Clausius statements

The Clausius statement of the Second Law involves cyclic processes and engines. It can be stated as m... 

When we apply these statements to the universe, the ultimate adiabatically enclosed system, the result is the Clausius statement... 

Carnot stated that the efficiency of a reversible Camot engine depends only on the temperatures of the heat reservoirs and is independent of the nature of the working substance. This theorem can be proved by showing that the assumption of a reversible engine with any but the known efficiency of a reversible Camot engine leads to a contradiction of the Clausius statement of the second law. 

The Clausius statement of the second law, although logically able to serve as a basis for the general equilibrium theory, was couched in terms of nonequilibrium processes that themselves lay outside the scope of such a theory. Attempts to derive the consequences of the Clausius statement were therefore tortuous and indirect, making further progress difficult. 

It was the principal genius of J. W. Gibbs (Sidebar 5.1) to recognize how the Clausius statement could be recast in a form that made reference only to the analytical properties of individual equilibrium states. The essence of the Clausius statement is that an isolated system, in evolving toward a state of thermodynamic equilibrium, undergoes a steady increase in the value of the entropy function. Gibbs recognized that, as a consequence of this increase, the entropy function in the eventual equilibrium state must have the character of a mathematical maximum. As a consequence, this extremal character of the entropy function makes possible an analytical characterization of the second law, expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, processes, perpetual motion machines, and the like. 

The inequality (5.26) merely says that the entropy function was at a maximum before the variation, which is the counterpart of the Clausius statement [cf. (4.48)]... 

Moreover, if is always positive and nonzero, W must be positive and nonzero except in the case when the two temperatures are equal. This observation results in the Clausius statement of the second law of thermodynamics Heat of itself will not flow from a heat reservoir at a lower temperature to one at a higher temperature. It is in no way possible for this to occur without the agency of some system operating as a heat engine in which work is done by the surroundings on the system. 

Clausius statement of the Second Law took the form Die Entropie der welt strebt einem maximum zu = The entropy of the universe tends towards a maximum and applies to an isolated system. 

Related to the above is the Clausius Statement It is impossible to devise a machine which, operating in cycles, transfers heat from a colder to a hotter body without producing any other effects in the universe. 

This may be demonstrated by noting what would happen if the Clausius statement were incorrect. If no other changes occur in the universe then the heat extracted from the cold reservoir must be transferred without loss to the heat reservoir. Equation (1.15.9) would then have to be altered to read - Qh/Th + Qc/Tc < 0, with the requirement that Q - Qc be the heat flow into the engine and that Qh — Qc. In these circumstances one would have Th/Tc < 1, which is nonsensical because, by definition, Th > Tc. 

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... 

The second-law statement of Rudolf Clausius (1822-1888) is that it is not po.ssible to construct a device that operates in a cycle and whose sole effect is to transfer lieat from a colder body to a hotter body. Show from the a.xiom. Soe > 0 that the process below is impossible, so that the Clausius statement of the second law is consistent with what has been presented in this book. 

Proof II. In this proof we start from a physical postulate which embodies Clausius statement of the second law. The physical postulate is A... 

The famous Clausius statement is as follows It is impossible to construct a device to work in a cyclic process whose sole effect is the transfer of heat from a body at a lower temperature to a body at a higher temperature. Clausius also stated the first and the second laws of thermodynamics combined together as The energy of the universe is constant, and The entropy of the universe tends toward a maximum. ... 

Prove that the Clausius statement is a necessary consequence of the second law, as formulated in this chapter. 

The minimum required work corresponds to reversible operation (Sge = o). If the cycle is irreversible, it will require more work. Positive work means that we (the surroundings) must supply this amount to the device. This is precisely what a refrigerator does It uses work to pump heat from lower to higher temperature (more about refrigeration in Chapter 6). Without supplying work from the surroundings, the process is impossible. This proves the Clausius statement. 

Comments The Clausius statement refers to a device, which we took it to mean cycle. A noncyclic process would leave the state of the working fluid altered, but this contradicts the stipulation that the sole effect of such device should be to transfer heat between two reservoirs. Therefore, the Clausius device, if it exists, must be a cycle. 

Example 4.14 The Clausius Statement as an Equivalent Statement of the Second Law... 

Starting with the Clausius statement, define entropy and prove the inequality in eq. 14.21. 

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