ENERGY AND THERMODYNAMICS

One of the goals of food protein research is to be able to explain the behavior of proteins in food systems. Further attempts are made to relate the structure of proteins to their functional properties. In order to obtain these goals there must be an appreciation for the forces involved in determining the functionality of proteins in various food products.

Much of the basic information required for this task can be found in the principles of thermodynamics. It is not the purpose of this exercise to present a rigorous treatment of thermodynamic principles, but rather to introduce some basic concepts that govern all reactions. Thermodynamics is an exact science. Simplifications will be made as much as is practical without losses of accuracy. When these simplifications are made, the assumptions made will be presented and their impact on the accuracy of the results discussed.

Useful Working Definitions

The following terms will be defined to aid the discussion:

System - A bounded area separated from the rest of the universe which is under study. A system may be open, closed or isolated
.

An open system allows for the transfer of both mass and energy between itself and its surroundings. Living organisms are examples of open systems.

A closed system is one that prohibits the transfer of mass with its surroundings, but is open to the exchange of energy. Most of the discussion that follows will concern closed systems.

An isolated system does not permit the transfer of either mass or energy to its environment. A calorimeter is an example of an closed system.

Heat (q) - is the energy transferred into or out of a system due solely to differences in temperature. Heat manifests itself by differences in the velocity of molecules that absorb it.

Work (W) - is defined as any change (besides thermal) of energy between a system and its surroundings. In most cases the discussions will be concerned with changes in volume and pressure.

Internal Energy (E) - is the total energy within a system.

Enthalpy and the First Law

Given these definitions, a quantity called enthalpy (H) will now be defined as the sum of the internal energy of a system plus the mechanical-volume energy associated with the space occupied by the system. The latter is the product of its volume and the external pressure exerted on the system.

H = E + PVEquation 1

The first law of thermodynamics states that energy is conserved. Energy may be neither created nor destroyed; only its form can change. This can be expressed as:

DE = DQ - DW Equation 2

That is, the change in energy of the system is equal to the change in heat added ( DQ ) minus the work done ( DW ) by the system.

For an isolated system there is no exchange of energy with the environment so DE must be equal to zero and the heat and work of the system must be equal. This is the basis of calorimetry, where the work done is equivalent to and is measured as the change in heat of the system.

For a change at constant pressure and performing only PdV work, the differential change in internal energy is:

dE = dQ - PdV Equation 3

By definition the differential enthalpy change would be the sum of the differential changes in the properties used in equation 1 to define enthalpy, i.e.:

dH = dQ - dW + PdV + VdP Equation 4

But, the work has been specified to be only PdV. Then dW = PdV and:

dH = dq + PdV Equation 5

If the change occurs at constant volume:

dE = dq Equation 6

While at constant pressure:

dH = dq Equation 7

That is, if change occurs at constant temperature and the volume is kept constant then the change in internal energy of the system is equal to the change in the heat of the system. If the change occurs at constant pressure, the change in enthalpy of the system is equal to the change in heat. In living organisms, the changes in pressure are very small so equation 7 nearly applies. Furthermore, in solids and liquids, the volume changes are small so that for biological systems the approximation that the enthalpy change of the system is approximately equal to the internal energy change can be made, i.e.:

DH is approximately equal to DE Equation 8

In practical terms when changes in enthalpy of reactions involving proteins are considered, it can be said that changes in the internal energy of the molecules are approximately equal to their heat content.

Entropy and the Second Law: A Conceptual View Based on Order of the System

If there was a container as shown in figure 1 which contained two components of equal volume separated by a door, each side could be filled with an equal amount of two different gases maintained at the same constant pressure and temperature.

Figure 1A

Figure 1B

If the door were opened for a time and then closed the situation that exists in figure 1B would be expected to occur. The number of molecules of A and B in each component would be expected to be nearly equal. If the door were opened again for a period of time and then closed a situation nearly like that in figure 1B would still be expected. Intuition and experience suggest that the molecules are not likely to appear as they did in 1A again.

In this example there were no changes of temperature, pressure or volume, but evidently some barrier exists to ordering of the molecules in nature. The second law of thermodynamics which states that the entropy (S) of the universe is increasing can be utilized to explain this. For this purpose it is convenient to define entropy as:

S = k ln W Equation 9

where k is Boltzman's constant and W is equal to the number of ways something can be arranged. There is only one way that the system in figure 1A can be arranged. Entropy then is a measure of the randomness of a system and the second law states that the randomness of the universe is increasing. The isothermal expansion of an ideal gas from volume 1 ( V1 ) to volume 2 ( V2 ) can be utilized to illustrate the concept of entropy. Volume V1 can be divided into n1 cells of volume V so that:

V₁ = n₁V Equation 10

The larger volume can be divided into n2 cells of volume V so:

V₂ = n₂V Equation 11

If one molecule is placed in the initial system there are n1 ways to do this. For two molecules there are n12 arrangements. For a mole of molecules in volume V1, the number of arrangements is:

W1 = n₁^N0 Equation 12

where N0 is equal to Avagadro's number. In the larger volume, V2 :

W2 = n₂^N0 Equation 13

The entropy change can now be defined in terms of increased randomness of the system. Substitution of equations 12 and 13 into equation 9 produces:

DS = k ln n₂^N0 - k ln n₁^N0 Equation 14

or:

DS = k ln(n²/n¹)N0 Equation 15

 

By definition, R, the gas constant, is equal to kN0 so :

Equation 16

and from 10 and 11:

and

Substitution of into equation 16 gives:

Equation 17

Which reduces to:

Equation 18

Equation 18 relates the change in entropy directly to the ratio of the two volumes which is related to the number of ways of arranging the molecules in these volumes.

An Alternate Definition of Entropy

Another way to visualize entropy is to consider the absorption of heat, q, by a system at a constant volume and pressure. The absorbed heat must have moved molecules into higher energy states. If this process is reversible indicated by qrev, we can define the change in entropy at a given temperature as:

Equation 19

 

Two observations should be made:

1. Energy changes in an isolated system are always zero
2. In all irreversible processes, the entropy of the system plus that of its surrounding increases.

In the "real" world there are no truly reversible processes because there is always some loss of energy to friction and thus to entropy. Thus the entropy of the universe is increasing and will reach its maximum value when no energy differences exist.

The processes protein chemists are interested in are thermodynamically unstable. Order is often introduced into a system. Given enough time, these systems must increase their entropy and proceed to a disordered state. To prevent this, energy must be expended as in living cells or the activation energy of the process must be so high that the rate of reaching equilibrium is small. The second approach is generally the one taken in food systems.

Free Energy and Useful Work

Gibb's free energy (G) can now be defined as:

G = H - TS Equation 20

From the definition of enthalpy equation 1 states that H = E + PV, then:

G = E + PV - TS Equation 21

Thus, the free energy is the total internal energy and mechanical energy of occupying a space in excess of that energy associated with creating the system.

For a differential change in G :

dG = dE + PdV + VdP - TdS - SdT Equation 22

With proteins work is often done at constant temperature and pressure so that two of the above terms vanish, i.e.:

SdT = 0 and VdP = 0

and from equation 19 :

TdS = dq Equation 23

Further from the first law ( equation 2 ):

dE - dq = -dW

consequently, substitution of these four terms into equation 22 yields:

dG = - dW + PdV Equation 24

which rearranges to:

-dG = dW - PdV Equation 25

That is, the decrease in free energy is equal to the work done by the system minus the mechanical work done on the system. For any set of conditions, the work done by the system is maximal and can be called qmax, then:

- dG = dq_max - PdV Equation 26

PdV is work that can not be harnessed in chemical processes as it is used to create the new space required by the system and can be termed as wasted work, or:

-dG = dW_useful Equation 27

Thus, changes in Gibb's free energy are a measure of the useful work that can be obtained from a system. A negative value for dG suggests that the reaction products have lower energy than do the reactants and that the reaction will proceed. A positive value suggests that energy must be applied to the system in order for the reaction to proceed. A useful definition of DG for a process in which a system goes from one condition ( state ) to another is:

DG = DH - TDS Equation 28

Thus, it has been established that DG is useful work. It is made up of a component related to the change in internal energy of the system ( DH ) and also an entropy term (TDS ) which measures the change in randomness of the system.

The Relationship between Energy and Equilibrium

For a chemical reaction we can define the thermodynamic equilibrium constant Keq as:

Equation 29

For the reaction A + B ¥ C + D at equilibrium:

Where the values in parentheses represent the active concentration of the corresponding component. This constant is independent of the starting concentrations and will always be the same at constant temperature and pressure. The equilibrium is the result of the system attempting to reach the lowest possible free energy. The free energy, DG, and the equilibrium constant, Keq, can be related by the following expression:

Equation 30

Which rearranges to:

DG = -RT ln Keq Equation 31

This is a useful over simplification as it should be obvious from the definition of Gibb's free energy that the value obtained will depend upon the temperature chosen. Further it has been assumed that Keq is independent of concentration. This is nearly true for very dilute solutions, but in most cases the use of concentrations in experiments to determine free energy can lead to serious errors.

To remedy this a number of Gibb's free energy states may be dfined. Two of these are especially important to this discussion. The first of these is the standard free energy, G0, which is obtained from Keq when the reactants are kept at one molal concentration. In living systems free energy changes generally occur around pH 7. The quantity, DG1, refers to free energy changes that occur at pH 7.

Non-Ideality, Chemical Potential and Activity

To help visualize the complications of changes in Keq with concentration imagine a fast moving object placed into a certain space. It is free to move wherever it chooses and to express its full energy including bouncing off the walls of the container. When 9 additional real objects are placed into the same container, the total energy will be less than its potential, i.e. ten times that for one particle. This is because there is now a real chance that the particles will interact in ways that will alter their full reaction potential. The more particles there are, the greater will be the reduction in energy from what would expected from ideal, non-interacting particles.

Partial Molar Quantities

An understanding of the concept of partial molar quantities will greatly aid in the discussion of chemical potential and activity. If one liter of water has n moles of a component added to it, a volume change will occur. If the number of moles of solvent is maintained constant and the change in volume per mole of added solute is measured:

V = ( dV/dn) Equation 32

where V is termed the partial molar volume.

Chemical Potential

This can be done for a number of different properties of a system. The one that is of most use for this discussion is the change in free energy of a component per change in concentration, U1:

Equation 33

Where the quantity G1 is the partial molar free energy of component one. This quantity G1 has been defined as the chemical potential, U1, of the component:

G1 = U1 Equation 34

Thermodynamically Active Concentration

Another term is required, the one to correct for the non-ideality of real solutions. This term is the thermodynamic activity, a. Quite simplistically, this can be considered to be related to the "real" energy of a system. While there may be one mole of solute in a solution it may, because of non-ideality, behave as though there were only 0.5 mole of ideal solute. The effective concentration of this non-ideal solute is defined as its activity, a, and the term, g,the activity coefficient, is used to relate activity to concentration such that:

Equation 35

Where m = molal concentration. The activity and chemical potential of a solute can be related by defining U01 as the chemical potenial of component one in a pure state. Then:

Equation 36

Equation 36 gives the exact definition of activity and shows that it is related to the free energy of a compound in the pure state, but it is not equal to neither the free energy nor the the chemical potential. Chemical potential, like any other expression of energy can only have meaning when it is compared to some reference state. For a mixture of components:

U₁ = U⁰₁ + RT ln X₁ Equation 37

Where X1 is the mole fraction of component one. For a pure substance X1 = 1 and:

U₁ = U⁰ Equation 38

which indicates that the chemical potential of a pure component is the reference state. For a solution, however, the mole fraction scale has little meaning and:

U₁ = U⁰₁ + RT ln C₁ Equation 39

It should be noted that whenever C1, the concentration of component C is equal to one, the reference state exists and U1 = U0. If C1 is 1 g/ liter, then U1 = U0 and that is a reference state. A more common reference state is when C1 = one molar. To correct for non-ideality:

U₁ = U⁰₁ + RT ln a₁ Equation 40

and then U1 = U0 when the activity of component one is equal to unity.