Thermodynamics mathematical statement

As we have seen earlier, the mathematical statement of the First Law of Thermodynamics in terms of the universe can be written as... 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

We can obtain an explicit equation for the entropy of an ideal gas from the mathematical statements of the two laws of thermodynamics. It is convenient to derive this equation for reversible changes in the gas. However, the final result will be perfectly general because entropy is a state function. 

Ions or molecules flowing down their concentration gradients is one aspect of a very general statement known as the Second Law of Thermodynamics. The Second Law is a mathematical statement to the effect that all real processes increase the disorder, captured in a quantity known as entropy, of the universe. Entropy is a measure of disorder or randomness and may be thought of as negative information. 

This relation is a direct consequence of the definition of enthalpy by Equation (I) and of the mathematical statement of the first law of thermodynamics, namely that the change in internal energy. AT. is equal to the heat adsorbed minus the work done q - PAY). It is clear that this thermodynamic relation does not define absolute values of enthalpy or internal energy. Changes in enthalpy, however, are readily measured by calorimetric techniques, and the relative enthalpy values nre sufficient for all therinochcmical calculations. 

Thermochemistry is concerned with the determination of the heat absorbed by a system when some process occurs within the system. The quantity of heat absorbed may be determined experimentally by the use of calorimeters or by calculation from prior knowledge of the thermodynamic properties of the system. The equations relating the heat absorbed by a system for a given process to the change of energy or enthalpy of the system for the change of state that occurs during the process are the mathematical statements of the first law of thermodynamics. They are Equations (2.26) and (2.30), written here as... 

The existence of an energy balance is not sufficient to answer all questions about a chemical reaction. Does a given reaction take place at all If so, to what extent does it proceed Questions relating to the processes and extent of chemical reactions require the introduction of some new thermodynamic functions which, like E and //, are properties of the state of the system. These new functions are entropy, S, and Gibbs free energy, G. In order to answer these and other questions, a mathematical statement of the second law of thermodynamics is required ... 

In thermodynamics, a system is the matter within a defined region. The matter in the rest of the universe is called the surroundings. The first law of thermodynamics, a mathematical statement of the law of conservation of energy, states that the total energy of a system and its surroundings is a constant ... 

Equation (2.1) is the mathematical statement of the first law of thermodynamics. It is to be noted that both sides of the equation should be expressed in the same units. Thus if internal energy and mechanical work are expressed in ergs, the heat absorbed must be converted to ergs by use of the mechanical equivalent of heat,... 

Equation (1.75) is a mathematical statement of the second law of thermodynamics for reversible processes. The introduction of the integrating factor for Sq causes the thermal energy to be split into an extensive factor. S and an intensive factor T. Introducing Eq. (1.75) into Eq. (1.56) yields the combined first and second laws... 

The first law of thermodynamics is a statement of the law of energy conservation. The change in the system energy when its state changes from A to B is written as the sum of the work W done on the system, and the heat flow Q into the system, during the process. The mathematical statement of the first law is then... 

The second law of thermodynamics states that the entropy of the universe increases in a spontaneous process and remains unchanged in an equilibrium process. The mathematical statement of the second law of thermodynamics is given by... 

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. 

The mathematical statement of the second law is The heat change dQ on Sl reversible path, regarded as a linear differential form in V and t, the temperature of the surroundings, possesses an integrating factor llT(t). T t) is a function of t alone and is identical for all thermodynamic systems. It is called the thermodynamic temperature and is identical with the absolute temperature. The function S, defined by dS = dQjT, is called the entropy and is a function of state. In all natural processes... 

The mathematical statement of the second law provides, as a corollary, a criterion for thermodynamic equilibrium. It is clear that a thermodynamic system must be in an equilibrium state if there are no natural processes by which it can proceed from its specified state to another state. Thus, a state of a thermodynamic system is an equilibrium state if all virtual variations in state fail to satisfy the inequality... 

Then (Sq/T) becomes a state function called entropy and T the absolute temperature. As a state function, entropy is path-independent. Eqn (1.25) is a mathematical statement of the second law of thermodynamics. The introduction of the integrating factor for 8q causes the thermal energy to be split into an extensive factor S and an intensive factor T. Clausius defined the entropy with the integrating factor of the inverse of absolute temperature in T 8q) = dS. Similarly, integrating factor 1/P in IP 6W) = dV leads to exact differential dV, which is formulated by Clapeyron in 1834. Introducing Eqn (1.25) into the first law of thermodynamics dU =8q + yields the combined first and second laws of thermodynamics... 

A rigorous mathematical statement can be formulated as follows If a reaction mixture consists of only two isomers, Aj and A2, the system moves toward the equilibrium, never overshooting it. The concentrations CAi(t) and ca2(0 approach the equilibrium concentrations CA,eq and CAj.eq monotonically without ever exceeding these concentrations. The type of dynamic trajectory is not arbitrary, but is governed by thermodynamic functions of the chemical composition, such as the Gibbs free energy function G, which decrease monotonously in time. Such a limitation on dynamic trajectories is eliminated if there are more than two chemical components in the system. 

Mathematical statement of the first law of thermodynamics. Calculating work done in gas expansion or compressioir Definition of enthalpy. 

Equation 6.0.1 is the mathematical statement of the Nernst heat theorem or third law of thermodynamics. It is true in general only if each reactant and product is a pure crystal with identical unit cells arranged in perfect spatial order. 

Clapeyron s paper used indicator diagrams and calculus for a rigorous proof of Carnot s conclusion that the efficiency of a reversible heat engine depends only on the temperatures of the hot and cold heat reservoirs. However, it retained the erroneous caloric theory of heat. It was not until the appearance of English and German translations of this paper that Clapeyron s analysis enabled Kelvin to define a thermodynamic temperature scale and Clausius to introduce enffopy and write the mathematical statement of the second law. 

The mathematical statement of the second law is associated with the definition of entropy S, dS = 8q /T. Entropy is a thermodynamic potential and a quantitative measure of irreversibility. For reversible processes, dS is an exact differential of the state function, and the result of the integration does not depend on the path of change or on how the change is carried out when both the initial and final states are at stable equilibrium. The entropy of a closed adiabatic system remains the same in a reversible process, and increases during an irreversible process. A system and its surrounding create an isolated composite system where the sum of the entropies of all reversible changes remains the same, and increases during irreversible processes. 

Kelvin s statement of the second law of thermodynamics is that heat put into a system that undergoes a cyclic process cannot be completely converted into work done on the surroundings. Clausius statement of this law is that heat cannot flow from a cooler to a hotter body if nothing else happens. The mathematical statement of the second law was shown to be a consequence of the Kelvin statement. It asserts that S, the entropy, is a state function if we define... 

The mathematical statement of the second principle of thermodynamics takes the form of the entropy inequality. The search for a corresponding inequality valid for liquid-gas mixtures easily reveals the inequality presently being used for a single material [11] as not sufficiently general. Indeed, the various entropy inequalities proposed in works on irreversible thermodynamics include an influx of entropy beyond the one usual for a singlematerial. In order to state the entropy inequality proposed here, the primitive properties of the mixture are ... 

The first law of thermodynamics states that energy can neither be created nor destroyed and hence that in any process the total quantity of energy of the system and its surroundings remains constant. The mathematical statement of the first law is given by the equation ... 

Cardwell continues "And then, in a moment of penetrating insight, Kelvin saw that this simple relationship (Eq.4.11.1) amounted to a quantitative, or mathematical statement of the second law of thermodynamics. If we consider the heat absorbed as positive and that transmitted as negative then the relationship becomes ... 

The second law of thermodynamics was actually postulated by Carnot prior to the development of the first law. The original statements made concerning the second law were negative—they said what would not happen. The second law states that heat will not flow, in itself, from cold to hot. While no mathematical relationships come directly from the second law, a set of equations can be developed by adding a few assumptions for use in compressor analysis. For a reversible process, entropy, s, can be defined in differential form as... 

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. 

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