Entropy as a state function

So, we can assign temperatures consistently with our experience of hotness. Now it s time to move on and prove both the existence of entropy as a state function and the Clausius inequality. This requires employing the concept of the reversible process. A reversible process is one each step of which may be exactly reversed by an infinitesimal change in the external conditions prevailing at the time of that step. The two requirements needed for a real process to approximate reversibility are (1) the process proceeds slowly compared to all internal... 

In Section 19.2 we introduced entropy as a state function. We saw that the entropy of a system is an indicator of its randomness or disorder. As chemists, we are usually interested in relating the macroscopic observations that we make on the system to the microscopic description of a system in terms of atoms and molecules. In Ihis section we will look more closely at how the structure and behavior of molecules affect their entropy. We will also learn about the third law of thermodynamics, which is concerned with the entropy of substances at absolute zero. 

Entropy as a State Function if a change results in a greater number of microstates, there are more ways to disperse the energy of the systan and the entropy increases ... 

As we have seen in this chapter, the foundation of the concept of entropy as a state function is entirely macroscopic. The validity of the Second Law is rooted in the reality of irreversible processes. In stark contrast to the macroscopic irreversible processes we see all around us, the laws of both classical and quantum mechanics are time symmetric, i.e., according to the laws of mechanics, a system that can evolve from a state A to a state B can also evolve from the state B to the state A. For example, the spontaneous flow of gas molecules from a part that has a higher density to a part that has a lower density and its reverse (which violates the Second... 

The statement of the first law of thermodynamics defines the internal energy and asserts as a generalization of experiment fact that it is a state function. The second law of thermodynamics establishes the entropy as a state function, but in a less direct way. 

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. 

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 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 entropy change in a process is given by eqn. (14) and it follows that entropies can be assigned to individual substances. As entropy is a state function, its value will depend on the state of the substance and, with the aid of eqn. (14), the entropy difference between any two states can be calculated. The third law enables a zero to be fixed for the entropy scale and there are tables [5—9] which give the entropies of many substances in their standard states at the reference temperature of 298 K. As long as there is no change of phase, the entropy at any other temperature can be calculated using... 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. Note that even though entropy is a state function, we cannot just disregard the rev subscript in Eq. 1 and calculate AS directly from q/T for any path. If we want to calculate the entropy difference between a pair of states joined by an irreversible path—such as between the initial and final states of an ideal gas that undergoes free expansion—then we can calculate that entropy difference from a reversible path between the same two states. For example, a slow, careful, reversible isothermal expansion of the gas would be a reversible path for which we can calculate AS from qrev/T. The values of q and w will differ for the two paths, but AS for the reversible path will be the same as for the irreversible path. 

Molecular randomness, or disorder, is called entropy and is denoted by the symbol S. Entropy is a state function (Section 8.3), and the entropy change AS for a process thus depends only on the initial and final states of the system ... 

As entropy is a state function, we are free to choose a path from the initial and final states. The path along which a process takes place reversibly would be most convenient for thermodynamic calculations, because the heat absorbed or released can be directly related to the entropy change ... 

Entropy is a state function, so AS is the same as for the reversible isothermal expansion, calculated earlier. Because less work is done than in the reversible process and A U is the same in both cases, less heat is withdrawn from reservoirs in the surroundings. Therefore, the decrease of entropy of the surroundings is less than in the reversible expansion. In the limit of expansion against a vacuum (Joule process), no work is done and no heat is withdrawn from the surroundings. In this case, AAsur = 0. 

Solution Because entropy is a state function, we can calculate AS by any path that takes the system from its initial to its final state. For the path, we choose isothermal compression at 0°C from 1.0 atm to 3.0 atm, followed by heating at a constant pressure of 3.0 atm from 0°C to 50°C. The CP of a monatomic ideal gas is (5/2)R, independent of temperature and pressure. Because step 1 is an isothermal compression of an ideal gas, V2/V1=P1/P2... 

Because we have achieved the same change of state as for mixing at constant pressure and entropy is a state function, this must be the entropy change for opening the valve between bulbs at the same pressure. Note that, as expected, entropy of mixing is always positive. 

Equation (1.60) illustrates a very important thermodynamic property in which the quantity dqrev / T becomes zero when the cyclic process is completed regardless of the paths taken from the initial to final states. Such a property is known as a state function, as are P, V, T, E, and H. Clausius suggested defining a new thermodynamic state function, called entropy and denoted as S, where dqrev / T = dS, so that... 

When a system undergoes an irreversible process from one equilibrium state to another, the entropy change of the system AS is stilt evaluated by Eq. (A). In this case Eq. (A) is applied to an arbitrarily chosen reversible process that accomplishes the same change of state. Integration is not carried out for the original irreversible path. Since entropy is a state function, the entropy changes of the irreversible and reversible processes are identical. 

This is the same as the result from part (a), an illustration of the fact that the entropy is a state function. By contrast, the amounts of heat for the two paths are different 28.5 and 18.6 kj. 

Entropy is a state function -> Changing the path does not change AS. 

The Second Law (the entropy law) introduces entropy, S, as a state function. More importantly, this law also describes the equilibrium state of a system as one of maximum entropy. For a system at constant temperature and pressure, the equilibrium condition is expressed as ... 

We have said that entropy is a state function but we must justify this statement before proceeding. Traditionally this was done from a consideration of the efficiency of heat engines—a consideration which was a major preoccupation with the pioneers of thermodynamics. As chemists, we shall allow ourselves a short cut by considering only a perfect gas. 

If one mole of a perfect gas is expanded from a volume of 0.01 m3 to 0.10m3, the entropy change is AS = 8.3 In 10 = 19.1 JK"1 mol" This relation will be correct for both reversible and irreversible changes as entropy is a state function and AS is independent of the path taken between the states A and B. The heat lost by the surroundings is equal to the heat gained by the gas and thus for a reversible expansion ASOTe , - 0 (as ASoverai, = qtJT - qteJT). 

Like energy and enthalpy, entropy is a state function. Consider a certain process in which a system changes from some initial state to some final state. The entropy change for the process, AS, is... 

Note One can arrive at this by using q from Path (a) as the reversible path, or one can simply use AS from Path (a), realizing that entropy is a state function.)... 

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