Boltzmann distribution equation

Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N, from which the emission line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

The calculation of the thermodynamic functions of a substance is based upon theuu Boltzmann distribution equation, which predicts the most probable distributionvv of molecules (or atoms) among a set of energy levels. The equation is... 

For the case of two spherical particles of radii a and a2, Stern potentials, iftdi and i//d2, and a shortest distance, H, between their Stern layers, Healy and co-workers195 have derived the following expressions for constant-potential, V, and constant-charge, Fr, double-layer interactions. The low-potential form of the Poisson-Boltzmann distribution (equation 7.12) is assumed to hold and Kax and xa2 are assumed to be large compared with unity ... 

In order to get from Equation 7.14 to the ideal gas law, we need to relate v2 to the temperature. As discussed in Chapter 4, we can use the Boltzmann distribution, Equation 4.26, to give the probabilities of observing different velocities ... 

A given population of free atoms exists at various electronic energy levels. The distribution of atoms in the energy levels is given by the Boltzmann distribution equation, i.e. 

Noting that the potential d>(jt) is symmetric with respect to the origin, the Boltzmann distribution (Equation 6.2) is integrated to give the numbers of small ions in the region R as... 

In the limiting case of small particle concentrations one can neglect the non-linear terms in equation (7.2.6) which gives us the simplest estimate of the correlation function for similar particles. At relatively long times it reaches the steady-state and gives just the Boltzmann distribution, equation (6.3.4), A (r, oo) exp(—[7aa( ))- The increasing in time deviation... 

Note that we now have Boltzmann distribution equations for each type of defect., and that the energy, E, to form the defect is like an activation energy. The fraction of defects present, either Ny / N or Ni / N, is an exponential function of this activation energy. 

It is advisable to look more closely at this exponential type of distribution law, since it occurs so frequently in physics and physical chemistry in a more general form as the Boltzmann distribution. Equation (2.36) is most informative in discussing the exponential distribution it can be written... 

The degree of excitation by a thermal source can be described by the Boltzmann distribution equation. If Ni is the number of atoms in the excited state and No is the number in the ground state, then the excited fraction is given by... 

The distribution of ions close to the surface (in the so-called diffused layer) is commonly given by the Boltzmann distribution equation (35,36),... 

The partition function approaches one at low temperatures and two at high temperatures. The relative populations of the two states are given by the Boltzmann distribution, Equation (10.9) ... 

Answer Focusing on the sodium D line at 589.3 run, we can calculate the energy gap as shown. The degeneracies for tiiis transition as 2 1, and the Boltzmann distribution (Equation 5.9) is used to calculate Ihe ratio and percentage of sodium atoms in the excited state at the temperatures given. Even at the highest temperature, less than 1 in 5 sodium atoms is in the excited state. This implies that, in flames, emission signals wiU be too low to be of practical value and that only in plasma will emission be feasible. 

The term In (gj/Nj) can be expressed in terms of the Boltzmann distribution, equation 17.20, if we take the logarithm of that equation. This introduces a term... 

The expression for Teiec embodies two competing trends of the solution phase potential at the reaction plane. An increase in solution phase potential results in a larger driving force for electron transfer in cathodic direction. This effect is proportional to the cathodic transfer coefficient c. At the same time, a more positive value of (y)- (l) - (po) corresponds to lower proton concentration at the reaction plane, following a Boltzmann distribution (Equation 3.77). The magnitude of this effect is determined by the reaction order yh+ K is> therefore, of primary interest to know the difference of kinetic parameters, - yh+ ... 

For example, consider a local Monte Carlo scheme for a Lennard-Jones liquid in the NVT (constant particle number, volume, and temperature) ensemble We choose one particle at random, and move it a fixed distance away from the previous position in an arbitrary direction. Eor the reverse move (from j to ij 0j = at the two prefactors cancel out. Pi oc exp[— 3 (i)] according to the canonical Boltzmann distribution. Equation (1.13) indicates that the move is always accepted if the energy of the system is lowered by the displacement. If the energy increases, the move is accepted with probability exp (— 3AE), that is, we draw a random number between 0 and 1 and accept the move if the random number is smaller than exp (—PA ). 

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