Molecule average energy

Phonon vibration spectrum was determined from force constant k which was determined from dependence of the calculated molecule average energy on volume ( a3), i.e. from compressibility k d2Etot(a,T)lda2. The pressure in the system was determined conventionally as P(a,T) = -dF(a,T) / 8V One can determine the lattice constant a(T) for every value of (P,T) by numerical inversion of the dependence P a,T) => a(P,T) ... 

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

Experimental access to the probabilities P(E ,E) for energy transfer in large molecules usually involves teclmiques providing just the first moment of this distribution, i.e. the average energy (AE) transferred in a collision. Such methods include UV absorption, infrared fluorescence and related spectroscopic teclmiques [11. 28. 71. 72, 73 and 74]. More advanced teclmiques, such as kinetically controlled selective ionization (KCSI [74]) have also provided infonnation on higher moments of P(E ,E), such as ((AE) ). 

This expression is correct, but not useful because we need information about the excited states of the molecule. Such information is hard to come by. In the old days, we would try to make progress by taking some average energy difference Ae between the excited states and the ground state. We can then write... 

A measure of the average energy of the molecules of a substance. The heat intensity. 

FIGURE 6.17 The translational and rotational modes of atoms and molecules and the corresponding average energies of each mode at a temperature T. (a) An atom or molecule can undergo translational motion in three dimensions, (b) A linear molecule can also rotate about two axes perpendicular to the line of atoms, (c) A nonlinear molecule can rotate about three perpendicular axes. 

FIGURE 7.10 More energy levels become accessible in a lx>x of fixed width as the temperature is raised. The change from part (a) to part (b) is a model of the effect of heating an ideal gas at constant volume. The thermally accessible levels are shown by the tinted band. The average energy of the molecules also increases as the temperature is raised that is, both internal energy and entropy increase with temperature. 

Thermal energy also can be transferred between objects. When a hot object is placed in contact with a cold object, thermal energy flows from the hot object to the cold object until the two reach the same temperature. At the molecular level, there Is a decrease in the average energy of the molecules in the hotter object, so its temperature decreases. There is a corresponding increase in the average energy of the molecules in the cooler object, so its... 

The average energy of the molecules follows by logarithmic differentiation ... 

As an exercise we leave the reader to show that the average energy of a diatomic molecule in the gas phase at temperatures where only the vibrational ground state is populated equals Sk T/l. What is it at high temperatures ... 

Calculate the average energy of the molecule and give the limiting values at low and high temperature. 

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