Maximum randomness

Fig. 7.4. Heterogeneous nucleation takes place at higher temperatures because the maximum random fluctuation of 10 atoms gives a bigger crystal radius if the atoms are arranged as a spherical cap.

As a guide with only rough experimental backing, the ratio of maximum random packing size to tower diameter is... 

The equilibrium state is a compromise between these two factors, minimum energy and maximum randomness. At very low temperatures, energy tends to be the more important factor. Then equilibrium favors the molecular substances with the... 

For each of the following reactions, state (0 whether tendency toward minimum energy favors reactants or products, (ii) whether tendency toward maximum randomness favors reactants or products. 

Why are these constants so different To see why, we must turn to the two factors that control every equilibrium, tendency toward minimum energy and tendency toward maximum randomness. 

The gaseous state is more random than the liquid state since the molecules move freely through a much larger space as a gas. Hence randomness decreases as a gas dissolves in a liquid. In this case, unlike solids, the tendency toward maximum randomness favors the gas phase and opposes the dissolving process. 

Thus we see that the equilibrium solubility of a gas again involves a balance between randomness and energy as it does for a solid, but the effects are opposite. For a gas, the tendency toward maximum randomness favors the gas phase, opposing dissolving. The tendency toward minimum energy favors the liquid state, hence favors dissolving. 

Fig. 10-3. Maximum randomness versus minimum energy—solubility of solids and gases.

Equilibrium in any reaction is determined by a compromise between tendency toward minimum energy f golf balls roll downhill ) and tendency toward maximum randomness. Reaction (29) and reaction (30) both involve increase in randomness since the regular solid lattice dissolves or melts to become part of a disordered liquid state. Both reactions produce ions. But reaction (29) proceeds readily at 25°Q whereas reaction (30) does not... 

We see (hat ° furnishes a basis for predicting the equilibrium state. In Chapter 9 and in subsequent chapters, equilibrium was treated in terms of two opposing tendencies—toward minimum energy and toward maximum randomness. What is the connection between ° and these two tendencies ... 

Consider two reactions for which ° shows that products are favored, one an exothermic reaction, and the other an endothermic reaction. For the exothermic reaction, when the reactants are mixed they are driven toward equilibrium in accord with the tendency toward minimum energy. Now contrast the endothermic reaction for which ° shows that equilibrium favors products. When these reactants are mixed, they approach equilibrium against the tendency toward minimum energy (since heat is absorbed). This reaction is driven by the tendency toward maximum randomness. 

Summarizing, ° measures quantitatively the difference between the tendency to minimum energy and tendency to maximum randomness under the standard state conditions. 

If the tendencies toward minimum energy and maximum randomness are in favor of opposite directions in a reaction, the reaction becomes an equilibrium reaction. 

In this reaction, the minimum energy is in favor of reactants and the maximum randomness is in favor of products. Thus, this is an equilibrium reaction. 

The minimum energy and the maximum randomness are both in favor of the products in the following reaction. 

Explain the tendency toward maximum randomness for the following reactions. 

The tendency toward maximum randomness is in favor of the side which has more gas particles. The tendency toward maximum randomness is in favor of products in reaction I. 

If the tendency toward minimum energy and maximum randomness both favor the same direction, the reactions are not reversible. Which of the following reactions is riQt an equilibrium reaction ... 

Since some spontaneous reactions are exothermic and others are endothermic, enthalpy alone can t account for the direction of spontaneous change a second factor must be involved. This second thermodynamic driving force is nature s tendency to move to a condition of maximum randomness or disorder (Section 8.13). 

The tendency of things to get "messed up" is common in everyday life. You may rake the leaves on your lawn into an orderly pile, but after a few windy days the leaves are again scattered randomly. The reverse process is nonspontaneous the wind never blows the randomly disordered leaves into a neatly arranged pile. Molecular systems behave similarly Molecular systems tend to move spontaneously to a state of maximum randomness or disorder. 

Why do systems tend to move spontaneously to a state of maximum randomness or disorder The answer is that a disordered state is more probable than an ordered state because the disordered state can be achieved in more ways. Suppose, for example, that you shake a box containing 20 identical coins and then count the number of heads (H) and tails (T). It s very unlikely that all 20 coins will come up heads that is, a perfectly ordered arrangement is much less probable than the totally disordered state in which heads and tails come up randomly. 

Precision measure of the maximum random error or deviation of a single observation. It may be expressed as the standard error or a multiple thereof, depending on the probability level desired. 

A simple expression for is obtained by taking p to be the probability that a primitive segment selected at random is at that moment a member of the surplus popula-ticm and by assuming that the simultaneous probabihties for segments on the same chain are uncorrelated. This st ulation of "maximum randomness produces directly the... 

This assumption of maximum randomness is admittedly crude, but in the absence of more detailed information we will use it when an explicit form for P is required. Other assumptions are possiUe. For example, in the primitive s ment model the surplus segments in each of the m + 1 possible local loops must ocoir in pairs in order to be selfcancelling. For that case the cmnbinatorial term K (mo) k (mo + m) /m (mo m/2)i for m even and o for m odd ( jpendix I). This leads to a different espresrion for P, but the approximation equations for large mo (Eqs. 52 and 53) have the same forms. 

Gr will arise merely as consequences of a Construction Principle (CP) and of the maximum randomness size-assignation procedure applied to simulate the network from these distributions. Second, a general Monte Carlo methodology will be implemented to represent three-dimensional (3D) networks where H, Co, C and Gr coexist. Third, a simulation of the condensation-evaporation characteristics of a fluid within the simulated 3D networks will be performed. 

A pair of tapered slots and hoops provide maximum randomness with minimal nesting. 

The probability that a sequence of interest is present in a library can be calculated as shown in Table 10.1. The optimal digestion of the target (e.g., partial digest with SamSA) to obtain maximum randomness of cloning is achieved when the most abundant insert size equals the vector capacity (Seed et al., 1982). A problem which may frequently occur is the change of representation of clones in the libraries during amplification. Some clones may be lost or become under-represented just because they reproduce slightly less rapidly. Independent libraries may then have to be screened. False positives also occur often, even if utmost care is taken to prevent contamination. 

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