Structural Biochemistry/Internal Energy

Overview
In Thermodynamics, the total energy of a specific system is called the Internal Energy. It is the total amount of energy within the system, excluding the energy in the surroundings. Internal Energy can be divided into two parts: Kinetic Energy and Potential Energy. We say that the change in Internal Energy equals to ∆U, where the final Internal Energy is subtracted from the initial Internal Energy to obtain the change in Internal Energy, ∆U.

Internal Energy is a state function because of the fact that its value depends on the present state of the system, as it is independent of the path. In calculating the change in Internal Energy, by whatever process, as long as the initial and final states are the same, the Internal Energy does not differ. Another way to look at the Internal Energy of a system is to take the perspective of work in relation to Internal Energy. Either mechanical work from the change in pressure or volume can result in the change in Internal Energy. Overall, the internal energy of the system does increases as mass is added into the system, hence making Internal Energy an extensive property as it directly proportional to the amount of material in the system at that time.

Internal energy is defined as the energy associated with the random, disordered motion of molecules. It is separated in scale from the macroscopic ordered energy associated with moving objects. It refers to the invisible microscopic energy on the atomic and molecular scale. For example, a room temperature glass of water sitting on a table has no apparent energy, either potential or kinetic. But on the microscopic scale it is a seething mass of high speed molecules traveling at hundreds of meters per second. If the water were tossed across the room, this microscopic energy would not necessarily be changed when we superimpose an ordered large scale motion on the water as a whole.

Internal energy involves energy on the microscopic scale. For an ideal monoatomic gas, this is just the translational kinetic energy of the linear motion of the "hard sphere" type atoms, and the behavior of the system is well described by kinetic theory. However, for polyatomic gases there is rotational and vibrational kinetic energy as well. Then in liquids and solids, there is potential energy associated with the intermolecular attractive forces. A simplified visualization of the contributions to internal energy can be helpful in understanding phase transitions and other phenomena which involve internal energy.

More generally, while external energy is energy due to macroscopic motion (of the system as a whole) or to external fields, internal energy is all other forms of energy, including random motion (relative motion of molecules within the system) and dipole moments and stress.

Molecular Interpretation
Each molecule has a specific number of degrees of freedom, which include translations, rotation, or vibration. Yet, as the equipartition theorem states, the increase in thermal energy is even spread between degrees of freedom of the molecule in question. According to the equipartition theorem, the average energy of each contribution to the total energy is ½*kT so for a monatomic gas that only contains translation the equation looks like this: U_m (T)=U_m (0)+3/2 RT. With the inclusion of rotational energy for linear molecules the equation then becomes: U_m (T)=U_m (0)+5/2 RT. Now, with a non-linear molecules, including both transitional and rotational energies, the equation then becomes: U_m (T)=U_m (0)+3RT.

Internal Energy for an Ideal Gas
With the assumption that the gas being studied is ideal, the definition of Internal Energy can change. Since there are absolutely no intermolecular interactions in a perfect gas, so long as the distance between the molecules take no effect on energy. Then the assumption can be made that the Internal Energy of a gas to be independent of its volume.

The internal energy of an ideal gas is a function of the temperature only.

An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. One can visualize it as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature.

An ideal gas can be characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T). The relationship between them may be deduced from kinetic theory and is called the Ideal Gas Law: PV=nRT

The ideal gas law can be viewed as arising from the kinetic pressure of gas molecules colliding with the walls of a container in accordance with Newton's laws. But there is also a statistical element in the determination of the average kinetic energy of those molecules. The temperature is taken to be proportional to this average kinetic energy; this invokes the idea of kinetic temperature.

Reference
Physical Chemistry by Atkins and de Paula (9th edition)