1) Maxwell relationships and their applications are explored, including Maxwell's equations which relate partial derivatives of thermodynamic properties like internal energy (U), entropy (S), volume (V), temperature (T), and pressure (P). 2) An example application shows the dependence of entropy (S) on temperature (T) and volume (V) for an ideal gas using Maxwell's equations. 3) It is shown that the difference between constant pressure (CP) and constant volume (CV) heat capacities can be expressed using Maxwell's equations in terms of the thermal expansion coefficient and isothermal compressibility of materials.

1. THERMODYNAMICS OF BIOLOGICAL SYSTEMS
LESSON 9:
MAXWELL RELATIONSHIPS AND THEIR APPLICATIONS
Biological Systems and Maxwell’s Demon All of the primary thermodynamic properties P, V, T, U, and S
Unlike most physical systems, biological systems typically seem are included in Eq. (2-40). Additional thermodynamic proper-
capable of spontaneously organizing themselves. And as a ties arise only by definition in relation to these primary
result, even the original statements of the Second Law talked properties. The enthalpy was earlier defined as a matter of
only about “inanimate systems”. In the mid-1860s James convenience by the equation:
Clerk Maxwell then suggested that a demon operating at a
microscopic level could reduce the randomness of a system such H ≡ U + PV (2-41)
as a gas by intelligently controlling the motion of molecules. Two additional properties, also defined for convenience, are the
For many years there was considerable confusion about Helmholtz energy
Maxwell’s demon. There were arguments that the demon must
use a flashlight that generates entropy. And there were extensive
A ≡ U − TS (2-42)
demonstrations that actual biological systems reduce their and the Gibbs energy,
internal entropy only at the cost of increases in the entropy of
G ≡ H − TS (2-43)
their environment. But in fact the main point is that if the
evolution of the whole system is to be reversible, then the Each of these defined properties leads directly to an equation
demon must store enough information to reverse its own like Eq.2-40. Upon multiplication by n.Eq. (2-41) becomes:
actions, and this limits how much the demon can do, prevent- nH = nU + P(nV)
ing it, for example, from unscrambling a large system of gas Differentiation gives:
molecules.
d(nH) = d(nU) + P d(nV) + (nV)dP
Roperty Relations for Homogeneous When d(nU) is replaced by Eq. (6.1), this reduces to:
Phases
d(nH) = Td(nS) + (nV)dP (2-44)
The first law for a closed system of n modes is:
Similarly, from Eq. (2-42),
d(nU) = dQ + dW (2-39)
d(nA) = d(nU) - T d(nS) - (nS)dT
For the special case of a reversible process. Eliminating d(nU) by Eq. (6.1) gives:
d(nU) = dQrev + dWrev d(nA) = -Pd(nV) – (nS)dT (2-45)
Equations (1.2) and (5.12) are here written: In analogous fashion, eqs. (6.3) and (6.4) combine to yield:
dWrev = - Pd(nV) dQrev = Td(nS) d(nG) = (nV)dP – (nS)dT (2-46)
Together these three equations give: Equations (2-44) through (2-46) are subject to the same
d(nU) = Td(nS) – Pd(nV) (2-40) restrictions as Eq. (2-40). All are written for the entire mass of
where U, S, and V are molar values of the internal energy, any closed system.
entropy, and volume. Our immediate application of these equations is to one mole
This equation, combining the first and second laws, is derived (or to a unit mass) of a homogeneous fluid of constant
for the special case of a reversible process. However, it composition. For this case, they simplify to:
contains only properties of the system. Properties depend on dU = TdS – PdV (2-47)
state alone, and not on the kind of process that leads to the dH = TdS + VdP (2-48)
state. Therefore, Eq. (2-40) is not restricted in application to
dA = - PdV – SdT (2-49)
reversible processes. However, the restrictions placed on the
nature of the system cannot be relaxed. Thus Eq. (2-40) applies dG = VdP - SdT (2-50)
to any process in a system of constant mass that results in a These fundamental property relations are general equations for a
differential change from one equilibrium state to another. The homogeneous fluid of constant composition.
system may consist of a single phase (a homogeneous system), Another set of equations follows from Eqs. (2-47) through (2-
or it may be made up of several phases (a heterogeneous 50) by application of the criterion of exactness for a differential
system); it may be chemically inert, or it may undergo chemical expression. If F = F(x, y), then the total differential of F is
reaction. defined as:
The only requirements are that the system be closed and that
the change occur between equilibrium states.
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2. For a reversible constant volume process TdS = q v =dU CvdT
THERMODYNAMICS OF BIOLOGICAL SYSTEMS
 ∂F   ∂F 
dF =   ∂y  dy
 dx +    ∂S  Cv
 ∂x  y  x and   =
 ∂T  V T
or dF = M dx + N dy (2-51)
 ∂P   ∂S 
 ∂F   ∂F  Using the Maxwell equation   =  and the ideal
where M =   ∂y 
N =   ∂T  V  ∂V  T
 ∂x  y  x
By further differentiation,  ∂P  R
gas law, PV = RT, which gives   =
 ∂T  V V
 ∂M  ∂2F  ∂N  ∂2F

 ∂y  =
   =
  x ∂y∂x  ∂x  y ∂x∂y dS =
CV R
dT + dV
Since the order of differentiation in mixed second derivatives is T V
immaterial, these equations combine to give: Integration between states 1 and 2 -
 ∂M   ∂N  T2 V

 ∂y  =
  (2-52) S 2 − S 1 = CV ln + R ln 2
  x  ∂x  y T1 V1
When F is a function of x and y, the right side of Eq. (2-51) is Another example application of Maxwell Equations:
an exact differential expression; since Eq. (2-52) must then be Cv vs. C P
satisfied, it serves as a criterion of exactness. Let’s continue our discussion of the difference between the
The thermodynamic properties U, H, A, and G are known to be constant pressure and constant volume heat capacities. We
functions of the variables on the right sides of Eqs. (2-47) have the following relation:
through (2-50); we may therefore write the relationship
expressed by Eq. (2-52) for each of these equations:  ∂V    ∂U  
C P − CV =   P+
   
 ∂T   ∂P   ∂T  P   ∂V  T 
  = −  (2-53)
 ∂V  S  ∂S  V
 ∂V 
While P   and can be directly measures in experiments,
 ∂T   ∂V   ∂T  P
  =  (2-54)
 ∂P  S  ∂S  P  ∂U 
is   not, and it would be useful to express this term
 ∂P   ∂S   ∂V  T
  =  (2-55) through other variables.
 ∂T  V  ∂V  T
Differentiating the combined 1st and 2nd laws, dU = TdS- PdV
by V at constant T we have,
 ∂V   ∂S 
  = −  (2-56)
 ∂T  P  ∂P  T  ∂U   ∂S 
  = T  −P
These are MAXWELL’S Equations  ∂V  T  ∂V  T
Equations (2-47) through (2-50) are the basis not only for
 ∂V   ∂S 
derivation of the Maxwell equations but also of a large number C P − CV = T    
of other equations relating thermodynamic properties. We  ∂T  p  ∂V  T
develop here only a few expressions useful for evaluation of
thermodynamic properties. Using the Maxwell Equation
Example application of Maxwell Equations  ∂P   ∂S 
Let us consider the dependence of the entropy of an ideal gas   = 
on the independent variables T and V: S = S (T,V)
 ∂T  V  ∂V  T
 ∂S   ∂S   ∂V   ∂P 
dS =   dT +   dV C P − CV = T     This is valid for any system,
 ∂T  V  ∂V  T  ∂T  p  ∂T  V
we have not used any approximations.
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2.202 37

3. THERMODYNAMICS OF BIOLOGICAL SYSTEMS
 ∂V   ∂V 
Considering V = V(P,T): dV =  dP +   dT
 ∂P  T  ∂T  P
And differentiating by T at constant V we have:
 ∂P  (∂V / ∂T )P
  =−
 ∂T  V (∂V / ∂P )T and
(∂V / ∂T )P 2
C P − CV = −T
(∂V / ∂P )T
It would be convenient to express CP-CV through intensive
properties of materials (rather than extensive ( δ V/ δ T)P and
( δ V/ δ P)T). For isotropic materials we can define –
1  ∂V 
  = α where is coeeficient of thermal expansion
V  ∂T  P
1  ∂V 
−   = K T where KT is isothermal compressibility
V  ∂P  T
CP - CV = TV is small for solids (eg. 1.7x10-5 K-1 for Cu, 1.0x10-6
K-1 for diamond at T = 300K)
References
1. J. M. Smith, H. C. Van Ness, M. M. Abbott, Adapted by B.
I. Bhatt, Introduction To Chemical Engineering
Thermodynamics, Sixth Edition, Tata McGraw-Hill
Publishing Company Ltd, New Delhi
2. www.people.virginia.edu
Notes
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