Kelvin-Planck statement of the second law

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

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. 

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. 

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. ... 

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. 

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. 

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). 

The Kelvin and Planck statements of the second law (Section 1.11) 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 such a procedure, but it does seem somewhat unsatisfactory to base a universally applicable law on principles pertaining to the operation of heat engines. The reverse procedure, outlined in Section 1.11 does provide what appears to be a better alternative here the characteristics of cychcaUy operated heat engines are derived as a consequence of the second law. 

An alternative statement of the second law, due to Lord Kelvin (William Thomson, 1824-1907) and Max Planck (1858-1947), is that it is not possible to construct a device operating in a cycle that results in no effect other than, the production of work by tran.sferring heat from a single body. A schematic diagram of a Kelvin-Planck device is shown below. 

Since T is positive (remember, we are using absolute temperature), for the entropy generation to be positive, Q must be negative. That is. to be consistent with our statement of the second law, the device cannot absorb heat and con -ert all of it to work. However, the reverse process, in which the device receives work and converts all that work to heat, is possible. Therefore, our statement of the second law is consistent with that of Kelvin and Planck, but again more general. ... 

Each of the three parts is an essential component of the second law, but is somewhat abstract. What fundamental principle, based on experimental observation, may we take as the starting point to obtain them Two principles are available, one associated with Clausius and the other with Kelvin and Planck. Both principles are equivalent statements of the second law. Each asserts that a certain kind of process is impossible, in agreement with common experience. 

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... 

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. 

Finally, a closing remark other equivalent statements of the second law are commonly cited, namely those of Kelvin and Planck these are based on the operation of heat engines, discussed in Section 1.11. For more information, see Note 2. 

Statements of the Second Law Thermodynamic Operation of Heat Engines Kelvin and Planck Statements Temperature Scale Operation of Heat Engines... 

We begin here the discussion of the Second Law of Thermodynamics. This law has been enunciated in many different forms, the most prominent being the formulations by Kelvin and by Planck. These will be presented later as consequences of the approach derived below. Undoubtedly, the most elegant statement of this Law was provided by Caratheodory in the following form ... 

This Statement was later refined by Planck. On first glance, the formulation of the second law of Clausius and the formulation of Kelvin and Planck do not seem to have much in common. However, it is shown in elementary texts that both formulations are equivalent. 

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 second law of thermod5mamics is stated through the Kelvin-Planck statement and the Clausius statement. The Inequality of Clausius is a consequence of the second law of thermod5mamics, and it is stated for a system undergoing a thermodynamic cycle as... 

In most treatises on thermodynamics, it is usual to refer to the laws of thermodynamics. The conservation of energy is referred to as the First La of Thermodynamics, and this principle was discus.sed in detail in Chapter 3. The positivc-dehniie nature of entropy generation used in Chapter 4, or any of the other statements such as those of Clausius or Kelvin and Planck, are referred to as the Second Law of Thermodynamics. The principle of consers ation of mass precedes the development of thermodynamics. and therefore is not considered to be a law of thermodynamics. 

There are several ways of defining the second law of thermodynamics, but a very useful statement, according to Kelvin and Planck, is as follows ... 

The second law of thermodynamics has historically been a mysterious concept, and the basic idea has been verbalized by Clausius, Kelvin, Planck, and others for those who think in words. One simple statement by Rudolph Clausius (1822-1888) was... 

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