The Boltzmann principle

We have seen that, strictly speaking, thermodynamics is not concerned with the atomic and molecular nature of matter, i.e., with microscopic properties. Instead, it deals with the interrelationships of macroscopic properties such as temperature, pressure, and work. 

A completely separate branch of science is statistical mechanics which is concerned with microscopic properties. This subject makes use of what quantum mechanics tells us about the energy levels of molecules, and allows us to calculate macroscopic properties on the basis of this information. The area of overlap between statistical mechanics and thermodynamics is known as statistical thermodynamics which allows us, for example, to calculate equilibrium constants for chemical reactions using the molecular properties obtained from quantum mechanics. 

A treatment of these matters is outside the scope of the present book, but there is one aspect which must be referred to, namely the way in which systems distribute themselves between different energy states. This is dealt with by the Boltzmann principle, which was developed about 1876 by the Austrian physicist Ludwig Boltzmann (1844-1906). 

Suppose that certain energy states Si, s, S3,.. Sj,. are possible for a molecule. The Boltzmann principle tells us that the nurfiber of molecules Nj in the jth state of energy is related to the total number of molecules N by the expression 

A modification to equation (5.211) is required if there is more than one state having a particular energy value g . If this is the case, the level is said to be degenerate and is assigned a statistical weight, gi, equal to the number of superimposed levels. Equation (5.211) is then modified to 

The state of the previously defined collection is constantly changing, such that over time the collection is distributed thereby its objects may find themselves in different states. The number of complexions is the number of distributions of objects between the different states that they are likely to take. Of all the possible distributions, there is one that is the maximum of the number of complexions. The Boltzmann principle states that the number of complexions corresponding to the most probably distribution type is almost equal to the total number of complexions and vice versa. The state of the collection therefore always corresponds to the maximum of complexions, which we call the thermodynamic probability or dominant probability. 

This means that the distribution curve of the number of complexions between the different states shows a sharp peak corresponding to the maximum containing almost all the states. 

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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... 

This equation can be derived from Equation IX-4 by including in the expression for the polarization the contribution due to preferential orientation of the permanent dipole moments mo in the field direction. The component of the dipole moment of a molecule in the field direction is mo cos 0, where 9 is the polar angle between the dipole-moment vector and the field direction, and the energy of interaction is —noE cos 0. The relative probability of orientation in volume element sin Qd8d (in polar coordinates) is given by the Boltzmann principle as sin 9d9dtj>. The average value of the component is hence... 

Weber in 1854 had attributed paramagnetism to the orientation of little permanent magnets in the substance (and diamagnetism to induced currents, as discussed above). A quantitative treatment was developed by Paul Langevin in 1895, by application of the Boltzmann principle. The theory is the same as for the orientation of electric dipoles (see App. IX). It leads to the equation... 

The Boltzmann principle states that the effects of mechanical history of a sample are linearly additive. This applies when the stress depends on the strain or rate of strain or alternatively, where the strain is considered a function of the stress or rale of change of stress. 

Given the effective potentials acting on all 1C possible rotamers of residue, i, the Boltzmann principle can be used to calculate the probability of a particular rotamer ... 

The Theory for Solutions of Uni-univalent Electrolytes. According to the Boltzmann principle the ionic distribution is a function of the ratio of the electrical energy to the thermal energy, such that in a volume, dV, around a selected ion there will be... 

Show that the form of the Boltzmann principle given in equation (2-45) reverts to the defining equation for the shear creep compliance, equation (2-9), when a sample, initially at rest, is subjected to an instantaneous increment of stress at t = 0, which is thereafter held constant. 

This will be done in two steps utilizing different forms of the Boltzmann principle. 

Starting off with equation (p), Appendix 1 of Chapter 2, an expression for the Boltzmann principle in Laplace space,... 

The Boltzmann principle will later be employed in two places in this book. On p. 269 it is used to give us the average number of positive and negative charges in a volume element close to an ion. Suppose that the average concentration of positive... 

The second application of the Boltzmann principle in this book is in relation to chemical kinetics (p. 388). It can be shown from (5.212) that in any system the fraction f of molecules having energy m excess of a specified value E (per mole) is equal to... 

Suppose that the average electric potential in the volume element dV is 4>. The work required to bring a positive ion of charge z+e from infinity up to this point is then z+e. (Note that in this derivation z+ and z- are taken to be the numerical values of the valences of the ions e.g., for Na", z+ == 1 for CP, z- = 1.) According to the Boltzmann principle, the time-average numbers of positive and negative ions present in the volume element are (see Section 5.15, especially equation (5.216)). 

As discussed earlier for a Hookean solid, stress is a linear function of strain, while for a Newtonian fluid, stress is a linear function of strain rate. The constants of proportionality in these cases are modulus and viscosity, respectively. However, for a viscoelastic material the modulus is not constant it varies with time and strain history at a given temperature. But for a linear viscoelastic material, modulus is a function of time only. This concept is embodied in the Boltzmann principle, which states that the effects of mechanical history of a sample are additive. In other words, the response of a linear viscoelastic material to a given load is independent of the response of the material to ary load previously on the material. Thus the Boltzmann principle has essentially two implications — stress is a linear function of strain, and the effects of different stresses are additive. 

Let us illustrate the Boltzmann principle by considering creep. Suppose the initial creep stress, Oo= on a linear, viscoelastic body is increased sequentially to 0, 02- -On at times tj, t2...t , then according to the Boltzmann principle, the creep at time t due to such a loading history is given by... 

If droplets of liquid are to form in the midst of vapour, or minute crystals in the midst of liquid, they must grow from nuclei which, in the first instance, have to be produced by the chance encounters of molecules with appropriate velocities and orientations. The incipient nuclei are subject to two opposing influences. In virtue of the attractive forces, they tend to grow, and in virtue of the thermal motion they tend to disperse. These tendencies exist at all temperatures, but above the point of condensation or crystallization they come into balance while the agglomerates are still few, minute, and transitory. The existence of the nuclei amounts to no more than an increased probability of finding small groups of molecules closer together than they would be in the absence of attractive forces, and is a direct consequence of the Boltzmann principle. 

Now suppose the compression from A to occur in complete absence of liquid. This time the point B possesses no special significance. From JB to X the average aggregates of molecules in the vapour (formed in conformity with the Boltzmann principle) are too small to grow. Although beyond B the free energy per unit mass of vapour... 

There is indeed a very general rule, known as the Boltzmann principle. It states the following. Suppose there are Q, ways in which molecules can occupy a certain state. (In our case, this number is proportional to the probability P(R) — see Equation (6.16)). Then we need to find the quantity... 

What exactly does the Boltzmann principle (7.2) mean Its main idea is that the quantity f/gfi = —TS defined by (7.3) and (7.2) can be regarded as some sort of potential energy. Indeed, if the system is left to itself, it is most likely to drop down into the most likely state (sorry for this tautology ) According to (7.2) and (7.3), this would mean an increase in entropy, and hence a decrease in Ugg, which is just what the principle of minimum potential energy predicts. 

According to the Boltzmann principle, we obtain then entropy proportional to L/X. Transforming this to the function of end-to-end distance R (which is geometrically related to A in the same way as R was related to for freely jointed chain, i.e., R L LD /2X, or given formula (7.38), R L LX/t ), we arrive at the free energy... 

The first term in this equation can be evaluated by use of the Boltzmann principle, with the assumption that the substance contains permanent magnetic dipole moments that can orient themselves in the magnetic field. This theoretical treatment was carried out by the French scientist Paul Langevin in 1905. He derived the equation... 

A typical strain-time diagram at constant stress is shown in Fig. 2-6a. As we indicated before, the strain at any time is the sum of three different mechanisms which can be added by the use of the Boltzmann principle to give the resultant strain. 

The above illustrates one consequence of the Boltzmann principle, viz. that the additional creep e c(t — ti) produced by adding the stress Oq is identical with the creep that would have occurred had this stress ao been applied without any previous loading at the same instant in time ti. 

Chapter 4). If not, or if the load is varying in a more complex manner, a more complete analysis of deflection behaviour based upon the Boltzmann principle may be necessary. Linearity can be assumed for strains up to about 0.5 x 10 . ... 

We are now going to apply the Boltzmann principle by trying to find in which case the number of complexions Q or InQ is at its maximum. 

When all is said and done, probably the best definition of a linear material is simply one that follows Boltzmann s principle. Thus, spring-dashpot models, which are linear, automatically follow Boltzmann s principle. However, it is important not to infer a dependence of the Boltzmann principle on spring-dashpot models. The Boltzmann principle applies to a hnear response regardless of whether it can be described with a spring-dashpot model. All that is needed are experimental GXt) or Jdt) data. Models are used here simply as a matter of convenience to illustrate application of the principle. 

Analyze the dynamic properties of a Voigt-Kelvin element, that is, obtain G and G" in terms of model parameters G and tj and the frequency w. Hint Unless you are a masochist, do not use the Boltzmann principle here. Just examine the response to a sinusoidal strain. 

Derive the expression for v of a gas molecule using the Boltzmann principle. 

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