Potentials energy

Potential energy is stored energy. Objects may have potential energy stored in terms of their position. A ball up in a tree has potential energy due to its height. If that ball were to fall, that potential energy would be converted to kinetic energy. 

Potential energy due to position isn t the only type of potential energy. Chemists are far more interested in the energy stored (potential energy) in chemical bonds, which are the forces that hold atoms together in compounds. 

Human bodies store energy in chemical bonds. When you need that energy, your body can break those bonds and release it. The same is true of the fuels people commonly use to heat their homes and run their automobiles. Energy is stored in these fuels — gasoline, for example — and is released when chemical reactions take place. 

In a potential system, the notion of potential energy can be introduced as a function of a 

In a system, first choose a state that we can arbitrarily admit as a point with zero potential energy (position Uo = 0). Further, suppose that we need to find an MP potential energy in another point of the system, which we assign as position 1 (i.e., find the value Uj). The potential energy of a system in position 1 is taken to be numerically equal to the work of the field force on transferring the system from position 1 to that position where the potential energy is chosen as zero  

If the field is potential, the work A q does not depend on the pathway 1-0. It characterizes the system in point 1 with respect to point 0. If one needs to define the potential energy in position 2, the work of the field force should be measured. Obviously, A20 = U2 andAj2 = C/2 -t/j. Because Aji 

Position 0 was chosen arbitrarily any point of the system can be accepted as the zero point. This signifies that the defined potential energy is accnrate to a constant valne C. This arbitrariness is not essential, since in the calculations of the difference of energy (refer, for instance, to eq. (1.4.25-27)) constants C are mutually canceled out. Also, the presence of the constant in the equation does not affect the derivative of the potential energy function in respect to the coordinates. 

The correlation obtained shows how one can determine the potential energy of a system at a certain position. There is no universal formula for such a calculation (as for the kinetic energy). Correlation (1.4.25) shows a way of determining the system s potential energy by calculating the force work which leads a system to the given zero point. 

FORMS OF ENERGY (PER UNIT MASS OF MATERIAL) 3.3.1. Potential energy 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Equation (1) Is the first term of a Taylor expansion valid for (6/T) 2rr. For the case of the non-existence of zero point energy, one predicts an isotope effect for the Pb/ ° Pb vapor pressure ratio two orders of magnitude larger than the prediction from Eq. (1) at 600 K. In addition the no zero point energy case predicts the Pb to have the larger vapor pressure at 600 K. 

Stem s estimate of the difference in vapor pressures of Ne and Ne at 24.6 K througih Eq. (1) led to the first separation of isotopes on a macro scale by Keesom and van Dijk ( ). The same theory, without the approximation (6/T) 1, was used by Urey, Brickwedde, and Murphy ( ) to design a Raleigh distillation concentration procedure to enrich HD in H2 five fold above the natural abundance level, which was adequate to demonstrate the existence of a heavy isotope of hydrogen of mass 2. 

It is important to look into the implications of Eq. (1) since the development of the quantum-statistical mechanical theory of Isotope chemistry from 1915 until 1973 centers about the generalization of this equation and the physical interpretation of the various terms in the generalized equations. According to Eq. (1) the difference in vapor pressures of Isotopes is a purely quantum mechanical phenomenon. The vapor pressure ratio approaches the classical limit, high temperature, as t . The mass dependence of the Isotope effect is 6M/M where 6M = M - M. Thus for a unit mass difference in atomic weights of Isotopes of an element, the vapor pressure isotope effect at the same reduced temperature (0/T) falls off as M 2. Interestingly the temperature dependence of In P /P is T 2 not 6X0/T where 6X.0 is the heat of vaporization of the heavy Isotope minus that of the light Isotope at absolute zero. In fact, it is the difference between 6, the difference in heats of vaporization at the temperature T from ( that leads to the T law. 

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The general analysis, while not difficult, is complicated however, the limiting case of the very elongated, essentially cylindrical drop is not hard to treat. Consider a section of the elongated cylinder of volume V (Fig. II-18h). The centrifugal force on a volume element is u rAp, where w is the speed of revolution and Ap the difference in density. The potential energy at distance r from the axis of revolution is then w r Apfl, and the total potential energy for the... 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

At distances far from the dipole, the length d becomes unimportant and the dipole appears as a point dipole. The potential energy for a point dipole in the held produced by a charge (Eq. VI-3) is... 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

Thus for equal-size spheres the force between them is just xn f/siab-siab(J ) d is directly related to the potential energy between two slabs [13]. This point is examined further in the problems at the end of the chapter. 

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

Fig. VI-5. The effect of electrolyte concentration on the interaction potential energy between two spheres where K is k in cm". (From Ref. 44.)...

From the data in Problem 13, calculate the interaction potential energy between... 

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

Examination of Fig. VII-4 shows that the mutual interaction between a pair of planes occurs once if the planes are a distance a apart, twice for those la apart and so on. If the planes are labeled by the index I, as shown in the figure, the mutual potential energy is... 

The density of dislocations is usually stated in terms of the number of dislocation lines intersecting unit area in the crystal it ranges from 10 cm for good crystals to 10 cm" in cold-worked metals. Thus, dislocations are separated by 10 -10 A, or every crystal grain larger than about 100 A will have dislocations on its surface one surface atom in a thousand is apt to be near a dislocation. By elastic theory, the increased potential energy of the lattice near... 

Fig. VIII-5. Schematic potential energy diagram for electrons in a metal with and without an applied field , work function Ep, depth of the Fermi level. (From Ref. 62.)...

The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

If the total energy associated with the state is equal to the potential energy at the equilibrium position, it follows that... 

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