Motions and Energy

The particles of an ideal gcis cire idealized points with no volume and no bonds, however, points that we visualize cis flitting ciroimd through space. Any real molecule can undergo three kinds of more complex motion. The entire molecule can move in one direction, which is the simple motion we visualize for an ideal particle and see in a macroscopic object, such as a thrown baseball. We call such movement translational motion. The molecules in a gas have more freedom of translational motion than those in a liquid, which have more freedom of translational motion than the molecules of a solid. 

A real molecule can also undergo vibrational motion, in which the atoms in the molecule move periodically toward and away from one another, and rotational motion, in which the molecule spins about an axis. T figure 19.8 shows the vibrational motions and one of the rotational motions possible for the water molecule. These different forms of motion are ways in which a molecule can store energy. 

The vibrational and rotational motions possible in real molecules lead to arrangements that a single atom cannot have. A collection of real molecules therefore has a 

Describe another possible rotational motion for this molecule. 

Chemists have several ways of describing an increase in the number of microstates possible for a system and therefore an increase in the entropy for the system. Each way seeks to capture a sense of the increased freedom of motion that causes molecules to spread out when not restrained by physical barriers or chemical bonds. 

---

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant flmd properties, the equations of motion must be solved before the energy equation is solved. If flmd properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. Con-... 

More involved analyses for circular tubes reduce the equations of motion and energy to the form... 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

Boiling at a heated surface, as has been shown, is a very complicated process, and it is consequently not possible to write and solve the usual differential equations of motion and energy with their appropriate boundary conditions. No adequate description of the fluid dynamics and thermal processes that occur during such a process is available, and more than two mechanisms are responsible for the high... 

Motch, C. Zavlin, V. Haberl, F. (2003), The proper motion and energy distribution of the isolated neutron star RX J0720.4-3125 , A A 408, 323. 

Hence, when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise... 

If a = 0, the non-Newtonian viscosity is temperature independent and the equations of motion and energy can be solved independently from each other if, however, a/0, they are coupled. Next, we assume that viscous dissipation is negligible Br —> 0, and that the moving plate at velocity Vo is T and the lower stationary plate is Tq. The equations of motion and energy reduce to... 

There is a close connection between molecular mass, momentum, and energy transport, which can be explained in terms of a molecular theory for low-density monatomic gases. Equations of continuity, motion, and energy can all be derived from the Boltzmann equation, producing expressions for the flows and transport properties. Similar kinetic theories are also available for polyatomic gases, monatomic liquids, and polymeric liquids. In this chapter, we briefly summarize nonequilibrium systems, the kinetic theory, transport phenomena, and chemical reactions. 

The ideas that are outlined in a qualitative v e/ above can also be cast into a useful mathematical form for computer calculation. The basic idea is to write down a (fairly simple and approximate)function that gives the energy of the system as a function of the positions (or coordinates) of its atoms. Because the derivative (or gradient) of this function yields the forces for Newf on s equations, such a function is often called a "force field" and because molecules are viewed as being made up of balls and springs (so that quantum effects are ignored), the term "molecular mechanics" is used to represent a concrete, mechanical picture cf molecular motions and energies. 

The equations of continuity, motion, and energy in vector notation are written below ... 

The kinetic-molecular theory explains the properties of gases in terms of the size, motion, and energy of their particles. 

The MO approach to molecular energy and other properties is fundamentally different than that of MM. In MM we assume that nuclei move and electrons are essentially stationary, that is, they are not explicitly considered in the force field calculations. Most MO approaches use the Bom-Oppenheimer approximation, which considers nuclei relatively stationary compared with fast-moving electrons. Thus, the MO approach must somehow address electron motion and energy. Because the motion of electrons is governed by the Heisenberg Uncertainty Principle, quantum mechanical rather than classical physical calculations must be used. 

The Schrddinger equation describes the motions and energies of submicroscopic particles. This equation launched quantum mechanics and a new era in physics. 

For an exothermic reaction AH ys < 0), heat is released by the system, which increases the freedom of motion and energy dispersed and, thus, the entropy of the surroundings (AA urr > 0)-... 

We will derive first the equations that govern the fluid density, temperature, and velocities in the lowest layer of the atmosphere. These equations will form the basis from which we can subsequently explore the processes that influence atmospheric turbulence. In our discussion we shall consider only a shallow layer adjacent to the surface, in which case we can make some rather important simplifications in the equations of continuity, motion, and energy. The equation of continuity for a compressible fluid is... 

In the equations of motion and energy (16.39), the dependent variables n p, and 0 are random variables, making the equations virtually impossible to solve. We modify our goal... 

Let us assume that at some initial time t = 0 relative to the adiabatic atmosphere. Then, from (16.A. 14) we see that if Q = 0, the condition of T = 0 is preserved for t > 0 even though there may be motion of air. Also, the equation of motion (16.A.11) reduces to the usual form of the Navier-Stokes equation for the dynamics of an incompressible fluid under the influence of a motion-induced pressure fluctuation p with no contribution from buoyancy forces since T = 0. Therefore, for an atmosphere with no sources of heat and initially having an adiabatic lapse rate, the temperature profile is unaltered if the atmosphere is set in motion. As a result, the adiabatic condition can be envisioned as one in which a large number of parcels are rising and falling, a sort of convective equilibrium. Thus, we have been able to derive the relation for the adiabatic lapse rate here from the full equation of continuity, motion, and energy, in contrast with the derivation presented in Section 16.1.1, which is based on thermodynamic arguments. 

---