Internal Energy
The total energy of a system, E, is given by
E = K + V + U
In addition to macroscopic kinetic energy K and potential energy V, we ascribe an internal energy to a body. The internal energy, U consists of: molecular translational, rotational, vibrational, and electronic energies; the relativistic rest-mass energy mrest c2 of the electrons and the nuclei; there are coulombic and gravitational forces that hold molecules together, and there is a potential energy due to such forces of attraction between molecules. It does not include the kinetic energy, K, of the system itself if it is in motion, nor does it include any gravitational potential energy of the system with respect to some other mass. We will ignore these energies in our treatment.
Heat and work are defined only in terms of processes. Before and after the process of energy transfer between system and surroundings, heat and work do not exist. Heat is an energy transfer between system and surroundings due to a temperature difference. Work is an energy transfer between system and surroundings due to a macroscopic force acting through a distance. Heat and work are forms of energy transfer rather than forms of energy. Work is energy transfer due to the action of macroscopically observable forces. Heat is energy transfer due to the action of forces at a molecular level.
The internal energy is a state function in the sense that its value depends only on the current state of the system and is independent of how that state has been prepared. For a transformation between two states we have
ΔU = U2 − U1
for a cyclic process in which the initial and final states are the same U2 = U1, we have
ΔU = 0
The internal energy of a system can be changed in several ways. Internal energy is an extensive property and thus depends on the amount of matter in the system. For a pure substance, the molar internal energy, Um is defined as
Um ≡ U/n
where n is the number of moles of the pure substance. Um is an intensive property that depends on P and T.
Internal energy of a perfect gas
An ideal gas is a particular moleculear model system consisting of independent particles without volume and without intermolecular forces. The molecules of an ideal gas are thus point particles, which are, when constitued by the same element indistinguishable. Such a model, in reality, is only an approximation. However, in the limit of very low density, the molecules of a real gas are, on the average, so far apart from each other that the foces between them and their volumes have no influence on most of the thermodynamic properties of the sytem. It is known experimentally that:
PV = nRT (∂U/∂V)T, n = 0
the internal energy
For a monoatomic gas made by points like particle, lthe only possible energy is the translational one. Considering an atom moving in space, its trajectory can be specified by three coordinates (x, y, z) known as degree of freedom. For example, the kinetic energy an atom of mass m as it moves through space is
EK = 1/2(mx2 + my2 + mz2)
The equipartition theorem (which can be applied when quantization effects are ignorable), states that, for a collection of particles at thermal equilibrium at a temperature T, the average value of each quadratic contribution to the energy is the same and equal to 1/2 kT, where k is Boltzmann’s constant (k = 1.381 × 10−23 J/K).
According to the equipartition theorem, the average energy of each term in the expression above is 1/2 kT. Therefore, the mean energy of the atoms is 3/2 kT and the total energy of the gas is 3/2NkT, or 3/2nRT (since N = nNA and R = NAk). We can therfore write
Um = Um(0) + 3/2 NAkT = 3/2 RT
where Um(0) is the molar internal energy at T = 0, when all translational motion has ceased and the sole contribution to the internal energy arises from the internal structure of the atoms.
The name equipartition theorem is then justified because every degree of freedom possesses a mean kinetic energy of 1/2 RT.