State function, 2.3, 2.16, 3.5, 3.8 3.9

A state is the physical condition of a system described by a specific set of thermodynamic property values. There are two types of properties that describe the macroscopic state of a system 1) extensive and 2) intensive. Extensive properties are proportional to the size of the system intensive properties are independent of the size of the system. If you combine two identical systems and a property is the same 

You need to know what a state function is. State functions are pathway in da pen dent. State properties describe the state of a system. In oilier words, the change in a state property going from Dne state to another is the same regardless of the process via which the system was changed. 

Three state properties, one being extensive, describe the state or a system unambiguously. 

The first law of thermodynamics postulates that the internal energy, which is a state function, consists of the heat (Q) and the work (w) 

In the most common presentation the work embraces the volume work dw = —p.dV. However, other forms of work are possible (the surface work dw = s da, the mechanic work dw = F dr, the electric work dw = q.dU) and the magnetic work 

The individual constituents of the last expression have their dimensions as follows /i,0 [JA 2m 1], H [Am-1] and M [Am 1] then the expression has the dimension of [J m 3] which refers to the volume energy. If M refers to the molar magnetisation, A/moI[Am2mor1], then the product dw = ix0H dM has the dimension of the molar energy [J mol-1]. It is assumed hereafter that the other energy quantities (17, E, F, G, TS, CT) possess the same dimension. 

The other thermodynamic functions are introduced in a similar way (Table 2.3) [2, 3]. Of three observables (H, M, T) only two are independently variable, which is expressed by the constraint 

Although we usually have no way of knowing the precise value of the internal energy of a system, E, it does have a fixed value for a given set of conditions. The conditions that influence internal energy include the temperature and pressure. Furthermore, the internal energy of a system is proportional to the total quantity of matter in the system because energy is an extensive property. (Section 1.3) 

Suppose we define our system as 50 g of water at 25 °C ( FIGURE 5.9). The system could have reached this state by cooling 50 g of water from 100 °C to 25 °C or by melting 50 g of ice and subsequently warming the water to 25 °C. The internal energy of 

Heat flows from sutroundings into S3 em (endotfaermic reaction), temperature of sutroundings drops, ihcrmomctcr reads temperature well below room temperature 

Heat flows (violently) from system into surroundings (exothermic reaction), tempetatuie of sutroundings increases 

Initially hot water cools to water at 25 °C once this temperature is reached, system has internal energy E 

An analogy may help you understand the difference between quantities that are state functions and those that are not Suppose you drive from Chicago, which is 596 ft above sea level, to Denver, which is 5280 ft above sea level. No matter which route you take, the altitude change is 4684 ft. The distance you travel, however, depends on your route. Altitude is analogous to a state function because the change in altitude is independent of the path taken. Distance traveled is not a state function. 

Some thermodynamic quantities, such as E, are state functions. Other quantities, such as q and w, are not. This means that, although A = q + w does not depend on how the change occurs, the specific amounts of heat and work depend on the way in which the change occurs. Thus, if changing the path by which a system goes from an initial state to a final state increases the value of q, that path change will also decrease the value of w by exactly the same amount. The result is that AE is the same for the two paths. 

Note that q and w have opposite signs, and that the overall change in internal energy is positive. Therefore, the total energy of our gaseous system increases. 

If a system is insulated well enough, heat will not be able to get into the system or leave the system. In this situation, q = 0. Such systems are called adiabatic. For adiabatic processes. 

This restriction, that = 0, is the first of many restrictions that simplify the thermodynamic treatment of a process. It will be necessary to keep track of these restrictions, because many expressions like equation 2.12 are useful only when such restrictions are applied. 

Have you noticed that we use lowercase letters to represent quantities like work and heat but a capital letter for internal energy There is a reason for that. Internal energy is an example of a state function, whereas work and heat are not. 

FIGURE 2.7 Analogy for the definition of a state function. For both path (a) straight up a mountain and path (b) spiraling up the mountain, the overall change in altitude is the same and so is path-independent the change in altitude is a state function. However, the overall length of the path is path-dependent, and so would not be a state function. 

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The relationship between this M avefunction (sometimes called state function) and the location of particles in the system fonus the basis for a second postulate ... 

There exists a state function S, called the entropy of a system, related to the heat Dq absorbedfrom the surroundings during an infinitesimal change by the relations... 

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

We can state the form of the conjugate relationship in a setting more general than 4 (a, t), which is just a particular, the coordinate representation of the evolving state. For this purpose, we write the state function in a more general way, through... 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

The terms AG, AH, and AS are state functions and depend only on the identity of the materials and the initial and final state of the reaction. Tables of thermodynamic quantities are available for most known materials (see also Thermodynamic properties) (11,12). 

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

State Functions State functions depend only on the state of the system, not on past history or how one got there. If r is a function of two variables, x and y, then z x,y) is a state function, since z is known once X and y are specified. The differential of z is... 

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U enthalpy, H and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows ... 

S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T andp) are specified, for example. Likewise,... 

V is specified once T and p are specified it is therefore a state function. 

Since the internal energy is a state function, then Eq. (3-44) must be satisfied. 

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

In conversion calculations between the state functions temperature (Tj, pressure (p) and volume (V), the ideal gas law states that... 

Define a state function. Name three thermodynamic quantities that are state functions and three that are not. 

Figure 4.3 Forming configurational state functions from Slater determinants...

The experimental data followed the predicted model and the line represents the above stated function. The presented data indicate that the range of concentrations in this study exhibited an observed substrate inhibition. The experimental data from the current studies were observed to be fit with the predicted model based on Andrew s modified equations. 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

As a result of simultaneous introduction of elastic, viscous and plastic properties of a material, a description of the actual state functions involves the history of the local configuration expressed as a function of the time and of the path. The restrictions, which impose the second law of thermodynamics and the principle of material objectivity, have been analyzed. Among others, a viscoplastic material of the rate type and a strain-rate sensitive material have been examined. 

This relation defines a time-dependent column vector a. Because = 1, Eq. (7-50) implies afa = 1 a is a unit vector. This is true of all state vectors that correspond to normalized state functions. Substitution of (7-50) into (7-49), subsequent multiplication by u, and integration yield the Schrodinger equation (sometimes called the equation of motion ) for the component ar... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

Representation theory for nonunitary groups.—Before proceeding we should consider what is meant by a unitary and an anti-unitary operator.5 -6 If the hamiltonian of a system commutes with the operators u and a of the group 0, and T and O are state functions of the system, u is unitary if... 

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