The Clausius inequality

For any engine we have only the possibilities expressed by (8.39). We have shown that the equality holds f or the reversible engine. Since the heat and work effects associated with an irreversible cycle are different from those associated with a reversible cycle, this implies that the value of j 4QIT for an irreversible cycle is different from the value, zero, associated with the reversible cycle. We have shown that for any engine the value cannot be greater than zero consequently, it must be less than zero. Therefore for irreversible cycles we must have 

Consider the following cycle A system is transformed irreversibly from state 1 to state 2, then restored reversibly from state 2 to state 1. The cyclic integral is 

The limits can be interchanged on the second integral (but not on the first ) by changing the sign. Thus we have 

If the change in state from state 1 to state 2 is an infinitesimal one, we have 

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- 

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

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... 

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... 

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... 

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 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... 

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... 

Recapitulating hitherto existing results we can see that by (1.26) and (1.1), relation (1.25) is in fact the Clausius inequality (1.20) for arbitrary cyclic process from B (i.e., starting from equilibrium state) and, by (1.24), equality in (1.20) is valid for any Carnot cycle. 

This is referred to as the Clausius inequality, and implies that in an irreversible process the entropy increases. The heat supply J Q is therefore not completely converted into the heat component of the internal energy T dS. This is the essential concept of the second part of the Second Law of Thermodynamics. 

This relation is known as the Clausius inequality. It is valid only if the integration is taken around a cyclic path in a direction with nothing but reversible and irreversible changes—the path must not include an impossible change, such as the reverse of an irreversible change. The Clausius inequahty says that if a cyclic path meets this specification, it is impossible for the cyclic integral (dg/Tb) to be positive. 

The starting point to calculate the equilibrium state of a chemical reaction is the Clausius inequality, introduced in Section 3.1.3 for physical transformations of pure... 

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