Clausius inequality 4.36

Still another statement of the second law is the Clausius inequality which states that... 

Claus process, 634 Clausius inequality, 288 Clausius-Clapeyron equation, 312 clay, 616... 

The combination of the Clausius inequality (eq. 1.30) and the first law of thermodynamics for a system at constant volume thus gives... 

Because this cycle includes an irreversible step (1), we know from the Clausius inequality (4.40) that... 

Let us first introduce a useful short-cut to the constrained optimization procedure employed in Section 5.2, based on the general Clausius inequality [cf. (4.43)] for spontaneous changes toward equilibrium ... 

Table 5.1 summarizes the various constraint conditions and the associated thermodynamic potentials and second-law statements for direction of spontaneous change or condition of equilibrium. All of these statements are equivalent to Carnot s theorem ( dq/T < 0) or to Clausius inequality ([Pg.164]

This means that A is independent of the external constraints acting on the system. It must be a state function like energy or entropy. Moreover, in terms of the entropy production per unit time, the Clausius inequality... 

To get at this quantity recall the Clausius inequality (1.15.6). In the present case the integration in (1.15.6) reduces to a sum of processes occurring during heat transfers Qh from the hot reservoir into the engine and - Qc from the... 

An alternative to analyze the Curzon-Ahlborn cycle, taking into account some effects that are nonideal to the adiabatic processes through the time of these processes, is the model proposed in [5] and in [7]. It allows to find the efficiency of a cycle as a function of the compression ratio, rc = Vmax/Vmin. When rc, Fmax>>Vmin, the Curzon-Ahlborn-Novikov-Chambadal efficiency is recovered. The non-endoreversible Curzon and Ahlbom cycle can be analyzed by means of the so-called non-endoreversibility parameter Is, defined first in [14] and later in [15] and in [16], which can be used to analyze diverse particularities of cycles. Furthermore, this parameter leads to equality instead of Clausius inequality [14]. 

Suppose a thermal engine working like a Curzon and Ahlborn cycle, in which an internal heat by internal processes of working fluid appears, assuming ideal gas as working fluid. The Clausius inequality with the parameter of non-endoreversibility becomes... 

Remarkably, the so called Clausius inequality (73) can be changed into an equality by averaging the exponential exp —(3W) instead of the work W [45,46] ... 

The classical definition of entropy goes through the Clausius inequality ... 

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... 

This T, the thermodynamic or absolute temperature, is here a function of S, V and x. But it s easy to show that if T were a function of temperature and entropy, or if it were a function of temperature and anything else, we could violate Kelvin s statement. So T depends only on the empirical temperature, and this dependence must be the same for all systems in order for the entropy of a composite to equal the sum of the entropies of the subsystem. In order for Clausius statement to hold in the case of irreversible processes, the equal sign of rfQ = TdS becomes <, and we have Clausius inequality TdS,rwKere T is the... 

This is known as the Clausius inequality and has important applications in irreversible processes. For example, dS > (dQ/T) for an irreversible chemical reaction or material exchange in a closed heterogeneous system, because of the extra disorder created in the system. In summary, when we consider a closed system and its surroundings together, if the process is reversible and if any entropy decrease takes place in either the system or in its surroundings, this decrease in entropy should be compensated by an entropy increase in the other part, and the total entropy change is thus zero. However, if the process is irreversible and thus spontaneous, we should apply Clausius inequality and can state that there is a net increase in total entropy. Total entropy change approaches zero when the process approaches reversibility. 

If we combine the Clausius inequality given in Equation (113) with Equation (170), we obtain for both irreversible and reversible systems... 

D3.4 All of these expressions are obtained from a combination of the first law of thermodynamics with the Clausius inequality in the form TdS > dry (as was done at the start of Justification 3.2). It may be written as... 

According to the Second Law and the Clausius inequality in particular (see Section 1.4.2) it follows for a closed system of constant composition, which can exchange heat with its surroundings at constant temperature, that... 

This is the Clausius inequality, which is a fundamental requirement for a real transformation. The inequality (8.44) enables us to decide whether or not some proposed transformation will occur in nature. We will not ordinarily use (8.44) just as it stands but will manipulate it to express the inequality in terms of properties of the state of a system, rather than in terms of a path property such as 2irr-... 

The Clausius inequality can be applied directly to changes in an isolated system. For any change in state in an isolated system, 4Qirr = 0- The inequality then becomes... 

Central to the thermodynamic discussion of irreversible processes is the concept of entropy production. Consider the Clausius inequality, dS > Q/T, which we can rearrange to the form... 

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