Energy: internal

The energy associated with molecular motion. The temperature T of a material is a 

Work is done when a force acts through a distance  

Work done on a system by its surroundings is conventionally taken as negative work done by the system on the surroundings as positive. 

Where the work arises from a change in pressure or volume  

To integrate this function the relationship between pressure and volume must be known. In process design an estimate of the work done in compressing or expanding a gas is 

The internal energy, U, may be calculated from the partition function as 

If we take the derivative with respect to temperature T of the natural logarithm of Z in Equation 5.13, we obtain 

Tabulated values of internal energy are given in the standard state of a chemical compound, which is the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25°C. Its symbol is AH.  

Everyone knows what energy is, but it is an elusive topic if you are looking for a deep understanding. In fact, a Nobel Prize-winning physicist has affirmed that 

It is important to realize that in physics today, we have no knowledge of what energy is. (Feynman et at., 1963, pp. 4-2) 

As we cannot give a general definition of energy, the principle of the conservation of energy simply signifies that there is something which remains constant. (Poincare, 1952, p. 166) 

If only things were that simple. However, we know that they are not, because the energy in a stick of dynamite on the table is not equal to the work expended in lifting it from the floor. Similarly, the energy in water is not the same as in ice, whether on the floor or the table. These complications are actually of two types. 

There are many ways of doing work, because there are many kinds of forces. We are particularly concerned with the work involved in chemical reactions. 

Now that we have the numerical forms of kinetic energy per unit mass and potential energy per unit mass, we can rewrite Eq. AAb as 

This is the semifinal form of the energy balance equation. 

Now suppose we take as our system the tank shown in Fig. 4.1, We close the valves in the inlet and outlet lines, so that dm- = dm = 0. We also stop the rotating shaft and do not move the volume-changing piston, so there will be no work done dW = 0). Now we transfer 100 Btu of [energy into the tank from the heating jacket i 

However, in this operation the elevation and velocity of the material in the tank did not change, so that d(gz) = d(y H) = 0, and since m remained 

The potential and kinetic energies per unit mass are expressed in units of foot-pounds-force per pound-mass (ft-lbf/lbm) or joules per kilogram (J/kg), Here we have a change in internal energy expressed in Btu per pound-mass or calories per kilogram. In our balance equation, we obviously need some way to interconvert these units so that the sum (u -t- gz + V 2) is in a consistent set of units. All efforts to calculate this conversion factor from 

The MCAT doesn t generally ash about internal energy directly, he study of nternal energy is beyond the MCAT. However, in order to understand and use conservation of energy you must consider internal energy. 

Tidbits of InfD that may help your Internal energy is a state function. For an ideal gas, any state function can be ex- 

energy is independent of volume and is a function of temperature only. 

Tne zerutti law of thermodynamics just states tnanemperatuie exists, it s called the zeroth law because after the first, second, and third laws were already established It was realized that they depended upon a law that established the existence of temperature. 

The zeroth law of thermodynamics states Two systems in thermal equilibrium with a third system are in thermal equilibrium with each other. The zeroth law declares that two bodies in thermal equilibrium share a thermodynamic property, and that this thermodynamic property must be a state function. The thermodynamic property described by the zeroth law is temperature. 

For a stationary material the change in the internal energy is equal to the dilference between the net amount of heal added to the system and the net amount of work done by the system on its sujTOundings. For an infinitesimal change  

In this expression consistent units must be used. In the SI system each of the terms in equation 2.1 is expressed in Joules per kilogram (J/kg). In other systems either heat units (e.g. cal/g) or mechanical energy units (e.g. erg/g) may be used. dU is a small change in the internal energy which is a property of the system it is therefore a perfect differential. On the other hand, 8q and are small quantities of heat and work they are not properties of the system and their values depend on the manner in which the change is effected they are, therefore, not perfect differentials. For a reversible process, however, both 8q and can be expressed in terms of properties of the system. For convenience, reference will be made to systems of unit mass and the effects on the surroundings will be disregarded. 

8F is then a measure of the degree of irreversibility of the process. It represents the amount of mechanical energy converted into heat or the conversion of heat energy at one temperature to heat energy at another temperature. For a finite process  

When a process is isentropic, q = —F, a reversible process is isentropic when q = 0, that is a reversible adiabatic process is isentropic. 

The increase in the entropy of an irreversible process may be illustrated in the following manner. Considering the spontaneous transfer of a quantity of heat 6q from one part of a system at a temperature T to another part at a temperature Ti, then the net change in the entropy of the system as a whole is then  

We now derive an important expression for the internal energy of a liquid. Consider a system in the T, V, N ensemble and assume that the total potential energy of the interaction is pairwise additive, namely, 

The factor is included in (3.15) since the sum over i j counts each pair interaction twice. 

The first term on the rhs includes the internal and the kinetic energy of the individual molecules. For instance, for spherical and structureless molecules, we have q = A-3 and hence 

The second term on the rhs (3.17) is the average energy of interaction among the particles. This can be seen immediately by performing the derivative of the configurational partition function  

The average potential energy in (3.20), with the assumption of pairwise additivity (3.15), fulfills the conditions of the previous section hence, we can immediately apply theorem (3.5) to obtain 

Although we all have an intuitive sense of the word heat, thermodynamics provides a rigorous definition. Heat is a form of energy that flows as a result of differences in temperature. For example, when a cup of hot coffee is placed in a cool room, energy moves from the coffee to the room by heat because of the temperature difference. Work is energy that can be harnessed to do something. For example the expansion of a gas, the formation of an electrical current and injection of molten polymer into a mold are all examples of work being done. 

In experiments such as those conducted by Joule, energy is added to a fluid as work, but is transferred from the fluid as heat. What happens to this energy between its addition to and transfer from the fluid A rational concept is that it is contained in the fluid in another fonil, called internal energy. 

The internal energy of a substance does not include energy that it may possess as a result of its macroscopic position or movement. Rather it refers to energy of the molecules internal to the substance. Because of their ceaseless motion, all molecules possess kinetic energy of translation except for monatomic molecules, they also possess kinetic energy of rotation and 

We have attempted to introduce the concept of temperature as a thermodynamic parameter without any necessary reference to the concept of heat, although not necessarily avoiding that term. This was done by saying that diathermal walls exist tliat allow two bodies to interact energetically while excluding mechanical interaction and mass transfer. Since we limit forms of energy to work and heat (as discussed next), then if energy is transferred and work is eliminated, it follows that diathermal 

Energy can be defined as the capacity to do work, where work is a force moving through a distance, but at first this hardly seems very satisfying. What exactly is a capacity This is most readily visualized by considering various types of energy, and situations where energy transfers occur. 

The capacity to do work arises from three sources, giving three types of energy. 

Rest Energy, E, is energy a body has due to its mass. This source of energy was not understood until the introduction of the Special Theory of Relativity by Einstein in 1905. He postulated the essential equivalence of matter and energy, the two being related by 

Thus the total energy possessed by any macroscopic body or system is 

This book uses the symbol E sys for the energy of the system measured in an earth-fixed lab frame. If during a process the system as a whole undergoes motion or rotation relative to this lab frame, then its energy in the lab frame depends in part on coordinates that are not properties of the system. In such situations E sys is not a state function, and we need the concept of internal energy. 

The internal energy, U, is the energy of the system measured in a reference frame that allows U to be a state function—that is, at each instant the value of U depends only on the state of the system. This book will call a reference frame with this property a local frame. A local frame may also be, but is not necessarily, an earth-fixed lab frame. 

Here is a simple illustration of the distinction between the energy Esys of a system measured in a lab frame and the internal energy U. Let the system be a fixed amount of water contained in a beaker. The beaker itself is part of the surroundings. We can define the state of this system by two independent variables the temperature, T, and pressure, p, of 

Thermodynamics and Chemistry, second edition, version 3 2011 by Howard De foe. Latest version mnr.chem.vmid.edu/thennobook 

Our choice of the local frame used to define the internal energy U of any particular system during a given process is to some extent arbitrary. Three possible choices are as follows. 

Time You have written more than 45 books. What has driven you to be so prolific  

William F. Buckley The fear that the enemy wiU write more than I do. 

The first definition is that of internal energy. This is the energy within the boundaries of a systan. Because the boundaries can usually be drawn anywhere we want them to be, although drawing them some places may make the enclosed systan easier to deal with than drawing them differently, the amount of internal energy depends upon the locations of the boundaries. 

Energy is the capacity of the system to perform mechanical work. Heat can be transformed into work. So, in a physical sense, the amount of internal energy is changed by adding either heat or work. By convention, the internal energy of a system will increase either by delivering heat to it or by doing work on it (Haynie, 2001). The First Law of Thermodynamics expresses this by 

Internal energy is a state function of a system, which means that internal energy differences can be calculated by subtracting values at any two points without knowing about events happening between the two points. If such knowledge were necessary, it would be called a path function (as we have 

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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. 

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

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. 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

Translational -> internal energy transfer Surface excitation (phonon, electron)... 

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.)... 

Each hamionic temi in the Hamiltonian contributes k T to the average energy of the system, which is the theorem of the equipartition of energy. Since this is also tire internal energy U of the system, one can compute the heat capacity... 

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... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

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 -... 

Lindinger W and Smith D 1983 Influenoe of translational and internal energy on ion-neutral reaotions Reactions of Small Transient Species ed A Fonti]n and M A A Clyne (New York Aoademio)... 

A reactive species in liquid solution is subject to pemianent random collisions with solvent molecules that lead to statistical fluctuations of position, momentum and internal energy of the solute. The situation can be described by a reaction coordinate X coupled to a huge number of solvent bath modes. If there is a reaction... 

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)... 

Rynbrandt J D and Rabinovitch B S 1971 Direct demonstration of nonrandomization of internal energy in reacting molecules. Rate of intramolecular energy relaxation J. Chem. Phys. 54 2275-6... 

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