Planck statement

The Kelvin-Planck statement of the Second Law also focuses on cyclic devices and limitations. It may be stated as ... 

The Clausius and Kelvin-Planck statements and the Carnot principle reflect a historical interest in increasing the efficiency of engines. While the... 

Like the engine-based statements, Caratheodory s statement invokes limitations. From a given thermodynamic state of the system, there are states that cannot be reached from the initial state by way of any adiabatic process. We will show that this statement is consistent with the Kelvin-Planck statement of the Second Law. 

We wish to show that no points to the leftbb of 2 on the isotherm 62 are accessible from point 1 via any adiabatic path, reversible or irreversible. Suppose we assume that some adiabatic path does exist between 1 and 2. We represent this path as a dotted curve in Figure 2.11a. We then consider the cycle I —>2 —> 1 — 1. The net heat associated with this cycle would be that arising from the last step 1 — 1, since the other two steps are defined to be adiabatic. We have defined the direction 1 — 1 to correspond to an absorption of heat, which we will call qy. From the first law, the net work vv done in the cycle, is given by w = —q, since AU for the cycle is zero. Thus, for this process, iv is negative (and therefore performed by the system), since qy is positive, having been absorbed from the reservoir. The net effect of this cycle, then, is to completely convert heat absorbed at a high temperature reservoir into work. This is a phenomenon forbidden by the Kelvin-Planck statement of the Second Law. Hence, points to the left of 2 cannot be reached from point 1 by way of any adiabatic path. 

Kelvin-Planck statement of Second Law 57 Klotz. I. M. 217. 254. 256 krypton, heat capacity 577-8... 

In addition to the statement we have been using, several alternative ways exist to express the second law. One that will be particularly useful is the Kelvin-Planck statement ... 

Furthermore, in the four steps of the cycle (Fig. 6.8) three are adiabatic (one irreversible, two reversible). Hence, Qcycie is identical with Q of the isothermal step, that is, Q of Equation (6.104). If g > 0, then W < 0 that is, work would have been performed by the system. In other words, if Q were positive, we would have carried out a cyclical process in which heat at a constant temperature had been converted completely into work. According to the Kelvin-Planck statement of the second law, such a process cannot be carried out. Hence, Q cannot be a positive number. As Q must be either negative or zero, it follows from Equation (6.104) that... 

An essential step in the Caratheodory formulation of the second law of thermodynamics is a proof of the following statement Two adiabatics (such as a and b in Fig. 6.12) cannot intersect. F rove that a and b cannot intersect. (Suggestion Assume a and b do intersect at the temperature Ti, and show that this assumption permits you to violate the Kelvin-Planck statement of the second law.)... 

An interesting alternative demonstration of Equation (7.75) can be carried out on the basis of isothermal cycles and of the Kelvin-Planck statement of the second law. Consider two possible methods of going from State a to State b, a spontaneous change of state, in an isothermal fashion (Fig. 7.1) (1) a reversible process and (2) an irreversible process. 

The Kelvin and Planck statements (see Section 1.10) deal with the impossibility of operating thermal engines under certain prescribed conditions from these assumptions the Second Law may then be deduced. There is no logical objection to this procedure, but it does seem somewhat unsatisfactory to base a universally applicable Law on principles pertaining to the operation of heat engines. The reverse... 

This may be demonstrated by noting what would happen if the Planck statement were incorrect. If no other changes occurred in the universe then the heat extracted from the cold reservoir must be transferred without loss to the heat reservoir. Equation (1.10.8) would then have to be altered to read —QhlTh + QdTc 0, with the requirement that Qc be the heat flow into the engine and that Oh = Qc-In these circumstances one finds that 7X/ 1, which contradicts the definition, Th > Tc. 

An alternative wording of the Kelvin-Planck statement is that heat cannot be completely converted to work in a cyclic process. However, it is possible, as shown above, to do the converse and completely conven work to heat. Since heat cannot be completely converted to work, heat is sometimes considered a less useful form of energy than work. When work or mechanical energy is converted to heat, for example, by friction, it is said to be degraded. 

Kelvin-Planck statement of the second law of thermodynamics is as follows It is impossible to construct an engine to work in a cyclic process whose sole effect is to convert all the heat supplied to it into an equivalent amount of work. ... 

Both Clausius and Kelvin-Planck statements are related to each other, and violation of one leads to the violation of the other. 

The second law of thermodynamics dictates that certain processes are irreversible. For example, heat travels in a direction of decreasing temperature. There are two commonly cited equivalent qualitative statements to the second law. The Kelvin-Planck statement states that it is impossible to construct any cyclic device that receives heat from a single thermal reservoir and converts it entirely into work. According to the Clausius statement, it is impossible to construct a device, which operates in a cycle that produces no other effect on the environment other than the transfer of heat from a low temperature reservoir to a higher temperature reservoir. Both of the Kelvin-Planck and Clausius statements of the second... 

The first statement is the Kelvin-Planck statement of the second law of thermodynamics. As a corollary, it is not possible to affect a cyclic process that can convert heat absorbed by a system completely into work done by the system. Mathematically stated, the second law of thermodynamics can be written as... 

The Kelvin-Planck statement of the second law It is impossible to constmct a heat engine whose only effect, when it operates in a cycle, is heat transfer from a heat reservoir to the engine and the performance of an equal quantity of work on the surroundings. Both the Clausius statement and the Kelvin-Planck statement assert that certain processes, although they do not violate the first law, are nevertheless impossible. 

The plan of the remaining sections of this chapter is as follows. In Sec. 4.3, a h)q)o-thetical device called a Carnot engine is introduced and used to prove that the two physical statements of the second law (the Clausius statement and the Kelvin-Planck statement) are equivalent, in the sense that if one is true, so is the other. An expression is also derived for the efficiency of a Carnol engine for Ihe purpose of defining thermodynamic temperature. Section 4.4 combines Carnot cycles and the Kelvin-Planck statement to derive the existence... 

We can use the logical tool of reductio ad absurdum to prove the equivalence of the Clausius and Kelvin-Planck statements of the second law. 

Thus, if the Kelvin-Planck statement is correct, it is impossible to operate the Clausius device as shown, and our provisional assumption that the Clausius statement is incorrect must be wrong. In conclusion, if the Kelvin-Planck statement is correct, then the Clausius statement must also be correct. 

We can apply a similar line of reasoning to the heat engine that the Kelvin-Planck statement claims is impossible (a Kelvin-Planck engine ) by seeing what happens if we assume this engine is actually possible. We combine a Kelvin-Planck engine with a Camot heat pump, and make the work performed on the Camot heat pump in one cycle equal to the work performed by the Kelvin-Planck engine in one cycle, as shown in Fig. 4.6(c). One cycle of the combined system, shown in Fig. 4.6(d), shows the system to be a device that the Clausius statement says is impossible. We conclude that if the Clausius statement is correct, then the Kelvin-Planck statement must also be correct. 

These conclusions complete the proof that the Clausius and Kelvin-Planck statements are equivalent the tmth of one implies the tmth of the other. We may take either statement as the fundamental physical principle of the second law, and use it as the starting point for deriving the mathematical statement of the second law. The derivation will be taken up in Sec. 4.4. 

The ratio Tc / Th is positive but less than one, so the efficiency is less than one as deduced earlier on page 111. This conclusion is an illustration of the Kelvin-Planck statement of the second law A heat engine cannot have an efficiency of unity—that is, it cannot in one cycle convert all of the energy transferred by heat from a single heat reservoir into work. The example shown in Fig. 4.5 on page 108, with e = 1/4, must have Td 7h = 3/4 (e.g., Tc = 300 K and T = 400 K). 

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