Lecture 33 – Finish chapter 9 and...
...The Third Law of Thermodynamics
Chapter 10 – Monday November 19th
Chapter 9
•Review of entropy of mixing
•Phase equilibrium
Chapter 10
•Statements of the Third Law
•Equivalence of statements
Reading:
Chapter 10 (pages 169 - 180)
Homework 9 is due on Mon. Nov. 26th
I will assign homework 10 on Mon. 26th
Entropy of mixing
final state f
initial state i
n1
T, P
n2
T, P
n = n1 + n2
T, P = P1 + P2
Definition of partial pressure Pj of a constituent in a mixture:
Pj = x j P.
Here, P is the pressure of the mixture, and xj is the kilomole fraction
of the jth constituent gas:
n
n
xj =
j
∑n
i
i
=
j
n
Entropy of mixing
final state f
initial state i
n2
T, P
n1
T, P
n = n1 + n2
T, P = P1 + P2
P is the pressure of the mixture; Pj is the partial pressure and xj the
kilomole fraction of the jth constituent gas, where
Pj = x j P
Then,
and
xj =
nj
∑n
i
i
ΔS = − nR [ x1 ln x1 + x2 ln x2 ]
=
nj
n
Entropy of mixing
0.8
xlnx-[(1-x)ln(1-x)]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.3
x
0.6
0.9
Statements of the third law
•The third law of thermodynamics is concerned with the
behavior of systems in equilibrium as temperature
approaches absolute zero.
S=∫
T
đQ
+ S0
T
•As long as we deal in differences in entropy, knowledge of
S0 is unnecessary.
•However, absolute entropy IS important, e.g.
dG = −SdT + PdV
U = ST – PV + μn
Statements of the third law
•The Gibbs-Helmholtz equation:
⎛ ∂G ⎞
G = H +T ⎜
⎟
∂
T
⎝
⎠P
•For an isothermal process:
⎛ ∂ ( ΔG ) ⎞
ΔG = ΔH + T ⎜
⎟
∂
T
⎝
⎠P
•These equations imply:
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•Nernst postulated the following:
⎡ ∂ ( ΔG ) ⎤
lim ⎢
= 0,
⎥
T →0
⎣ ∂T ⎦ P
⎡ ∂ ( ΔH ) ⎤
lim ⎢
=0
⎥
T →0
⎣ ∂T ⎦ P
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•Nernst postulate implies that:
⎡ ∂ ( G2 − G1 ) ⎤
⎡ ∂G2 ∂G1 ⎤
lim ⎢
= lim ⎢
−
= lim [ S1 − S2 ] = 0
⎥
⎥
T →0
∂T
∂T ⎦ T →0
⎣
⎦ P T →0 ⎣ ∂T
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•The Nernst formulation of the Third Law:
‘All reactions in a liquid or solid in thermal equilibrium
take place with no change of entropy in the neighborhood
of absolute zero.’
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•Planck later postulated that:
⎛ ∂G ⎞
lim ⎜
= 0,
⎟
T →0 ⎝ ∂T ⎠
P
⎛ ∂H ⎞
lim ⎜
=0
⎟
T →0 ⎝ ∂T ⎠
P
⇒
lim S = 0
T →0
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•Planck’s statement of the Third Law:
‘The entropy of a true equilibrium state of a system at
absolute zero is zero.’
Statements of the third law
lim G = lim H
T →0
T →0
lim ΔG = lim ΔH
T →0
T →0
•Another statement of the Third Law is:
‘It is impossible to reduce the temperature of a system to
absolute zero using a finite number of processes.’
Adiabatic cooling
S
B
B1
M
B2 > B1
T3 T2
T1
T
dU = TdS – PdV = TdS – MdB = đQ – MdB
Adiabatic cooling
S
B1
M
B2 > B1
T3 T2
T1
T
dU = TdS – PdV = TdS – MdB = đQ – MdB
Equivalence of the 3rd law statements
S
B1
B2 > B1
0 T3
T2
T1
T
dU = TdS – PdV = TdS – MdB = đQ – MdB
Ice violates the Planck 3rd Law