Lindemanns Hypothesis

Obviously, thermal vibrations of atoms in a solid are strongest on the verge of melting. Sutherland was the first (1891) to suggest that melting occurs when the amplitude of vibrations reaches a certain fraction (equal for all the elements) of the atomic size [13]. In 1910, Lindemann [14] developed this idea and related the critical amplitude to the temperature of melting (Tm) and atomic oscillation frequency v proportional to the characteristic Debye temperature ( ). In its modern form [15] the Lindemann s rule states that a material melts at the temperature at which the amplitude of thermal vibration exceeds a certain critical fraction of the interatomic distance, and this fraction depends somewhat on the crystal structure, position in the Periodic Table, and perhaps other unspecified physical quantities. These works initiated numerous 

It seems natural to calculate L-factors directly from the thermochemical characteristics of substance fusion, namely, from the sum of the enthalpy of substance preparation for melting AH = H Tjn) - H 0) and the enthalpy of melting proper Am//, e.g. from // = A//m + A//m. Since the energy of a harmonic oscillator is = Af, AR, where/ is the force constant and AR is the change in the bond length for E = Hm (when AR = hR), we obtain 

What is the physical meaning of the critical amplitude of atomic vibrations in a melting metal Evidently, bond lengths can be stretched only in a frameworks of a certain stable structure. As shown by Goldschmidt, the relative interatomic distances in the structures of metals with different coordination munbers change as follows  

the change of interatomic distances according to L-factor, exceeds by 13 to 15 % the limits of the stability of the metal structures at any phase transition that leads to the destruction of the crystalline order, i.e. to amorphization (melting) of the solid. 

It is evident that the distances are sharply elongated when helium is crystallized, while they remain constant in the molecules and crystals of other rare gases. The situation is similar in liquid rare gases Ac = 8.9 on average in the liquid state, whereas, in liquid helium, Ac = 4, and the bond length is longer by 6 % (cf. 0.3 % 

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The formulation of reaction rate theory used in the previous sections applies to bimolecular and higher-order reactions in general, but to unimolecular reactions only at high pressures. We shall, therefore, reconsider the problem of isotope effects in unimolecular gas reactions. We start with the recent elaboration of the Lindemann hypothesis given by Marcus.42... 

Figure 6.2 also makes another key quantitative point the hfetunes of energy-rich larger polyatomics are exceedingly long compared with the time scale of vibrational motion. As is consistent with the Lindemann hypothesis, the lifetimes are also longer than the time between collisions at ordinary pressures. This raises a question for a molecule that lives for so many vibrational periods, will it remember how it was formed ... 

According to the Lindemann-Christiansen hypothesis, formulated independently by both scientists in 1921, all molecules acquire and lose energy by collisions with surrounding molecules. This is expressed in the simplified form of the Lindemann mechanism, in which we use an asterisk to indicate a highly energetic or activated molecule, which has sufficient energy to cross the barrier towards the product side, and M is a molecule from the surroundings M may be from the same type as A ... 

According to Lindemann s hypothesis, the activation by collision can still give rise to first order kinetics if the activated molecules decompose only slowly compared to the rate at which they are deactivated. There is a time-log between the moment of activation and the moment of decomposition and in such a case, a stationary concentration of the activated molecules gets built up. Since the activated molecules will be in equilibrium with the normal molecules, their concentration will be proportional to that of normal molecules. The activated molecules disappear through two parallel processes, i.e. through deactivation and decomposition, represented as follows ... 

Letter from Perrin to Einstein, 28 August 1919 Einstein to Perrin, 5 November 1919, Einstein Collection, Institute for Advanced Study, Princeton (now housed in Jerusalem), cited originally in Nye, Molecular Reality, n. 93, 177. See F. A. Lindemann, "Note on the Significance of the Chemical Constant and Its Relation to the Behaviour of Gases at Low Temperatures," Phil.Mag. 39 (1920) 2125, cited in M. Christine King and Keith T. Laidler, "Chemical Kinetics and the Radiation Hypothesis," Archive for History of Exact Sciences 30 (1984) ... 

This problem was resolved in 1922 when Lindemann and Christiansen proposed their hypothesis of time lags, and this mechanistic framework has been used in all the more sophisticated unimolecular theories. It is also common to the theoretical framework of bimolecular and termolecular reactions. The crucial argument is that molecules which are activated and have acquired the necessary critical minimum energy do not have to react immediately they receive this energy by collision. There is sufficient time after the final activating collision for the molecule to lose its critical energy by being deactivated in another collision, or to react in a unimolecular step. 

We have seen that the radiation hypothesis is not supported by experiment and that it can not be used to explain the fact that unimolecular reaction rates are uninfluenced by collisions. When investigators found this avenue of explanation closed they resumed consideration of the collision hypothesis. As early as 1922 Lindemann suggested that since a time interval exists between activation, of a molecule and its dissociation the apparent connection between the two phenomena would ordinarily be lost. This view was received with increasing favor as the radiation hypothesis became more and more discredited. Rodebush7 in 1923 howed that the known facts could be explained on the basis of collisions... 

As the frequencies of the atoms are independent of the temperature, the influence of increasing temperature can only become apparent in solid bodies when the amphtudes of the vibrating atoms become considerable. A temperature will ultimately be reached at which neighbouring atoms come into contact at their maximum displacement from the centre of equihbrium. When this occurs, they will no longer return to their original position, and a fixed configuration in space becomes impossible, i.e. the sohd becomes a hquid. The melting point is, therefore, the temperature at which the amplitude of vibration of the atoms is equal to the mean distance between them. From this hypothesis, Lindemann deduces the formula. 

Tensile properties are obviously whole hber properties, as opposed to surface properties, and it has been suggested that they are cortical properties not related to the cuticle. This is because of experimental evidence and is, in part, due to the importance of the alpha to beta transformation that occurs on stretching [5]. Wolfram and Lindemann [6] have suggested that the cuticle does contribute to the tensile properties, especially in thin hair. However, Scott [7] has provided support for the no cuticle involvement hypothesis, by evaluating the tensile properties of hair hbers that were abraded under controlled conditions. In no instance could he demonstrate a signihcant change in tensile properties where only cuticle had been abraded. 

Recognition and understanding of this behavior is due to Lindemann and Hinshelwood (LH) and quantification is due to Rice, Ramsperger, and Kassel (RRK). According to the LH hypothesis for activation, collision of a reactant molecule. A, with one or more hot bath molecules, M, gives an activated reactant, A, which either reacts unimolecularly to give product, B, or collides with a cool bath molecule, M, and loses its energy (Eq. (1.1)). 

Lindemann s hypothesis, as later interpreted and developed by C. N. Hinshelwood (1897-1967) and others, is as follows. In a unimolecular process an energized molecule A is first formed by a collision between two molecules of the reactant A ... 

If this Lindemann-Hinshelwood hypothesis is correct, unimolecular gas reactions should be first order at high pressures and should become second order at low pressures. This behavior has now been confirmed for a large number of reactions. In its original form the hypothesis had some difficulty in interpreting results quantitatively, but a number of extensions of the original hypothesis have been made, notably by R. A. Marcus whose treatment is consistent with transition-state theory. 

Lindemann s Hypothesis Table 6.3 Critical amplitudes of atomic vibrations determined by different methods 339... 

Figure 5.4 sketches the potential energy curves for dissociation and isomerization in one dimension. In both cases the molecule has to gain sufficient energy to undergo reaction. The Lindemann-Christiansen hypothesis, formulated independently by both scientists around 1921, says the molecule acquires the necessary energy by collisions with other molecules, after which it can either lose its energy in a subsequent collision, or cross the transition barrier to form products. The process is represented schematically by... 

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