20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #9
page 1
Gibbs Free Energy, Multicomponent Systems,
Partial Molar Quantities, and the Chemical
Potential
Comments on
the special role of G(T,p):
If you know G(T,p), you know all other thermodynamic quantitites.
⎛ ∂G ⎞
⎟ ,
⎝ ∂T ⎠ p
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
S = −⎜
V =⎜
⎛ ∂G ⎞
⎟
⎝ ∂T ⎠ p
H = G +TS
⇒
H = G −T ⎜
U = H − pV
⇒
U = G −T ⎜
A = U −TS
⇒
A =G − p⎜
∂S ⎞
C p =T ⎛⎜
⎟
⎝ ∂T ⎠ p
⇒
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟
⎟ − p⎜
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
⎛ ∂2G ⎞
C p = −T ⎜
2 ⎟
⎝ ∂T ⎠ p
We can get all the thermodynamic functions from G(T,p).
•
p-dependence of G(T,p)
From
⇒
for liquids, solids, and gases (ideal)
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
V =⎜
p2
G (T , p2 ) = G (T , p1 ) + ∫ Vdp
p1
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
•
Lecture #9
Liquids and solids
V is small
⇒
G (T , p2 ) = G (T , p1 ) +V ( p2 − p1 ) ≈ G (T , p1 )
•
page 2
⇒
G (T
)
Ideal gases
G (T , p2 ) = G (T , p1 ) + ∫
p2
p1
p
RT
dp = G (T , p1 ) + RT ln 2
p
p1
Take p1 = p o = 1 bar
G (T , p ) = G o (T ) + RT ln
From
p
p0
⎛ ∂G ⎞
⎟
⎝ ∂T ⎠ p
S = −⎜
or G (T , p ) = G o (T ) + RT ln p
(p in bar)
⇒
S (T , p ) = S o (T ) − R ln p
Multicomponent systems, the chemical equilibrium, partial molar
quantitites.
So far we’ve worked with fundamental equations for a closed (no
mass change) system with no composition change.
dU =TdS − pdV
dA = −SdT − pdV
dH =TdS +Vdp
dG = −SdT +Vdp
How does this change if we allow the composition of the system to
change? Like in a chemical reaction or a biochemical process?
• Consider Gibbs free energy of a 2-component system
G (T , p ,n1 ,n2 )
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
⇒
Lecture #9
page 3
⎛ ∂G ⎞
⎛ ∂G ⎞
⎛ ∂G ⎞
⎛ ∂G ⎞
dG = ⎜
dT + ⎜
dp + ⎜
dn1 + ⎜
dn2
⎟
⎟
⎟
⎟
∂p ⎠T ,n ,n
∂n1 ⎠T , p ,n
∂n2 ⎠T , p ,n
⎝ ∂T ⎠ p ,n ,n
⎝
⎝
⎝
1
2
−S
We define
2
1
2
µ1
V
⎛ ∂G ⎞
⎟
⎝ ∂ni ⎠T , p ,nj ≠i
µi ≡ ⎜
1
µ2
as the chemical potential of species “i”
µi (T , p , nj ) is an intensive variable
This gives a new set of fundamental equations for open systems
(mass can flow in and out, composition can change)
dG = −SdT +Vdp + ∑ µi dni
i
dH =TdS +Vdp + ∑ µi dni
i
dU =TdS − pdV + ∑ µi dni
i
dA = −SdT − pdV + ∑ µi dni
i
⎛ ∂G ⎞
⎛ ∂H ⎞
⎛ ∂U ⎞
⎛ ∂A ⎞
=⎜
=⎜
=⎜
⎟
⎟
⎟
⎟
⎝ ∂ni ⎠T , p ,nj ≠i ⎝ ∂ni ⎠S , p ,nj ≠i ⎝ ∂ni ⎠S ,V ,nj ≠i ⎝ ∂ni ⎠T ,V ,nj ≠i
µi = ⎜
• At equilibrium, the chemical potential of a species is the same
everywhere in the system
Let’s show this in a system that has one component and two parts,
(for example a solid and a liquid phase, or for the case of a cell
placed in salt water, the water in the cell versus the water out of
the cell in the salt water)
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #9
page 4
Consider moving an infinitesimal amount dn1 of component #1 from
phase a to phase b at constant T,p. Let’s write the change in state.
dn1 (T , p ,phase a ) = −dn1 (T , p ,phase b )
dG = −S dT
0
0
+V dp + ∑ µi dni = ⎡⎣ µ1(b ) − µ1(a ) ⎤⎦ dn1
i
µ1(b ) < µ1( a )
⇒ dG < 0
⇒
spontaneous conversion from (a) to (b)
µ1( a ) < µ1(b )
⇒ dG > 0
⇒
spontaneous conversion from (b) to (a)
At equilibrium there cannot be any spontaneous processes, so
µ1( a ) = µ1(b ) at equilibrium
e.g. liquid water and ice in equilibrium
ice
water
µ ice (T , p ) = µ water (T , p )
at coexistence equilibrium
For the cell in a salt water solution, µ water (cell ) (T , p ) > µwater (solution ) (T , p ) and
the cell dies as the water flows from the cell to the solution (this is
what we call osmotic pressure)
The chemical potential and its downhill drive to equilibrium will be
the guiding principle for our study of phase transitions, chemical
reactions, and biochemical processes
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #9
page 5
• Partial molar quantities
µi is the Gibbs free energy per mole of component “i”, i.e. the
partial molar Gibbs free energy
⎛ ∂G ⎞
= µi = Gi
⎜
⎟
⎝ ∂ni ⎠T , p ,nj ≠i
G = n1 µ1 + n2 µ2 + " + ni µi = ∑ ni µi = ∑ ni Gi
i
i
Let’s prove this, using the fact that G is extensive.
G (T , p , λn1 , λn2 ) = λG (T , p , n1 , n2 )
dG
(T , p , λn1 , λn2 ) = G (T , p ,n1 ,n2 )
dλ
⎛ ∂G ⎞
⎛ ∂G ⎞
⎛ ∂ ( λn1 ) ⎞
⎛ ∂ ( λn2 ) ⎞
+⎜
=G
⎜⎜
⎟⎟
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝ ∂ ( λn1 ) ⎠T , p ,λn2 ⎝ ∂λ ⎠T , p ,λn2 ⎝ ∂ ( λn2 ) ⎠T , p ,λn1 ⎝ ∂λ ⎠T , p ,λn1
n1
n2
λ is arbitrary, we can choose λ = 1
⇒
n1 µ1 + n2 µ2 = G
We can define other partial molar quantities similarly.
⎛ ∂A ⎞
= Ai
⎜
⎟
∂
n
⎝ i ⎠T , p ,nj ≠i
⇒
A = n1A1 + n2A2 + " + ni Ai = ∑ ni Ai
i
partial molar Helmholtz free energy
note what is kept constant
⎛ ∂H ⎞
= Hi
⎜
⎟
⎝ ∂ni ⎠T , p ,n j ≠i
⇒
⎛ ∂A ⎞
⇒ not to be confused with ⎜ ⎟
⎝ ∂ni ⎠T ,V ,n
H = n1H1 + n2H2 + " + ni Hi = ∑ ni Hi
i
= µi
j ≠i
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #9
page 6
partial molar enthalpy
⎛ ∂U ⎞
= Ui
⎜
⎟
⎝ ∂ni ⎠T , p ,n j ≠i
⇒
U = n1U1 + n2U2 + " + niUi = ∑ niUi
i
partial molar energy
__________________________________________________
Let’s compare µ of a pure ideal gas to µ in a mixture of ideal gases.
• Chemical potential in a pure (1-component) ideal gas
From
G (T , p ) = G o (T ) + RT ln
p
p0
⇒
µ (T , p ) = µ o (T ) + RT ln
• Chemical potential in a mixture of ideal gases
onsider the equilibrium
pA′ + pB′ = ptot
mixed pure
A
A,B
p'A, p'B pA
Fixed Partition
\
At equilibrium
µA ( mix ,T , ptot ) = µA ( pure ,T , pA )
Also
pA ( pure ) = pA′ ( mix ) = ptot XA
Dalton’s Law
So
p
p0
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #9
page 7
µA (mix ,T , ptot ) = µA ( pure ,T , ptot XA )
⎛p X ⎞
= µAo (T ) + RT ln ⎜ tot A ⎟
⎝ p0 ⎠
= µAo (T ) + RT ln
ptot
+ RT ln XA
p0
µA ( pure ,T , ptot )
∴
Note
µA ( mix ,T , ptot ) = µA ( pure ,T , ptot ) + RT ln XA
XA < 1
⇒
µA (mix ,T , ptot ) < µA ( pure ,T , ptot )
The chemical potential of A in the mixture is always less than the
chemical potential of A when pure, at the same total pressure. This
is at heart a reflection about entropy, the chemical potential of “A”
in the mixture is less than if it were pure, under the same (T,p)
conditions, because of the underlying (but hidden in this case)
entropy change!