Entropy state function

While the first law allows us to calculate the energy change associated with a given process, it says nothing about whether or not the process itself will take place spontaneously. This is the province of the second law of thermodynamics and leads to the introduction of another state function, entropy, S. The entropy change in a system which moves from state 1 to state 2 is defined by... 

The thermodynamic state function entropy, S, is a measure of the disorder of the system. The greater the disorder of a system, the higher is its entropy. For any substance, the particles are more highly ordered in the solid state than in the liquid state. These, in turn. 

Equation (5.25) suggests that the ratio between the heat (received or rejected) and the temperature might be a system property that might characterise the reversibility of the heat exchange in a cyclic process. By more rigorous reasoning, this observation leads to a new state function, entropy, defined as ... 

Note that thermodynamic interpretation of e is essentially different for compressible and incompressible fluid. Compressible fluid can be considered as a two-parametric system. According to the second law of thermodynamics, there is a state function - entropy s, playing the role of a thermodynamic potential. If e and specific volume 1 jp are taken as independent parameters, then the equation of state of compressible gas will be s = s(e, 1/p), and the perfect differential of the entropy is... 

Such a form of entropy inequality (1.42) and likewise the energy balance (1.5) will be used (in fact by further simplifications) in Chap. 2 where uniform systems without space gradients are treated The process is a time sequence of the states and we may expect the validity of (1.5), (1.42) for arbitrarily close time instants. Therefore we formulate these basic laws for the rate (time derivative) of the state functions (entropy, energy) with heatings (rate of heat exchange) and power, cf. (2.1),... 

Finally, as a state function, entropy can be tabulated. The steam tables in the appendix include the specific entropy of water in the saturated region (liquid, vapor) and in the superheated vapor region. If tabulated values are not available, then entropy changes maybe calculated from eg. (4.8). The calculation requires the amount of heat that is exchanged along the path and this is generally obtained by application of the first law. This methodology is applied below to a number of special cases. 

When AH is negative, the reaction is exothermic, whereas a positive value of AH points to an endothermic reaction. What can we say about a reaction s spontaneity based on its enthalpy change If we were to stop and list spontaneous processes that we observe around us and then determine whether those processes are exothermic or endothermic, chances are that a majority would be exothermic. This implies that there is some relationship between enthalpy and spontaneity. The relationship is not exclusive, however. If you think for a moment you should be able to point out some endothermic reactions that obviously occur spontaneously. The melting of an ice cube at room temperature is one simple example. So at this point we might conclude that exothermic reactions seem to be preferred in some way. But clearly there must be things other than energy or enthalpy at work in determining whether or not a process is spontaneous. To develop a way to predict the spontaneity of a reaction, we must first introduce an additional thermodynamic state function—entropy. 

To predict spontaneity, we introduced two new concepts and state functions entropy and free energy. We can define entropy in either of two ways as the ratio of the heat flow to the temperature or as a measure of the number of ways that a system can have the same energy. This latter definition, for practical purposes, is a measure of the extent... 

Then (Sq/T) becomes a state function called entropy and T the absolute temperature. As a state function, entropy is path-independent. Eqn (1.25) is a mathematical statement of the second law of thermodynamics. The introduction of the integrating factor for 8q causes the thermal energy to be split into an extensive factor S and an intensive factor T. Clausius defined the entropy with the integrating factor of the inverse of absolute temperature in T 8q) = dS. Similarly, integrating factor 1/P in IP 6W) = dV leads to exact differential dV, which is formulated by Clapeyron in 1834. Introducing Eqn (1.25) into the first law of thermodynamics dU =8q + yields the combined first and second laws of thermodynamics... 

At the third stage the equilibrium thermodynamics was created by Clausius, Helmholtz, Boltzmann and Gibbs. Since that time the equilibrium principles started to develop as applied to macroscopic systems of any physical nature. The main, second law of thermodynamics was discovered by Clausius (Clausius, 2008). He found out the existence of the state function, entropy (S), that can change in the isolated systems exclusively towards increase. The inequality that shows such monotonicity of change... 

In this chapter, we have introduced a new state function entropy. It will have a unique impact on our study of thermodynamics. It is not an energy, like internal energy or enthalpy It is a different kind of state function, a different quantity. One way to think of it, as introduced by Boltzmann, is as a measure of the number of states available to a system. 

In applying the first law, two important state functions have been introduced and defined, namely the internal energy U and the enthalpy H. The second law introduces and defines a new fundamental state function entropy S. Before introducing this new state function, however, it is necessary to define some previously mentioned concepts. 

The first law of thermodynamics was connected with the definition of a state function internal energy U. Similarly, the second law leads to the definition of a state function entropy S. 

As formulated by Clausius, the Second Law introduces another state function, entropy (symbol S). Since entropy is a state function, its nmnerical value is determined solely by the state of the system and not by how that state is reached. Nevertheless, the prescription to calculate entropy depends on one particular path. According to the mathematical statement of the second law ... 

The feasibility of a process should be tested against both, the first and the second law. To this purpose the latter introduces the state function entropy, defined by Eq.3.10.1, and requires that Eq.3.11.1 be satisfied. Note that the entropy change of both, system and surroundings, should be considered. 

The second law refers to the state function entropy (M= S). In contrast to the internal energy, entropy can certainly be created inside the system, and this entropy creation is always positive. In fact, it only disappears at equilibrium ... 

There exists a state function S, called the entropy of a system, related to the heat Dq absorbedfrom the surroundings during an infinitesimal change by the relations... 

The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T andp) are specified, for example. Likewise,... 

Entropy S like internal energy, volume, pressure, and temperature is a fundamental property of a system. As such, it is a function of the state of the system and a state function so that... 

The significance of this relationship is that although qx and q2 are not state functions, q/T is a state function, since the sum of the q/T terms in the cycle add to zero. This important observation led to the formulation of the entropy function. 

The implication of equation (2.37) is that 8qre /T is the differential of a state function. This state function is the entropy S, with the differential dS given byq... 

Equation (2.66) indicates that the entropy for a multipart system is the sum of the entropies of its constituent parts, a result that is almost intuitively obvious. While it has been derived from a calculation involving only reversible processes, entropy is a state function, so that the property of additivity must be completely general, and it must apply to irreversible processes as well. 

The solution surfaces cannot intersect. If they did, states located at the points of intersection would have multiple values of entropy, and this would violate a fundamental property of state functions. Thus, the surfaces can be expected to be ordered monotonically, either systematically increasing (or decreasing) as one proceeds in a given direction from surface to surface. For our purposes, let us assume Si >S2>S3 in Figure 2.12. 

A very important characteristic of entropy that is not immediately obvious from Eq. 1 but can be proved by using thermodynamics is that entropy is a state function. This property is consistent with entropy being a measure of disorder, because the disorder of a system depends only on its current state and is independent of how that state was achieved. 

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. 

Entropy is a measure of disorder according to the second law of thermodynamics, the entropy of an isolated system increases in any spontaneous process. Entropy is a state function. 

To calculate a change in entropy for a process we find a reversible path between the initial and final states. It is immaterial whether the actual process is irreversible or reversible. Because entropy is a state function, the change for that path will be the same as that for the irreversible path. 

STRATEGY Because the entropy is a state function, we can calculate the change in entropy by choosing a reversible path that results in the same final state. In this case, consider the following ... 

We can break the overall calculation down in this way because entropy is a state function. For an example of this type of calculation, see Exercise 7.39. 

STRATEGY Because entropy is a state function, the change in entropy of the system is the same regardless of the path between the two states, so we can use Eq. 3 to calculate AS for both part (a) and part (b). For the entropy of the surroundings, we need to find the heat transferred to the surroundings. In each case, we can combine the fact that AU = 0 for an isothermal expansion of an ideal gas with AU = w + q and conclude that q = —tv. We then use Eq. 4 in Chapter 6 to calculate the work done in an isothermal, reversible expansion of an ideal gas and Eq. 9 in this chapter to find the total entropy. The changes that we calculate are summarized in Fig. 7.21. 

Calculate AU and AS for this entire cycle, (b) What are the values of q and w for the entire cycle (c) What are A.S slin. and AStota for the cycle If any values are nonzero, explain how this can be so, despite entropy being a state function, (d) Is the process spontaneous, nonspontaneous, or at equilibrium ... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

We see that the total change in entropy is a positive quantity for both these spontaneous processes, even though one process is exothermic and the other is endothermic. When this type of calculation is carried out for other processes, the same result is always obtained. For any spontaneous process, the total change of entropy is a positive quantity. Thus, this new state function of entropy provides a thermod3mamic criterion for spontaneity, which is summarized in the second law of thermodynamics ... 

There is no single criterion for the system alone that applies to all processes. However, if we restrict the conditions to constant temperature and pressure, there is a state function whose change for the system predicts spontaneity. This new state function is the free energy (G), which was introduced by the American J. Willard Gibbs and is defined by Equation G = H - T S As usual, H is enthalpy, T is absolute temperature, and S is entropy. 

Why do some reactions go virtually to completion, whereas others reach equilibrium when hardly any of the starting materials have been consumed At the molecular level, bond energies and molecular organization are the determining factors. These features correlate with the thermodynamic state functions of enthalpy and entropy. As discussed In Chapter 14, free energy (G) is the state function that combines these properties. This section establishes the connection between thermodynamics and equilibrium. 

Clearly, the differential obtained, namely, d S = SqJT is exact and S, the entropy, is a thermodynamic state function, that is, it is independent of the path of integration. While Eq. (88) was obtained with the assumption of an ideal gas, the result is general if reversible conditions are applied. 

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