Physical Statements

The second law of thermodynamics, like the first, represents a generalization of the results of a large number of experiments. In Sec. 4-1 we present two equivalent physical statements of the second law. In Sec. 4-2 we present the mathematical statement of the second law and determine how a criterion for equilibrium can be set up, making use of the mathematical statement. In Sec. 4-3 the mathematical statement of the second law is shown to be equivalent to the physical statements. The argument proceeds by demonstrating that Caratheodory s principle can be derived from the physical statements. 

We consider two equivalent physical expressions of the second law, one based on Kelvin s principle and the other a generalization of Clausius postulate. 

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The internal partition function [ ] does not appear in Eq. (3.35), though it does appear explicitly in Eqs. (3.18) and (3.19), p. 40. Start with Eq. (3.35), and show that this implies Eq. (3.19). Give a physical statement and interpretation of this distinction. 

This ignition criterion is immanently reasonable. It can also be shown to be roughly equivalent to a number of other reasonable physical statements—for example, that the rate of liberation of heat by chemical reactions inside the slab must approximately balance the rate of heat loss from the slab by thermal conduction. The rule also clearly represents an over-simplified approximation discrepancies exceeding a factor of 2 or 3 will not be surprising. A formula expressing the rule is... 

For n > 2, the Gegenbauer functions of the second kind are infinite at the points = 1, which correspond to 6 = 0 and 9 = it. Therefore, if there are no singularities in the physical statement of the problem, then all the constants with tildes in (2.1.5) must be zero. Moreover, for n = 0 and n = 1, the remaining constants, if nonzero, result in infinite values of the tangential velocity Vg on... 

The boundary conditions on a surface where a chemical reaction occurs depend on the specific physical statement of the problem. In the special case of an infinitely rapid heterogeneous chemical reaction, the corresponding boundary condition has the form... 

A complete mathematical statement of a problem requires specification of boundary and initial conditions. Boundary conditions are based on a physical statement or principle (for example for viscous flow the component of velocity parallel to a stationary surface is zero at the wall for an insulated wall the derivative of temperature normal to the wall is zero etc.). 

MATHEMATICAL STATEMENT DERIVED FROM THE PHYSICAL STATEMENTS... 

In this section, we present two derivations of Caratheodory s principle from the physical statements of the second law one is based on physical statement a and the other is based on physical statement b. Then we derive the mathematical statement of the second law from Caratheodory s principle. 

If Qc were less than zero, the cyclic process would be in contradiction to Clausius physical statement of the second law. From Eqs. (4-40) and (4-41), we find... 

In order to complete the derivation of the mathematical statement of the second law from the physical statements, we must prove that... 

It is relatively easy to derive the physical statements of the second law from the mathematical statements and thus complete the verification of their equivalence. 

The main purpose of this book is to present a rigorous and logical discussion of the fundamentals of thermodynamics and to develop in a coherent fashion the application of the basic principles to a number of systems of interest to chemists. The concept of temperature is carefully discussed, and special emphasis is placed on the appropriate method for the introduction of molecular weights into thermodynamics. A new treatment of the second law of thermodynamics is presented which demonstrates that Caratheodory s principle is a necessary and sufficient consequence of the physical statements of Clausius and Kelvin. 

In this review, the rte<4dnetic ai )toach is presented, complete mathematical and physical statements of the preplan are given, and the operation of a tubular polymerization reactor is analyzed as an example. The fundamental necessity of using the rheokinetic approach, whenever there is a sharp growth in the viscosity, is demonstrated. The trends of further investigation are presented. 

THE EXTENDED JENCKEL EQUATION, AN EFFICIENT VISCOSITY TEMPERATURE FORMULA. I. PROPERTIES AND APPLICABILITY OF THE EQUATION ON THE NUMERICAL INTERPOLATION. II. PHYSICAL STATEMENTS FROM THE NUMERICAL VERIFICATION. 

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... 

The physical statement of (4.147) is that the number of neutrons slowing past lethargy u (in an infinite medium, per unit volume per unit time) is given by the number of neutrons produced at u, S(w ) du multiplied by the probability p(w w ) that a neutron at w will reach u without being absorbed and the product integrated over all w < w [for which jS(w ) has a nonzero value]. 

If J(t) continues to increase indefinitely, G(t) must tend to zero, according to (1.2.40). In particular, this is true for a viscoelastic liquid. It corresponds to the physical statement that the stress resulting from a suddenly applied strain relaxes to zero, even though the strain is maintained. 

Definition The BP is physically defined as the temperature at which the first bubble comes out as vapor. One can express this physical statement by saying that Z yi = 1, where yj is the composition of component i in the vapor phase. 

This equation represents the mathematical function for the physical statement of the BP. Therefore, the solution for the BP becomes a trial and error problem by solving the function, since K, is a function of T or K = f(Ti). 

There are two principal physical statements of the second law of thermodynamics (1) If a system undergoes a cyclic process it cannot turn heat put into the system completely into work done on the surroundings. (2) Heat cannot flow spontaneously from a cooler to a hotter object if nothing else happens. 

No violation of either physical statement of the second law of thermodynamics has ever been observed in a properly done experiment. We regard the second law as a summary and generalization of experimental fact. A machine that would violate the Kelvin statement of the second law and turn heat completely into work in a cyclic process is called a perpetual motion machine of the second kind. 

Caratheodory devised a three-part proof that the mathematical statement of the second law follows from a physical statement of the second law. The first part is to establish that in the state space of the system only one reversible adiabat passes through any given point. This was shown in Chapter 3. The second part of the argument is to show that this fact implies that a function S exists whose differential vanishes along the reversible adiabat on which also vanishes. This implies that d rev possesses an integrating factor, which is a function y that produces an exact differential dS when it multiplies an inexact differential ... 

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