Kinetic energy of a molecule

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

Thermal neutrons, or neutrons in thermal equilibrium with the substance in which they exist most commonly, neutrons of kinetic energy about 0 025 eV, which is about of the mean kinetic energy of a molecule at 15°C. 

The basis of the application of group theory to the classification of the normal vibrations of a molecule lies in the fact that the potential and kinetic energies of a molecule are invariant to symmetry operations. A symmetry operation is a physical transformation of the molecule, such as reflection in a mirror plane of symmetry or rotation through 120° about... 

The well-known Maxwell-Boltzmann distribution for the velocity or momentum associated with the translational motion of a molecule is valid not only for free molecules but also for interacting molecules in a liquid phase (see Appendix A.2.1). The average kinetic energy of a molecule at temperature T is, accordingly, (3/2)ksT. For the molecules to react in a bimolecular reaction they should be brought into contact with each other. This happens by diffusion when the reactants are dispersed in a solution, which is a quite different process from the one in the gas phase. For fast reactions, the diffusion rate of reactant molecules may even be the limiting factor in the rate of reaction. 

All the F(Z.ys which one uses in the kinetic theory of gases to approximate the exact values of continuous phase functions are also in this category. Let us denote by i the kinetic energy of a molecule when its phase point is exactly at the center of cell < >< in the p-space. Since very small total kinetic energy of all a< molecules whose phase point lies in the cell au can be approximated very well by... 

From MaxwelFs distribution of velocities, either in the form (2.4) or (2.6), we can easily find the moan kinetic energy of a molecule at temperature T. To find this, we multiply the kinetic energy p2/2m by the fraction of molecules in a given range dpx dpy dpgy and integrate over all values of momenta, to get the weighted mean. Thus we have... 

Q The average kinetic energy of a molecule is proportional to the absolute temperature. 

Postulate 5 states that the average kinetic energy of a molecule is proportional to the absolute temperature. This also means that the kinetic energy of a sample of molecules will be proportional to the absolute temperature. Equation 8.15 that describes this behavior is ... 

The mechanism of thermal conduction in a gas is a simple one. We identify the kinetic energy of a molecule with its temperature thus, in a high-temperature region, the molecules have higher velocities than in some lower-temper-... 

The rate at which the momentum transfer takes place is dependent on the rate at which the molecules move across the fluid layers. In a gas, the molecules would move about with some average speed proportional to the square root of the absolute temperature since, in the kinetic theory of gases, we identify temperature with the mean kinetic energy of a molecule. The faster the molecules move, the more momentum they will transport. Hence we should expect the viscosity of a gas to be approximately proportional to the square root of temperature, and this expectation is corroborated fairly well by experiment. The viscosities of some typical fluids are given in Appendix A. 

From the kinetic theory of gases we know that the average translational kinetic energy of a molecule of an ideal gas is given... 

Let s examine this equation in two ways. First, we note that the kinetic energy of a molecule of mass m moving at speed u is equal to j mu, so the erage kinetic energy of molecules (1 mol), which we denote by E, is Nppiu. This quantity is exactly the same as that in the left side of Equation 9.12, with the factor j replacing... 

In order to find the equation of motion of the atoms in a molecule we need to express the kinetic and potential energies as a function of the atomic coordinates. The coordinates that we shall use describe the displacements of the atoms from their equilibrium positions, . Here ui), (ui)y, (ui) 2 are the magnitudes of displacements of the atom I in a molecule from its equilibrium position, referred to the Cartesian frame. Using the time derivatives of these coordinates, we can write the kinetic energy of a molecule, E]f° containing Adatom atoms with masses mi. 

In addition to translation, molecules can possess other kinds of motion. Because a chemical bond acts as a kind of spring, the two atoms in H2 will have a natural vibrational frequency. In more complicated molecules, many different modes of vibration become possible, and these all contribute a vibrational term KEvib to the total kinetic energy. Finally, a molecule can undergo rotational motions which give rise to a third term KErot. Thus the total kinetic energy of a molecule is the sum... 

We shall now make particular application of the transport equation to the three phenomena mentioned above. We begin with heat conduction. Here A stands for the kinetic energy of a molecule, i.e. idn = const. -f where c m is the specific heat of the gas at... 

Surface energy is defined as the increase in energy per unit area of the surface in an isothermal change. According to the classical theory, the mean value of the kinetic energy of a molecule is the same whether it is in the interior or on the surface of the liquid. Therefore we have only to calculate the increase in the total potential energy of the system which results from an increase of. the area of the surface. 

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