S, state function

Note that the system s state function need not be an eigenfunction of the operator in (3.30) that corresponds to the physical property B of the system. Thus, the particle-in-a-box stationary-state wave functions are not eigenfunctions of p,. Despite this, we still must get one of the eigenvalues (3.37) of p, when we measure p, for a particle-in-a-box stationary state. 

The time-dependent perturbation changes the system s state function from exp -iE t/hyif to the superposition (9.122). Measurement of the energy then changes to one of the energy eigenfunctions exp -iE i,t/h)il/ i, (reduction of the wave function, Section 7.9). Tie net result is a transition from stationary state n to stationary state m, the probability of such a transition being... 

Schrodinger s state function contains all the information that can be known about a quantum system. Among the properties of a system it predicts is the distribution of charge. The quantity dxj dx2 - dx , where dxi denotes an infinitesimal volume element dii) and the spin component for electron i, is the probability that each of the N electrons in a system will be in some particular volume element with a particular spin component. If this quantity is summed over all spins and integrated over the spatial coordinates of all electrons but those of electron number one, what remains is the probability that electron number one is in some particular volume element dz. That is. 

The thermodynamic potentials, being a system s state functions of the corresponding (natural) parameters, arc of special importance in the system state description, their partial derivatives being the parameters of the system as well. The equalities between th( second mixed derivatives are a property of the state functions and lead to relation-ship.s between the system parameters (the Gibbs-Helmholtz equations). Hence, once any thermodynamic potential (usually, the Gibbs or the Helmholtz one) has been evaluated, by means of either simulation or experiment, this means the complete characterization of the thermodynamic properties of the system. 

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... 

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 ... 

Hooke s law functional form is a reasonable approximation to the shape of the potential gy curve at the bottom of the potential well, at distances that correspond to bonding in md-state molecules. It is less accurate away from equilibrium (Figure 4.5). To model the se curve more accurately, cubic and higher terms can be included and the bond- ching potential can be written 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,... 

We now know that electrons in atoms can hold only particular energies and that their probable whereabouts are described by Schrodiiiger s wave function. The energies and probable locations depend on integer numbers, or quantum numbers. Quantum numbers describe the energy and geometry of the possible electronic states of an atom. These states, in turn, deteriiiilie the chemical behavior of the elements—that is, how chemical bonds can form. 

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. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

Entropy S like internal energy, volume, pressure, and temperature is a fundamental property of a system. As such, it is a function of the state of the system and a state function so that... 

As we have seen earlier, the thermodynamic variables p, V, T, U, S, H, A, and G (that we will represent in the following discussion as W, X, T, and Z) are state functions. If one holds the number of moles and hence composition constant, the thermodynamic variables are related through two-dimensional Pfaffian equations. The differential for these functions in the Pfaff expression is an exact differential, since state functions form exact differentials. Thus, the relationships that we now give (and derive where necessary) apply to our thermodynamic variables. 

The implication of equation (2.37) is that 8qre /T is the differential of a state function. This state function is the entropy S, with the differential dS given byq... 

According to the Caratheodory theorem, the existence of an integrating denominator that creates an exact differential (state function) out of any inexact differential is tied to the existence of points (specified by the values of their x, s) that cannot be reached from a given point by an adiabatic path (a solution curve), Caratheodory showed that, based upon the earlier statements of the Second Law, such states exist for the flow of heat in a reversible process, so that the theorem becomes applicable to this physical process. This conclusion, which is still another way of stating the Second Law, is known as the Caratheodory principle. It can be stated as... 

The coefficients in front of the d , s are ratios of 0 functions that individually depend upon different variables. Since E is a state function, its differential dE... 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

Figure 4.5 The correction to ideal behavior AS rr = Sm., — 5m r is given by ASjx>rr + AS. 2 + ASm.3 since S is a state function, (p is a very low pressure.)...

Quantities like V, U, S, H< A, and G are properties of the system. That is, once the state of a system is defined, their values are fixed. Such quantities are called state functions. If we let Z represent any of these functions, then it does not matter how we arrive at a given state of the system, Z has the same value. If we designate Z to be the value of Z at some state l, and Z to be the value of Z at another state 2, the difference AZ = Z2 - Z in going from state l to state 2 is the same, no matter what process we take to get from one state to the other. Thus, if we go from state l through a series of intermediate steps, for which the changes in Z are given by AZ, AZ . AZ,-. and eventually end up in state 2,... 

The lattice enthalpy of a solid cannot be measured directly. However, we can obtain it indirectly by combining other measurements in an application of Hess s law. This approach takes advantage of the first law of thermodynamics and, in particular, the fact that enthalpy is a state function. The procedure uses a Born-Haber cycle, a closed path of steps, one of which is the formation of a solid lattice from the gaseous ions. The enthalpy change for this step is the negative of the lattice enthalpy. Table 6.6 lists some lattice enthalpies found in this way. 

Calculate AU and AS for this entire cycle, (b) What are the values of q and w for the entire cycle (c) What are A.S slin. and AStota for the cycle If any values are nonzero, explain how this can be so, despite entropy being a state function, (d) Is the process spontaneous, nonspontaneous, or at equilibrium ... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

There is no single criterion for the system alone that applies to all processes. However, if we restrict the conditions to constant temperature and pressure, there is a state function whose change for the system predicts spontaneity. This new state function is the free energy (G), which was introduced by the American J. Willard Gibbs and is defined by Equation G = H - T S As usual, H is enthalpy, T is absolute temperature, and S is entropy. 

By omitting time-dependent terms, as in the preceding paragraph, the liP ) function may be read as the sum of the unperturbed wavefunction ) and a term which is the product of this function by a linear combination of the electronic coordinates, i.e. the Kirkwood s j) function. Thus, the (r) dipolar factor ensures gauge-invariance. But the role of the dipolar factor g f) in this mixed method is essential on the following point its contribution in the a computation occurs in a complementary (and sometimes preponderant) way to that calculated only from the n) excited states, the number of which is unavoidably limited by the computation limits. But before discussing their number, we have to comment the description of these states. 

Figure 4. Potential energy curves for the Sq and S, states of BMPC as a function of the angle of rotation...

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