The internal energy

The constant is of unspecified size, and is the constant referred to by Bridgman in the quote at the head of this chapter. 

Since we do not use absolute values of U or U, we cannot use absolute values of any quantities having U in their equations of definition. This may become a point of some regret if you find yourself puzzling over some unfamiliar standard states later on. Somewhat paradoxically, in spite of being possibly the most fundamental of thermodynamic quantities. Internal Energy or even changes in U are little use d in geochemical applications. It is never listed in tables of thermodynamic values. 

To understand how biological processes can store and release energy, we need to be familiar with a very important law that relates work and heat to changes in the energy of all the constituents of a system. 

Change in internal energy in terms of heat and work 

We have seen that a feature of a perfect gas is that for any isothermal expansion the total energy of the sample remains the same and that q = -w. That is, any energy lost as work is restored by an influx of energy as heat. We can express this property in terms of the internal energy, for it implies that the internal energy remains constant when a perfect gas expands isothermally from eqn 1.6 we can write 

Change of internal energy during isothermal expansion of a perfect gas. 

The definition of AH in terms of w and q points to a very simple method for measuring the change in internal energy of a system when a reaction takes place. We have seen already that the work done by a system when it pushes against a fixed external pressure is proportional to the change in volume. Therefore, if we carry out a reaction in a container of constant volume, the system can do no expansion 

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Joule s law The internal energy of a gas depends only on its temperature (being independent of its pressure and volume). Like the other gas laws, it is only approximately true. At high pressures it is invalidated by the existence of inlermolecular forces. 

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

If the adiabatic work is independent of the path, it is the integral of an exact differential and suffices to define a change in a function of the state of the system, the energy U. (Some themiodynamicists call this the internal energy , so as to exclude any kinetic energy of the motion of the system as a whole.)... 

One can trivially obtain the other thennodynamic potentials U, H and G from the above. It is also interesting to note that the internal energy U and the heat capacity Cy can be obtained directly from the partition fiinction. Since V) = 11 exp(-p , ), one has... 

The presence of tln-ee-body interactions in the total potential energy leads to an additional temi in the internal energy and virial pressure involving the three-body potential / 2, r, and the corresponding tlnee-... 

In this fonnalism, which is already far from transparent, the internal energy is given by IJ -... 

The thennalization stage of this dissociation reaction is not amenable to modelling at the molecular dynamics level becanse of the long timescales required. For some systems, snch as O2 /Pt(l 11), a kinetic treatment is very snccessfiil [77]. However, in others, thennalization is not complete, and the internal energy of the molecnle can still enliance reaction, as observed for N2 /Fe(l 11) [78, 79] and in tlie dissociation of some small hydrocarbons on metal snrfaces [M]- A detailed explanation of these systems is presently not available. 

Now let us write down explicit expressions for p Q), -Pr(v,) and g-j-. Denoting the internal energy for a given state as e. and the relative translational energy as = I we have (in tluee dimensions)... 

Inelastic scattering produces a pennanent change in the internal energy and angrilar momentum state of one or both structured collision partners A and B, which retain their original identity after tire collision. For inelastic = (a, P) — /= (a, P ) collisional transitions, tlie energy = 1 War 17 of relative motion, before ( ) and after... 

The quantum internal energy (fi /2m )(VY ) /p depends also on the derivative of the density, unlike in the fluid case, in which internal energy is a function of the mass density only. However, in both cases the internal energy is a positive quantity. 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

Q is given by Equation (6.4) for a system of identical particles. We shall ignore any normalisation constants in our treatment here to enable us to concentrate on the basics, and so it does not matter whether the system consists of identical or distinguishable particles. We also replace the Hamiltonian by the energy, E. The internal energy is obtained via Equation (6.20) ... 

The pressure often fluctuates much more than quantities such as the total energy in constant NVE molecular dynamics simulation. This is as expected because the pressure related to the virial, which is obtained as the product of the positions and the derivativ of the potential energy function. This product, rijdf rij)/drij, changes more quickly with than does the internal energy, hence the greater fluctuation in the pressure. 

Information about the structure of a molecule can frequently be obtained from observations of its absorption spectrum. The positions of the absorption bands due to any molecule depend upon its atomic and electronic configuration. To a first approximation, the internal energy E oi a, molecule can be regarded as composed of additive contributions from the electronic motions within the molecule (Et), the vibrational motions of the constituent atoms relative to one another E ), and the rotational motion of the molecule as a whole (Ef) ... 

The chemical potential p, of the adsorbate may be defined, following standard practice, in terms of the Gibbs free energy, the Helmholtz energy, or the internal energy (C/,). Adopting the last of these, we may write... 

Ion/neutral reaction. Interaction of a charged species with a neutral reactant to produce either chemically different species or changes in the internal energy of one or both of the reactants. 

The classical formulation of the first law of thermodynamics defines the change dU in the internal energy of a system as the sum of heat dq absorbed by the system plus the work dw done on the system ... 

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