Concept of Internal Energy (E)

Energy is defined as the capacity to do work. Every system containing some quantity of matter possesses a store of energy. It may occur in various forms such as kinetic energy due to the motion of the body; potential energy by virtue of its position in a force field; thermal energy due to the temperature of the body; chemical energy due to constitution of the compounds; nuclear energy; mechanical energy, electrical energy, etc.

The energy possessed by a system due to translation and rotational motions of the molecules along with the energy of the electrons, nuclear and molecular interactions is known as its internal or intrinsic energy or energy content of the system. The energy acquired by a system in a force field like electrical, magnetic, gravitational surface, etc., are termed as external energies and are usually not considered as part of the internal energy of the system. The energy content of a system is denoted by E and depends on the internal structure and the constitution of the material composing the system. It is independent of the previous history of the system. The magnitude of energy is determined by the state of the system and in turn by the variables of the system like pressure, volume and temperature. As these variables are related to one another by an equation of state, the energy of a system of fixed composition may be described in terms of any two of these state variables, i.e.,

E = f(P, T) = f(V, T) = f(P, V)

Energy is an extensive property, i.e., its magnitude depends upon the quantity of material in the system. In SI system, it is expressed in J or kJ.

If in a transformation the system changes from state A to some other state B, the energy change ΔE = (E_A -E_B) is determined by the values of variables in the two states. The change in the state of the system can be brought about by a number of paths but the change in E is always the same, viz., E_B-E_A is determined by the values of variables in the two states. The change in the state of the system can be brought about by a number of paths by which the transformation is carried out. Energy is therefore, a state function. In a cyclic process, the initial state of the system is restored and change would be zero.

Properties of internal energy:

There are several properties of internal energy:

(I) The internal energy is an extensive property. 5 moles of a substance in a specified state have five times the internal energy possessed by one mole of the same.

(II) When system changes from a thermodynamics state A to a thermodynamic state B, its internal energy will also change. If E_A and E_B denote the internal energies in the two states, then the change in internal energy,* ΔE = E_B-E_A will be governed by the magnitude of the variables in the final and in the initial states. ΔE will be independent of the process or path along which the transformation has been carried out. A particular change of a system can be effected in different ways but ΔE would be the same. To illustrate: A system of carbon and oxygen may be directly converted to carbon dioxide, or through the intermediate formation of carbon monoxide.

(a) C + O₂ = CO₂

(b) C + ½ O₂ = CO

(c) CO + ½ O₂ = CO₂

But ΔE in both cases would be the same, provided the pressure, temperature etc. in the initial state and in the final state be the same in both the transformations.

Such a function, like E, whose magnitude is governed only by the state of the system and nothing else, is called a state function or a characteristic function. The change in the value of the state function for a specified transformation of the system is independent of the path of the transformation. Internal energy, E, is a state function.

(III)  If a system suffers a series of changes so as to come back to its initial state, i.e., when a cyclic process is completed, the internal energy of the system assumes its original magnitude, regardless of what might have happened to the system meanwhile or previously. Hence, if all the internal energy changes (ΔE) in the different stages of the cyclic process be summed up, these will vanish; i.e.,

 ΣΔE = 0

Therefore we can say

∮dE = 0 [The cyclic integral denotes cyclic process]

Later on, we shall come across other functions associated with the system, such as entropy, enthalpy etc., which are also state functions. Processes that describe how a system transitions between states are called path functions. Examples of path functions are the work and heating that are done when preparing a state. We do not speak of a system in a particular state as possessing work or heat. In each case, the energy transferred as work or heat relates to the path being taken between states, not the current state itself.

Reference

1) Peter Atkins, Julio de Paula, Atkins’ physical chemistry book, eighth edition.