Changes in Entropy

To determine AGmix of a polymer/solvent mixture, expressions for the change in entropy, A5mix and change in enthalpy, need to be derived. 

The Boltzmann equation is generally used to obtain an expression for AS of simple mixtures (mixtures of solvent-solvent or solvent-simple solute molecules) from the number of different arrangements ft (or the thermodynamic probabilities) of the solute and solvent molecules in the system For simple systems, the volume elements of solution are modeled by a three-dimensional lattice, where solute or solvent molecules can occupy any cell within the 

An equation corresponding to equation (2.10) can be written for the polymer solution  

As the polymer can take many different conformations, fl2 1 both fli2 and fl2 have to be evaluated in order to determine A5mix. 

Consider the two-dimensional lattice shown in Fig. 2.4, where the cells within it can be occupied either by a single repeat unit of the polymer molecule or by a solvent molecule. The first segment of a polymer chain can be placed at any of the (Ax -f A2X) lattice positions. The second segment 

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Applied to a two-phase system, this says that the change in pressure with temperature is equal to the change in entropy at constant temperature as the total volume of the system (a + P) is increased, which can only take place if some a is converted to P ... 

Lewis acid and the ligand. By contrast, soft-soft interactions are mainly enthalpic in origin and are characterised by a negative change in entropy" ... 

The phenomenological definition for the change in entropy associated with the isothermal, reversible absorption of an element of heat dq is... 

Note that in arriving at (2.58) it was assumed that there was no change in entropy along the rarefaction. This assumption is equivalent to stating that the rarefaction must be propagating into a uniform state. The Riemann Invarient has been defined in terms of the Riemann function... 

Throttling The expansion of a fluid through a constricted passage (across which there is a pressure difference), during which no external work is done. The initial and final velocities of the fluid are equal, and there is no heat exchange with external sources. A change in entropy will, however, take place. 

Obtain the energies of formaldehyde and 1,3,5-trioxane. What is AH n for trioxane cracking Is the process endothermic or exothermic Is ASrxn for the cracking reaction likely to be positive or negative Explain. Given the direction of the change in entropy, should one crack trioxane at higher or lower temperatures Explain (use equation 1). 

For any reversible process, the sum of the changes in entropy for the system and its surroundings is zero. All natural or real processes are irreversible and are accompanied by a net increase in entropy. 

Alternatively, for an ideal reversible process, the sum of all the changes in entropy-must be zero or... 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

In fact, the mean-field estimate for H, given by equation 3.64, predicts a second-order transition. While abrupt changes in entropy characterize sudden transitions between regions of periodic and chaotic rules, smooth changes in entropy as A is increased instead suggest that the rule path sometimes passes through a region of... 

We have already pointed out that, in general, wAB will vary with temperature. In this respect Wab resembles the quantities D, L, Y, and J, all of which are sensitive to temperature. When, for any process, we differentiate (60) with respect to the temperature, we obtain the change in entropy AS for the process. Now at, all temperatures the unitary term in (60) is independent of the composition of the solution and obviously, if we differentiate it with respect to the temperature, the quantity so obtained will necessarily be independent of the composition of the solution, and so will provide a unitary term in AS. We must write then... 

In any process the change in entropy is, of course, equal to the temperature coefficient of the change in free energy, taken with opposite sign ... 

AS, the change in entropy (Section 17.2) a positive value of AS tends to make a reaction spontaneous. 

In the previous chapter, we saw that entropy is the subject of the Second Law of Thermodynamics, and that the Second Law enabled us to calculate changes in entropy AS. Another important generalization concerning entropy is known as the Third Law of Thermodynamics. It states that ... 

Coefficient of Expansion The change in entropy with pressure is related to the coefficient of expansion by... 

Application of the first law of thermodynamics permits us to relate the change in entropy with the work output from the chain ... 

T must be greater than T2 and d.S is therefore positive. If the process had been carried out reversibly, there would have been an infinitesimal difference between Ti and 1 and the change in entropy would have been zero. 

EXAMPLE 7.1 Sample exercise Calculating the change in entropy when a system is heated... 

A large flask of water is placed on a heater and 100. J of energy is transferred reversibly to the water at 25°C. What is the change in entropy of the water ... 

Self-Test 7.1A Calculate the change in entropy of a large block of ice when SO. J of energy is removed reversibly from it as heat at 0°C in a freezer. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

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