PHYS3410 BIOPHYSICS II
Lecture Notes

Section I: Thermodynamics of Biological Systems

Lecture 4 : Extremum principle and thermodynamic potentials

For centuries we have been motivated by the belief that the laws of Nature are simple. The laws of mechanics, gravitation, electromagnetism and thermodynamics support this approach. The current search for a theory that unifies all the known fundamental forces between elementary particles is hoped to simplify things further.

Natural phenomena often occur in such a way that some physical quantity is minimized or maximized (extremized). (see, for instance, Fermat’s approach to light propagation or principle of least action for deriving equations of motion).

In thermodynamics, thermodynamic potentials are extremized as equilibrium is approached under different conditions.

Maximum entropy

When U and V are constant, system evolves to a state of maximum entropy.

Minimum energy

At constant S and V, system evolves to a state of minimum energy.

For a closed system:

dU = dQ - pdV  =  Td_eS  - pdV = TdS - pdV - Td_iS  (4.1)

but S and V are constant, so

dU =  - Td_iS  £ 0

Thus, in a system whose entropy is maintained at a fixed value, driven by irreversible processes, the energy evolves to the minimum possible value. To keep entropy constant, d_iS has to be removed from the system.

For systems at constant T and V, thermodynamic quantity called Helmholtz free energy, F, evolves to its minimum value.

F = U - TS  (4.2)

dF = dU - TdS = dU - Td_eS - Td_iS = dQ - pdV  - Td_eS - Td_iS = -Td_iS £ 0

The minimization of Helmholtz free energy is a useful principle. Many interesting features, such as phase transitions and formation of complex patterns in equilibrium systems can be understood, using this principle.

A schematic representation of the conditions for equilibrium in a system containing an unconstrained internal thermodynamic force X (from “Thermodynamics of Life processes).

The potential F illustrates the competition between energy and entropy, with temperature being the relative weight of the two factors. At low temperature energy prevails and ordered (low entropy) structures, such as crystals, form. Potential energy is dominant over kinetic energy, each particle “imprisoned” by the interaction with its neighbours. At high temperatures, entropy is dominant and so is molecular disorder: crystals melt into liquid and evaporate into gas, where potential energy diminishes.

For a reversible process dQ = TdS and the work available is given by:

dW_max = -(dU -TdS) = -dF  (4.3)

Thus the maximum work available during an isothermal process is a negative change in F. For irreversible processes the actual work available is always less.

If both p and T of a closed system are maintained constant, the quantity that is minimized at equilibrium is the Gibbs free energy.                                                                       

G = U + pV -TS = H - TS  (4.4)

Where H is called enthalpy = U + pV

(show that dG  = - Td_iS  £ 0 )

The Gibbs free energy is mostly used to describe chemical processes, because the usual laboratory situation corresponds to constant p and T.

Chemical potential and Affinity: the driving force of chemical reactions

Theophile De Donder (1872 – 1957) formulation of chemical affinity was founded on the concept of chemical potential (introduced by Josiah Willard Gibbs, 1839 – 1907).

Gibbs considered a heterogeneous system, each containing various substances: s₁, s₂….s_n of masses m₁, m₂…m_n. Initially, he considered simply the exchanges of the materials between different parts of the system. He argued that the change in energy dU of each homogeneous part must be proportional to changes in the masses of the substances, dm₁, dm₂ …dm_n.

dU =  TdS – pdV + m₁dm₁ + m₂dm₂ …….+ m_ndm_n  (4.5)

The coefficients m_k are called the chemical potentials.

It is more convenient to describe chemical reactions by the change in mole numbers of the reactants rather than the change in their masses:

  (4.6)

While Gibbs did not think in terms of chemical reactions, the equation (4.6) provided excellent basis for consideration of irreversibility and and entropy production in chemical processes. To make a distinction between irreversible chemical reactions and reversible exchange of moles with the exterior:

        dN_k =d_iN_k + d_eN_k  (4.7)

So, for reversible exchange of entropy with environment:

 (4.8)

 (4.9)

This is the “uncompensated heat” of Clausius for chemical reactions.

For a closed system d_eN_k = 0 and the rate of entropy production becomes:

  (4.10)

To write the equation (4.10) in a more elegant form, De Donder defined affinity of chemical reactions. The equation can then be written in terms of a thermodynamic force and thermodynamic flow.

Consider a chemical reaction:

X + Y G 2Z   (4.11)

In this case the changes in the mole numbers, dN_x, dN_y and dN_z, of the components are related by stoichiometry:

  (4.12)

Here the stoichiometric coefficients are taken as negative for the reactants and positive for the products (reaction proceeds to the right).

where dx is the change in the extent of reaction x. Substituting into eqn. (4.10), the entropy production for chemical reaction given by eqn. (4.11) can be written as:

(4.13)

De Donder defined a new state variable called affinity:

A = (m_x + m_y – 2m_z)  (4.14)

The affinity is the driving force for chemical reactions. A non-zero affinity implies that the system is not in thermodynamic equilibrium and that chemical reactions will continue to occur, driving system towards equilibrium.

In terms of affinity A, the rate of entropy increase can be written as a product of thermodynamic force A/T and a thermodynamic flow dx/dt.

  ( 4.15)