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  • Author: Herbert Zimmermann x
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Abstract

Ethidiumbromide (1) has two amino groups in 2-and 7-position which are protonated in acidic water solution. Both pKa-values of 1 are determined at 20 °C by means of the pH-dependence of the electronic spectra using a iterative calculating procedure, pKa1 = 0.713, pKa2 = 2.43. Acetylation of 1 and quantum mechanical calculations lead to the conclusion that the electronic density at the 7-amino group is greater than in 2-position. Therefore with decreasing pH preferably the 7-amino group is protonated (pKa2). followed by the protonation of the 2-amino group (pKa1).

The pKa of 7-amino-9-phenyl-10-ethyl-phenanthridinium-bromide in water solution at 20 °C is determined to pKa= 1 .2 5 .

Abstract

We consider the temporal evolution of an isolated system of N particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy, S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to dS/dt = 0: The entropy S does not change in time t. Thereby it is irrelevant, whether we consider a non-equilibrium state or an equilibrium state. Consequently, the system cannot irreversibly change by entropy production dS/dt > 0 from S′ to S max. This is a paradoxial result, which contradicts the experience. It can be traced back to the von Neumann equation, which in principle describes reversible processes and hence is unsuitable for calculating the irreversible evolution of the entropy S in time t. Each irreversible process is accompanied by a positive entropy production P = dS/dt ≥ 0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P, which is defined by the eigenvalue equation P|u〉 = P|u〉 with the state vector |u〉 of the N-particle system. There was an extensive discussion about the relation between the production of entropy P on one hand and the progression of time t on the other. Making use of this concept, we combine the operator of entropy production P with the time-development operator U(t, t 0) of the system and finally deduce the infinitesimal unitary operator U(t+τ,t) = 1 + (i/kP by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = tt 0 and k the Boltzmann constant, representing the atomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t|u〉 = t|u〉. We apply the infinitesimal time-evolution operator U(t+τ,t) to the operator of time t and finally obtain the Pt commutation relation i[P,t] = k, which is independent of τ. It shows that the operators P and t do not commute, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt on measuring P and t, which are given by the Pt uncertainty relation ΔPΔtk/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S′ to S max. Now, the irreversible steps are correctly described by the entropy production P = dS/dt > 0, while the thermal equilibrium is given by P = 0, ΔP = 0, and thus the lifetime of the equilibrium state Δt = ∞. According to the Pt uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible processes like Planck′s constant h to usual quantum mechanics.

Abstract

We consider a substance X with two monotropic modifications 1 and 2 of different thermodynamic stability ΔH1 < ΔH2. Ostwald´s rule states that first of all the instable modification 1 crystallizes on cooling down liquid X, which subsequently turns into the stable modification 2. Numerous examples verify this rule, however what is its reason? Ostwald´s rule can be traced back to the principle of the shortest way. We start with Hamilton´s principle and the Euler-Lagrange equation of classical mechanics and adapt it to thermodynamics. Now the relevant variables are the entropy S, the entropy production P = dS/dt, and the time t. Application of the Lagrangian F(S, P, t) leads us to the geodesic line S(t). The system moves along the geodesic line on the shortest way I from its initial non-equilibrium state i of entropy Si to the final equilibrium state f of entropy Sf. The two modifications 1 and 2 take different ways I1 and I2. According to the principle of the shortest way, I1 < I2 is realized in the first step of crystallization only. Now we consider a supercooled sample of liquid X at a temperature T just below the melting point of 1 and 2. Then the change of entropy ΔS1 = Sf 1 - Si 1 on crystallizing 1 can be related to the corresponding chang of enthalpy by ΔS1 = ΔH1/T. Now it can be shown that the shortest way of crystallization I1 corresponds under special, well-defined conditions to the smallest change of entropy ΔS1 < ΔS2 and thus enthalpy ΔH1 < ΔH2. In other words, the shortest way of crystallization I1 really leads us to the instable modification 1. This is Ostwald´s rule.

Abstract

We investigate the temporal evolution of the entropy S(t) of an isolated system from an initial non-equilibrium state i of entropy Si to the final equilibrium state of maximum entropy Smax > Si . The equations of classical thermodynamics and of phenomenological irreversible thermodynamics are unsuited for handling this problem. Hence, in order to determine S(t), we make use of Hamilton´s principle of mechanics and of the Euler-Lagrange equation of motion, which were adapted to thermodynamics. Now the Lagrangian F(S, P, t) depends on the entropy S, the entropy production P = dS/dt, and the time t. Application of variational calculus leads us straightforward to the time-dependent equations S(t) and P(t), which represent the geodesic line and the geodesic slope on a S,t diagram. Both equations agree with results of former calculations on the basis of quantum thermodynamics. The geodesic line S(t) describes the shortest way of the system from its initial state i to the final equilibrium state. This way is irreversible.

Es wurden arylsubstituierte 1.1′-Bis-imidazyle dargestellt durch Umsetzung der Natriumsalze entsprechender Imidazole mit Brom. 1.1′-Bis-imidazyle dissoziieren in Lösung in Radikale. Die Dissoziation wurde mit Hilfe der Absorptionsspektren gemessen. Die Gleichgewichtskonstanten, Dissoziations-Enthalpien und Entropien werden mitgeteilt und diskutiert.

Es wurde die Bewegung eines Teilchens untersucht, das sich in einem Doppelminimumpotential bewegt und durch Tunneleffekt von einer zur andern Mulde gelangen kann. Die Verweilzeiten des Teilchens in den Potentialmulden (a) und (b) wurden für symmetrische und unsymmetrische Doppelminimumpotentiale berechnet.

Zunächst wurden die Eigenlösungen eines Systems untersucht, bei dem ein Übergang des Teilchens zwischen den Mulden unmöglich ist. Dieses entkoppelte System läßt sich durch einen Hamilton-Operator ℋ0 beschreiben, der sich vom Hamilton-Operator ℋ des Doppelminimumproblems durch eine Teilchensperre T im Maximum der Potentialbarriere unterscheidet. ℋ0 setzt sich aus zwei kommutierenden Operatoren ℋ(a) und ℋ(b) zusammen, deren Eigenfunktionen jeweils nur über einen der Bereiche (a) oder (b) nichtverschwindende Amplituden aufweisen. Ein Eigenzustand von ℋ0 beschreibt somit ein Teilchen, das mit Sicherheit in einer der Mulden anzutreffen ist. Das gilt ebenso für eine beliebige statistische Mischung der Eigenzustände von ℋ(a)oderℋ(b). Damit kann für eine Gesamtheit von N0 Teilchen eine Anfangsbedingung formuliert werden, nach der zu einer bestimmten Zeit t0 sämtliche Teilchen z. B. in der Mulde (a) lokalisiert sind. Die zeitliche Änderung der statistischen Mischung wird nicht durch den Hamilton-Operator ℋ0, sondern durch ℋ bestimmt. Damit nimmt bei Anwendung des ℋ-abhängigen, unitären Operators U (t, t0) auf den statistischen Anfangszustand die Wahrscheinlichkeit W(b) (t), ein Teilchen für t &gt; t0 in der Mulde (b) zu finden, einen endlichen Wert an. Die Zeit t′, nach der W(b) (t) erstmal seinen zeitlichen Mittelwert W̄(b) annimmt, ist dann charakteristisch für die Teilchenbewegung zwischen den Mulden. Die wahrscheinliche Verweilzeit τ(a) eines Teilchens in der Mulde (a) ist dann näherungsweise zu τ(a) = t′/ W̄(b) gegeben. Die Beziehung stimmt im Falle eines symmetrischen Doppelminimumpotentials mit früheren Rechnungen überein. Im Falle unsymmetrischer Potentiale ist sie zur Bestimmung von Verweilzeiten besser geeignet als die quasiklassische Methode.