The internal energy, in thermodynamics, is the property that defines the energy of a thermodynamic system usually in the absence of effects from electric, magnetic or any other external fields which can be source of potential energy. It also excludes the kinetic energy of motion. The notion of internal energy is present in the expression of the first law of thermodynamics as a state variable, as originally formulated by the german scientist Rudolf Clausius, one of the founders of thermodynamics who is also the author of the notion of entropy as a state variable.
The total internal energy of a system cannot practically be measured. Thermodynamics concerns changes in the internal energy, not its absolute value. The processes that change the internal energy are transfers of energy or matter, into or out of the system, as heat or by thermodynamic work.Born, M. (1949), Appendix 8, pp. 146–149. These processes can be measured as changes in the system's properties, such as temperature, entropy, volume, electric polarization, and molar constitution. The internal energy depends only on the internal state of the system and not on the particular choice from many possible processes by which energy may pass into or out of the system. It is a state variable, a thermodynamic potential, and an extensive property.
Without a thermodynamic process, the internal energy of an isolated system does not change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics. Without transfer of matter, internal energy changes equal the algebraic sum of the heat transferred and the thermodynamic work done.
Thermodynamics defines internal energy at macroscopic scale, for the body as a whole. In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, rotations, and vibrations, and of the potential energies associated with microscopic forces, including chemical bonds.
The unit of energy in the International System of Units (SI) is the joule (J). The internal energy relative to the mass with unit J/kg is the specific internal energy. The corresponding quantity relative to the amount of substance with unit J/mol is the molar internal energy.
Cardinal functions
The internal energy of a system depends on its entropy S, its volume V and its number of massive particles: . It expresses the thermodynamics of a system in the energy representation. As a function of state, its arguments are exclusively extensive variables of state, which is why it is sometimes called a cardinal function of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, , of the same list of extensive variables of state, except that the entropy, , is replaced in the list by the internal energy, . It expresses the entropy representation.Tschoegl, N.W. (2000), p. 17.Callen, H.B. (1960/1985), Chapter 5.Münster, A. (1970), p. 6.
Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example , that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, for , to get .
In contrast, Legendre transformations are necessary to derive fundamental equations for other thermodynamic potentials and Massieu functions. The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation of Massieu functions. It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.Münster, A. (1970), Chapter 3.Bailyn, M. (1994), pp. 206–209.
For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics.
Description and definition
The internal energy of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state:
where denotes the difference between the internal energy of the given state and that of the reference state, and the are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state. From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, , and microscopic kinetic energy, , components:
The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles (often expressed as thermal engergy/ temperature). The microscopic potential energy algebraic summative components are those of the chemical and nuclear particle bonds (e.g. intermolecular forces), and the physical force fields within the system, such as due to internal induced electric or magnetic dipole moment (e.g. intramolecular forces), as well as the energy of deformation of solids (stress-strain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.
Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external gravitational, electrostatic, or electromagnetic fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the system with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter.
For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass. Typically, descriptions only include components relevant to the system under study. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy.
The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains.
At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). In the classical picture of thermodynamics, kinetic energy vanishes at zero temperature and the internal energy is purely potential energy. However, quantum mechanics has demonstrated that even at zero temperature particles maintain a residual energy of motion, the zero point energy. A system at absolute zero is merely in its quantum-mechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable entropy.
The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics relates the pseudo-random kinetic energy of individual particles to the mean kinetic energy of the entire ensemble of particles comprising a system. Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. While temperature is an intensive measure, this energy expresses the concept as an extensive property of the system, often referred to as the thermal energy,Leland, T. W. Jr., Mansoori, G. A., pp. 15, 16.Thermal energy – Hyperphysics. The scaling property between temperature and thermal energy is the entropy change of the system.
Statistical mechanics considers any system to be statistically distributed across an ensemble of microstates. In a system that is in thermodynamic contact equilibrium with a heat reservoir, each microstate has an energy and is associated with a probability . The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence:
This is the statistical expression of the law of conservation of energy.
Internal energy changes Thermodynamics is chiefly concerned with the changes in internal energy .
For a closed system, with mass transfer excluded, the changes in internal energy are due to heat transfer and due to thermodynamic work done by the system on its surroundings.This article uses the sign convention of the mechanical work as often defined in engineering, which is different from the convention used in physics and chemistry; in engineering, work performed by the system against the environment, e.g., a system expansion, is taken to be positive, while in physics and chemistry, it is taken to be negative. Accordingly, the internal energy change for a process may be written
When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible.
A second kind of mechanism of change in the internal energy of a closed system changed is in its doing of work on its surroundings. Such work may be simply mechanical, as when the system expands to drive a piston, or, for example, when the system changes its electric polarization so as to drive a change in the electric field in the surroundings.
If the system is not closed, the third mechanism that can increase the internal energy is transfer of substance into the system. This increase, cannot be split into heat and work components. If the system is so set up physically that heat transfer and work that it does are by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy:
If a system undergoes certain phase transformations while being heated, such as melting and vaporization, it may be observed that the temperature of the system does not change until the entire sample has completed the transformation. The energy introduced into the system while the temperature does not change is called latent energy or latent heat, in contrast to sensible heat, which is associated with temperature change.
Internal energy of the ideal gas