Probability distribution Boltzmann

Fig. III. 16. In light symmetric top molecules with reasonably large electric dipole moments such as for instance methylfluoride the change of the absorption spectrum due to the translational Zeeman effect occurs at comparatively low perpendicular velocities. The spectrum shown here corresponds to the absorption of a group of molecules moving at 267 m/sec (maximum of the Maxwell-Boltzmann probability distribution) perpendicular to the magnetic field. The dotted line gives the spectrum calculated neglecting the translational Zeeman effect. The Lorentz cross field has caused considerable mixing of Mj substates resulting in considerable changes in the selection rules...

The partition function can be considered as an average excited state number-operator, since it is the probability-weighted sum of energy states, each counted with a factor of 1. It may also be viewed as the normalization factor for the Boltzmann probability distribution. 

All possible contacts have been taken into account, and flie normalized Boltzmann probability distributions, as well as the Shannon information entropies, have been calculated for each pair of the molecules (i.e., AE, BE, CE, DE) as functions of temperature T within flie range from 100 K to 400 K. The results (Fig. 15.9) show that flie molecule A (wifli flie simplest structure) does not recognize molecule E at any temperature T under study. The ignorance (Shannon s entropy) of A with respect to E is of the order of 5 bits. Transforming molecule A into B improves the... 

All static properties can be calculated from the Boltzmann probability distribution (eqn [12]), which is in this case Gaussian ... 

The probability distribution for molecular velocities is the Maxwell-Boltzmann probability distribution ... 

Eor N2 molecules at 298.15 K, calculate the probability that V exceeds the speed of light, 2.997 x 10 ms according to the Maxwell-Boltzmann probability distribution. 

We now assert without proof that the Boltzmann probability distribution holds for this total energy ... 

The first model system designed to represent a dilute gas consists of noninteracting point-mass molecules that obey classical mechanics. We obtained the Maxwell-Boltzmann probability distribution for molecular velocities ... 

If we identify Sa as a minimum energy required to break out of a cage (a molecular activation energy), then from the Boltzmann probability distribution of Eq. (9.3-42) we see that the probability for a molecule to have this energy is... 

Arrhenius postulated that only activated molecules (those with high energy) can react and that the numbers of such activated molecules would be governed by the Boltzmann probability distribution of Eq. (9.3-41). This assumption leads to the Arrhenius relation ... 

The empirical Arrhenius formula for the temperature dependence of elementary rate constants was presented. This empirical formula was based on an idea that activated molecules with high energy are necessary for the reaction to occur and that the population of molecules with a characteristic activation energy is given by the Boltzmann probability distribution. We presented the collision theory of bimolecular reaction rates, using first the assumption that all collisions with a relative kinetic energy greater than a critical value would lead to reaction. 

The population of molecular states is governed by the Boltzmann probability distribution. 

So far in this chapter we have studied the quantum-mechanical states of isolated atoms or molecules. In a dilute gas, the molecules do not significantly interfere with each other, and we can apply these states to the individual molecules. However, all molecular states will not be occupied by the same numbers of molecules in a dilute gas at equilibrium. The Boltzmann probability distribution gives the probability of a molecular state of energy e in a system at thermal equilibrium ... 

In Chapter 9 there is a derivation of the Boltzmann probability distribution for classical dilute gases. There is a derivation of this probability distribution for a quantum dilute gas in Part 4. For now, we introduce it without derivation. The important fact about the Boltzmann probability distribution is that states of energy much larger than k T are... 

The states of polyatomic molecules are governed by the same Boltzmann probability distribution as those of atoms and diatomic molecules. The rotational levels of polyatomic molecules are generally large enough that many rotational states are occupied. The rotation of a linear polyatomic molecule such as acetylene or cyanogen is just... 

Write a computer program to calculate the relative intensities of the spectral fines in the fundamental band of the vibration-rotation spectrum of a diatomic molecule, assuming that the absorbance is displayed in the spectrum. Set the maximum absorbance of the first line of the P branch equal to 1. Assume the Boltzmann probability distribution and assume that the transition dipole moments for all transitions are equal. Use your program to calculate the relative intensities for the first 15 lines in each branch of the HCl spectrum... 

Table 25.2 Average, Most Probable, and Boltzmann Probability Distributions for the Vibrational States of Four Harmonic Oscillators...

Obtain the Boltzmann probability distribution for our model system that corresponds to an average energy per oscillator of hv. 

The Boltzmann distribution of Eq. (22.5-1) is not normalized. The normalized Boltzmann probability distribution is given by... 

Note that the Boltzmann probability distribution is not restricted to the states that we have listed, so that the upper limit of the sum is infinite. The formula for z is the sum of a geometric progression, a well-known sum ... 

In this equation, now represents the flux of a single incident particle. It is usually the case, however that the scatterer is not initially in one of its pure eigenstates. Instead, the target is normally in thermal equilibrium at some known temperature T. In this case, the scatterer is in a mixed quantum state and it is necessary to perform a summation over possible initial states, j), weighted according to the Maxwell-Boltzmann probability distribution. Pi = exp(- /fegT)/2y exp(- y/fegT), where is... 

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