observed: (1840 ∼)
A direct proportionality existed between the work done and the resultant
temperature rise. The same proportionality existed no matter what means
were employed in the work production
· Rotating a paddle wheel immersed in the water
· A current through a coil immersed in the water
· Compressing a cylinder of gas immersed in the water
· Rubbing together two metal blocks immersed in the water
※ Mechanical equivalent of heat with unit calorie
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - First Law
“The change of a body inside an adiabatic enclosure from a
given initial state to a given final state involves the same
amount of work by whatever means the process is carried
out”
 It was necessary to define some function which depends only on the
internal state of a body or system
– Internal Energy.
 For adiabatic process:
UB – UA = -w
 Generally:
UB – UA = q - w
dU = δq - δw
 dU  0
: as a state function
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Special processes
Absolute value of U is not known: Necessity of Special Paths
1. Constant-Volume Process:
ΔU ＝ qv
2. Constant-Pressure Process:
ΔH ＝ qp
,
⇒ concept of heat capacity:
C
q
,
T
C
q
dT
3. Reversible Adiabatic Process: q ＝ 0
4. Reversible Isothermal Process: ΔU ＝ ΔH ＝ 0
※ Importance of the identification of state functions
→ justification of the analysis of unrealistic reversible processes
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Some issues
 q   dU 
Cv  
 

dT
dT

v 

or dU = Cv dT
 q 
 dH 
Cp  




dT
dT

p 

or dH = Cp dT
 V 
C p  Cv  P

 T  P
 U 

 0
 V T
 V  
 U  
C p  Cv  
 P  
 
 T  P 
 V T 
or
or
 U 

 0
 V  T
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Special Processes
Reversible Adiabatic Process: q ＝ 0
dU  W
cv dT   PdV
for ideal gas
RTdV
c v dT  
V
 T2

 T1
  V1
  
  V2



R / Cv
 V1
 
 V2



 1
 P2V2
 
 P1V1



PV  = constant
Reversible Isothermal Process: ΔU ＝ ΔH ＝ 0
 V2 
 P1 


w  q  RT ln    RT ln  
 V1 
 P2 
Byeong-Joo Lee
http://cmse.postech.ac.kr
First Law of thermodynamics - Numerical Example
Byeong-Joo Lee
http://cmse.postech.ac.kr
The Second
Law
Byeong-Joo Lee
POSTECH - MSE
[email protected]
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Introduction
Spontaneous (or Natural or Irreversible) Process
▷ mixing of two gases
▷ Equalization of temperature
▷ A + B = C + D : criterion for equilibrium?
Entropy as a measure of the degree of irreversibility
▷ Lewis and Randall’s Consideration: A weight-pulley-heat_reservoir
▷ q/T = △S
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Reversible vs. Irreversible
△S = measurable quantity + un-measurable quantity
=
q/T
+
△Sirr
=
qrev/T
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Evaluation of Entropy Change
▷ Reversible Isothermal Compression of an Ideal Gas
q rev  wmax
 VB 
  P dV  RT ln  
VA
 VA 
VB
▷ Reversible Adiabatic Expansion of an Ideal Gas
Isentropic process: ΔU = -w
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Engines and Referigerators
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Historical Background
▷ Carnot, 1824 - 열기관의 효율은 이를 구성하는 두 온도만의 함수.
(caloric 이론에 의거)
▷ Joule, 1847 - 에너지는 보존되고, 여러 형태가 서로 변환이 가능함을
실험적으로 제시 → Mayer, Helmholtz 등의 에너지보존법칙에 final touch.
▷ Thomson - Carnot와 Joule 사이에 모순이 있음을 지적
▷ Clausius, 1850 - Joule을 인정하면서 Carnot의 원리 증명.
같은 일을 하면서 더 적은 열을 흡수(q2’)하고 방출(q1’)하는 엔진과
정상적인 Heat Pump를 결합, q2 - q2’ = q1 – q1’.
열이 낮은 온도에서 높은 온도로 흐르지 않는다.
따라서 Carnot의 원리는 성립한다.
▷ Thomson, 1851 - Carnot의 원리 증명
열을 흡수해서 모두 일로 바꾸는 것이 불가능
같은 열을 흡수하면서 더 많은 일과(w’) 더 적은 열을 방출(q1’)하는
엔진과 정상적인 Heat Pump를 결합, w’- w = q1 – q1’
열을 100% 일로 바꿀 수는 없다. 따라서, Carnot의 원리는 성립한다.
▷ Thomson, 1852 - 현재 물질 세계에는 역학적 에너지의 낭비를 향한
일반적 경향이 존재한다.
▷ Clausius, 1865 - 우주의 에너지는 일정하다. 우주의 엔트로피는 항상 증가한다.
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Thermodynamic Temperature Scale
Concept of Absolute Temperature
▷ The maximum efficiency is independent of the working substance
and is a function only of the working temperatures, t1 and t2.
q1
F (t1 ) T1
 f (t1 , t 2 ) 

q2
F (t 2 ) T2
Kelvin Scale (Absolute Thermodynamic Temperature Scale, K)
q1 T1

q2 T2
→
q
T  273.16
qTP
0K is the temperature of the cold reservoir which makes the efficiency
Of a Carnot cycle equal to unity
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Equivalence of temperature scales
Equivalence of Kelvin Scale and Ideal Gas Temperature Scale
w T2  T1

q2
T2
▷ Efficiency of Carnot Cycle:
▷ Carnot cycle이 두 개의 reversible isothermal process와 두 개의 reversible
adiabatic process로 이루어졌다고 가정하고 ideal gas temperature scale에
기초하여 효율을 계산하면 (T2-T1)/T2라는 같은 결과나 나온다.
V
V 
T2 ln  B   T1 ln  D
VA 
VC
w



q2
 VB 
T2 ln  
 VA 
VB VC

V A VD



q2  q1 T2  T1

q2
T2
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Entropy as a State Function
For a Carnot Cycle
q2  q1 T2  T1

q2
T2

q 2 q1
 0
T2 T1
For an arbitrary Cyclic process which can be broken into a large number
of small Carnot cycle.

※ dS 
q
T
q
T
0


dS  0
로 정의되는 entropy S는 state function이고 adiabatic system에서
감소할 수 없다.
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Entropy and Irreversibility
▷ Processes exhibiting Mechanical Irreversibility
Coming to rest of a rotating or vibrating liquid in contact
with a reservoir
Ideal gas rushing into a vacuum
▷ Processes exhibiting Thermal Irreversibility
Conduction or radiation of heat from hotter to cooler system/reservoir
▷ Processes exhibiting Chemical Irreversibility
Mixing of two dissimilar inert ideal gases
(※ example: k ln Ω, ln x! = x ln x – x )
Freezing of supercooled liquid
(※ example: freezing of supercooled Pb)
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Maximum Work
UB U A  q  w
q  dU system  w
dS system 
dS system 
q
T
 dS irr
dU system  w
T
 dS irr
w  TdS system  dU system  TdS irr
w  wmax  TS system  U system
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Entropy as a Criterion of Equilibrium
※ for an isolated system of constant U and constant V,
(adiabatically contained system of constant volume)
equilibrium is attained when the entropy of the system is maximum.
※ for a closed system which does no work other than work of
volume expansion,
dU = T dS – P dV (valid for reversible process)
U is thus the natural choice of dependent variable for S and V
as the independent variables.
※ for a system of constant entropy and volume, equilibrium is attained
when the internal energy is minimized.
w  TdS system  dU system  TdS irr
PdV  TdS system  dU system  TdS irr
0  dU system  TdS irr
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics - Condition for Thermodynamic Equilibrium
※ Further development of Classical Thermodynamics results from the fact
that S and V are an inconvenient pair of independent variables.
+ need to include composition variables in any equation of state and
in any criterion of equilibrium
+ need to deal with non P-V work
(e.g., electric work performed by a galvanic cell)
※ Condition for Thermodynamic Equilibrium of a Unary two phase system
dSisolated_ system  dS   dS 
1    P P          
 1
     dU      dV      dn
T 
T 
T 
T
T
T
The same conclusion is obtained using minimum internal energy criterion.
Byeong-Joo Lee
http://cmse.postech.ac.kr
Second Law of thermodynamics – Numerical Example
Byeong-Joo Lee
http://cmse.postech.ac.kr