Structural Biochemistry/Molecular Thermodynamics

Different from classical thermodynamics, essentially a deductive science, molecular thermodynamics focuses on the properties of individual chemical species and their mixtures at molecular level.

History of Development
By the end of the nineteenth century, most laws and postulates of classical thermodynamics were well-established. However, the rapid pace of development in scientific fields relating to chemistry, physics, and chemical engineering urges scientists and engineers to have a more holistic view on the subject of thermodynamics, which has given rise to the birth of molecular thermodynamics. More often than not, the source of property values is experiment. For instance, the ideal gas equation was evolved as a statement of observed physical behavior of gases and their interconnecting relationships between volume, pressure, temperature, and number of moles of gas present.

At the turn of the century, physicists and chemists who worked with principles of classical thermodynamics increasingly realized that experiments at macroscopic level often failed to provide any insight into why substances exhibit their observed properties. By conducting further experiments, they found that the basis for insight should rather be established on a microscopic view of matter.

Intermolecular Forces
Intermolecular forces are relatively weak forces between molecules in random motion. The energy resulted from this random motion is referred as internal energy.

The ideal gas model was the very first model introduced when internal energy between molecules was studied. An ideal gas is characterized by the absence of molecular interactions. However, ideal gas also possesses internal energy. Real gases, on the other hand, are composed of molecules that have not only the energy of individual molecules, but also energy shared between molecules due to their interactions. The intermolecular potential energy is associated with collections of molecules, and the intermolecular forces are reflected by the existence of energy in this form.

Pair-Potential Function
Two molecules attract each other when they are far apart and repel each another when they are close together. This fact was also established through the study of molecular thermodynamics. A sketch of intermolecular potential energy may reveal that the potential energy for an isolated pair of spherically symmetric neutral molecules is solely dependent on the distance which separates them.

If we let U denote intermolecular potential energy, F denote intermolecular force, and r denote the distance separating the two spherically symmetric neutral molecules, the intermolecular force may be expressed as a function of intermolecular potential energy and distance separating the two spherically symmetric neutral molecules as:

F(r) = - dU(r) / dr

The negative sign shown in the above equation signifies an intermolecular attraction, whereas a positive sign indicates an intermolecular repulsion.

The above differential equation is also referred as the pair-potential function. Specific values of U and r in this form may appear as species dependent parameters in a pair-potential function.

Internal Energy from A Microscopic View
Kinetic theory and statistical mechanics are the two theories that relate the behavior of molecules from microscopic level to macroscopic level. Many thermodynamic properties, such as internal energy, enthalpy, and entropy were able to be explained after the development of the two theories mentioned above. Together these two theories represent nearly all the knowledge we possess about molecular thermodynamics.

Before further exploring molecular thermodynamics from a molecular level, it is crucial to understand how the energy associated with each individual molecules of an ideal gas relates to the macroscopic internal energy of a system defined.

Energy, no matter external or internal, is quantized from a quantum mechanics point of view. In other words, the total amount of internal energy of a system may be treated and analyzed as tiny measurable units that carry discrete amount of energy. One of such an energy unit is often referred as quanta. Thus there are enormous numbers of quanta contained in a system, and the sum of the quanta determines the energy level of the system. The set of energy levels allowed to exist to a closed system, as specified by quantum theory, is determined by its volume. Each energy level of a system has a quantum states associated with its energy level, which is also known as the degeneracy of the level.

Statistical Mechanics' Contribution
From a molecular thermodynamic point of view, the state of a system is firmly established if and only if the temperature and volume of the system are defined. Nevertheless, a fixed temperature and a fixed volume do not guarantee an equilibrium is reached within the system. In the case of ideal gas, the random motion and collision of the gas molecules with each other and the wall of the container result in exchanges of energy with the surroundings. Momentary fluctuations caused by these random motions and collisions shift the energy level back and forth within the system. Therefore, it makes sense to define an average value over the discrete set of energy levels of the allowed quantum states. Moreover, statistical mechanics is sufficient in providing the means for arriving at the proper average value.

One of the fundamental postulates of statistical mechanics for a system with defined volume and temperature is that the probability of a quantum state depends only on its energy. The importance of this postulate is that it relates the energy level of a system with its probability. In other words, all quantum states with the same energy have the same probability. Following the same logic as stated above, a value for the thermodynamic internal energy may be obtained as the average of the energies of the quantum states, which is equivalent to its probability.