Specifying the State of a System

Classically we can specify the state of a particle by its position and velocity or momentum coordinates (x, p). These coordinates define a point in a six-dimensional (6-D) phase space. To specify the state of a system containing many particles, rather than specify the coordinates of every particle, it is simpler to divide phase space up into many cells and specify the number of particles in each cell or better yet, to specify the number of particles Ni that occupy those cells that correspond to a given energy, Ej. A function that specifies the number of particles for a given energy range is an energy distribution fimction. We will now derive distribution fimctions for various types of particles. 

we must decide on a logical choice for the size of the cell in phase space. We know from Heisenberg s imcertainfy principle that we cannot specify position and momentum of a particle to a precision greater than h, AyAp h, and AzAv h therefore, the 

Now consider a group of cells that correspond to the same energy, Ej. Denote the number of cells in this group, g,. Let N be the number of particles in this state. How many ways can we arrange Nj particles in gi cells  

We can write a sequence representing the number of particles in each cell by starting with a cell number and let the following letters represent the particles in that cell. 

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Intensive properties that specify the state of a substance are time independent in equilibrium systems and in nonequilibrium stationary states. Extensive properties specifying the state of a system with boundaries are also independent of time, and the boundaries are stationary in a particular coordinate system. Therefore, the stationary state of a substance at ary point is related to the stationary state of the system. 

One important characteristic of a state function is that once we have specified the state of a system by giving the values of some of the state functions, the values of all other state functions are fixed. Thus, in the example just given, once we have specified the mass, temperature, and pressure of the water, the volume is fixed. So, too, is the total energy of the system, and energy is therefore another state function. When the concept of entropy is discussed in the next chapter we shall see that entropy is also a state function. 

Degrees of freedom Variables which must be determined to specify the state of a system. 

Incomplete. Additional functions could be constructed, such as Uip, V,N) or S(U,p,N) but because these involve conjugate pairs p and V, or p andN, and are missing other variables, they do not uniquely specify the state of a system. Such functions cannot be obtained by Legendre transforms of the fundamental equations. 

The quantum average 0) of the dynamical variable O is actually a conditional average, i.e., it is conditional on the fact that the system of interest has been prepared in the state ir). In general, we lack sufficient infonnation to specify the state of a system. Thus, additional probabilistic concepts must be introduced that are not inherent in the formal structure of quantum mechanics. 

The concept of equilibrium is central in thermodynamics, for associated with the condition of internal eqmlibrium is the concept of. state. A system has an identifiable, reproducible state when 1 its propei ties, such as temperature T, pressure P, and molar volume are fixed. The concepts oi state a.ndpropeity are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates I and 3 following) is recognized much more indirectly. The number of properties for wdiich values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. 

In thermodynamics the state of a system is specified in terms of macroscopic state variables such as volume, V, temperature, T, pressure,/ , and the number of moles of the chemical constituents i, tij. The laws of thermodynamics are founded on the concepts of internal energy (U), and entropy (S), which are functions of the state variables. Thermodynamic variables are categorized as intensive or extensive. Variables that are proportional to the size of the system (e.g. volume and internal energy) are called extensive variables, whereas variables that specify a property that is independent of the size of the system (e.g. temperature and pressure) are called intensive variables. 

Although the change in state of the heat bath, hence the value of Q, usually is determined by measuring a change in temperature, this is a matter of convenience and custom. For a pure substance the state of a system is determined by specifying the values of two intensive variables. For a heat bath whose volume (and density) is hxed, the temperature is a convenient second variable. A measurement of the pressure, viscosity, or surface tension would determine the state of the system equally as well. This point is important to the logic of our development because a later dehnition of a temperature scale is based on heat measurements. To avoid circularity, the measurement of heat must be independent of the measurement of temperature. 

The number of independent variables required to specify the state of a mechanical or thermodynamic system. Degrees of freedom arise from the possible motions of molecules or particles in a system. (The term generalized coordinates is also used in physics to designate the minimal number of coordinates needed to specify the state of a mechanical system.) 2. The number of independent or unrestricted random variables constituting a statistic. See Statistics (A Primer)... 

The state of a system is specified if we specify the amounts of each component (in mole fraction) in each phase, the pressure, and the temperature. If all mole fractions but one are specified in each phase, the remainder can be calculated, because the sum of the mole fractions equals unity. Thus there are (c — 1) variables for the specification of the composition in each phase. The total number of variables of the system is, therefore, equal to p x (c 1) 4- 2, including the freedoms of pressure and temperature. The number of degrees of freedom of the system (denoted F) is equal to the number of possible variables (it is possible to take the chemical potentials as variables, instead of the mole fractions) required to specify the system, p X (c — 1) 4- 2, minus the number of variables not being independently changed or the number of restrictions, (p — 1) x c (see eqn (1.45)). Therefore... 

Shapiro and Brumer [S3] have examined a system in which the eigenstates of the Hamiltonian are subdivided into three sets, with dimensions Mo, M and M2, and ask if it is possible to transform a specified initial state of a system that lies in the subset of states with dimension Mo into a specified final state of the system that lies in the subspace of states with dimension M without passage through the states of the system that lie in the subspace with dimension M2. It is shown that if M2 Mo stringent restrictions are required to prevent involvement of the states in M2 in the specified transformation. This inability to direct the evolution of the state of the system away from a specified set of substates does not contradict the Huang-Tam-Clark theorem, since that theorem does not admit constraints on the evolution pathway of the state of the system. 

Figure 5.11 shows one of the most counterintuitive results of quantum mechanics. There are fundamental limits to our ability to make certain measurements—the act of determining the state of a system intrinsically perturbs it. For example, it is impossible to measure position and momentum simultaneously to arbitrarily high accuracy any attempt to measure position automatically introduces uncertainty into the momentum. Similarly, a molecule which is excited for a finite period of time cannot have a perfectly well-defined energy. As a result, classical determinism fails. It is not possible, even in principle, to completely specify the state of the universe at any instant, hence the future need not be completely defined by the past. These results are usually phrased something like ... 

In classical mechanics the state of a system is completely defined by specifying the positions and momenta of all the particles simultaneously. This means that 6 coordinates (3 position coordinates and 3 momentum coordinates) must be specified for each particle. The 6N dimensional space spanned by the coordinates required for a system of N particles is called phase space. 

The state of a system with two phases, and hence two degrees of freedom (divariant system), is not determined until two of its variables have been arbitrarily specified. The most important case is the coexistence of water vapour with a solution of calcium chloride. Here temperature and concentration determine the vapour pressure. Hence the vapour pressure of the solution at constant temperature is a function of its concentration. From this again it follows that the vapour pressure of a solution is always different from the vapour pressure of the. pure solvent at the same temperature. 

Since the state of a system of fixed content is completely determined by the values of P and F, the temperature is unambiguously defined by them. Thus r is a function of P and V, written T = T(P, V), Similarly, when P and T are known, it is implied that V has a unique related value, written V = V Py T), Such relationships embody an equation of state in which one variable is a function of the two independent variables or, since the latter specify the state, a function of state. Thus, for a gas, the equation of state is py = a - -bP cP, where a, b,Cy, depend only on the empirical temperature. In terms of absolute temperature a has the value RTy where R is the gas constant for the quantity of gas considered. When a gas behaves ideally the other terms on the right are negligibly small. 

The macroscopic state of any one-component fluid system in equilibrium can be described by just three properties, of which at least one is extensive. All other properties of the state of the same system are necessarily specified by the chosen three properties. For instance, if for a single component gas in equilibrium, pressure, temperature, and volume are known, all other properties which describe the state of that gas (such as number of moles, internal energy, enthalpy, entropy, and Gibbs energy) must have a specific single value. Since the state of a system can be described exactly by specific properties, it is not necessary to know how the state was formed or what reaction pathway brought a state into being. Such properties that describe the state of a system are called slate functions. Properties that do not describe the state of a system, but depend upon the pathway used to achieve any state, are called path functions. Work and heat are examples of path functions. 

How many variables need to be specified in order to fix the state of a system In order to fix the state of a one-phase system, the composition of the phase must be specified as well as two additional... 

The state of an equilibrium phase is specified by its chemical composition and a relatively few intensive properties, generally two, e.g., temperature and pressure, temperature and mass density, or density and pressure. The number and kind of intensive properties required are determined by experience. To specify the state of a nonequilibrium system generally requires more than the values of a few intensive properties. Thus a system undergoing heat transfer may require a mathematical function to describe its temperature distribution and thus its state. 

If arbitrary values are assigned to any two of the three variables p, V, and T, the value of the third variable can be calculated from the ideal gas law. Hence, any set of two variables is a set of independent variables the remaining variable is a dependent variable. The fact that the state of a gas is completely described if the values of any two intensive variables are specified allows a very neat geometric representation of the states of a system. 

State Variable,. A state variable is one that has a definite value when the state of a system is specified—... 

The smallest number of independent variables that must be specified in order to describe completely the state of a system is known as the number of degrees of freedom,/. For a fixed mass of a gas, / = 2, because one can vary any two variables (e.g., pressure and temperature) the third variable is fixed by the equation of state (e.g. volume). Hence, only two properties of a fixed mass of gas are independently variable. Stated mathematically, for a system at equilibrium, the number of degrees of freedom, /, equals the difference between the number of chemical components, c, and the number of phases, p (22) ... 

An (A, (p) dynamic system is deterministic if knowing the state of the system at one time means that the system is uniquely specified for all r 6 T. In many cases, the state of a system can be assigned to a set of values with a certain probability distribution, therefore the future behaviour of the system can be determined stochastically. Discrete time, discrete state-space (first order) Markov processes (i.e. Markov chains) are defined by the formula... 

In Chapter 3, the states of a system were specified by p, V, and other variables. It was shown that information in the statistical sense was low in most cases and indeed bordered on zero. The reason is that fluctuations wield only tiny impacts for large volume, multiparticle systems under equilibrium conditions. Matters are different when structured programs are applied. All the programmed pathways of Chapters 4 and 5 featured extended collections of states. For a given collection, there was appreciable information allied with the variables in query-and-measurement exercises. The exceptions were n for closed systems, and p, T, S, and so forth for isobaric, isothermal, and adiabatic—the special transformations of thermodynamics. 

The complete description of the state of a system must include the value of an extensive variable of each phase (e.g., the volume, mass, or amount) in order to specify how much of the phase is present. For an equilibrium system of P phases with a total of 3 independent variables, we may choose the remaining 3—P variables to be intensive. The number of these intensive independent variables is called tbe number of degrees of freedom or variance, F, of the system ... 

Either equation. S = S(U, V,N) or U = U S,V,N), will completely specify the state of a simple system. Thermodynamics does not tell you the specific mathematical dependence of S on ([/, V,N) or U on (S, V,N). Equations of state, such as the ideal gas law, specify interrelations among these variables. Equations of state must come either from experiments or from microscopic models. They are imported into thermodyoiamics, which gives additional useful relationships. 

In thermodynamics, the state of a system is specified in terms of macroscopic state variables such as volume V, pressure p, temperature T, mole numbers of the chemical constituents N, which are self-evident. The two laws of thermodynamics are founded on the concepts of energy U, and entropy S, which, as we shall see, are functions of state variables. Since the fundamental quantities in thermodynamics are functions of many variables, thermodynamics makes extensive use of calculus of many variables. A brief summary of some basic identities used in the calculus of many variables is given in Appendix 1.1 (at the end of this chapter). Functions of state variables, such as U and S, are called state functions. 

We say that the state of a system is known if the three independent variables are specified. For example, 1 mol of gas is in state 1 when we know the exact values of Pi, Vi, and Pi. Similarly, the same 1 mol of gas is in state 2 if we know the exact values of P2, V2, and T2. The change in a system occurs when the values change from Pi, Vi, and Pi to P2, V2, and P2. We say that the system changes from state 1 to state 2. In calculus, the term path is used to mean the route of change from state 1 to state 2. This term is important to our discussion because the two quantities dw and Aq both depend on the path, as shown in Figure 4.2. 

With this understanding, how many degrees of freedom must be specified in order to know the state of a system at equilibrium Consider a system that has a number of components C and a number of phases P. To describe the relative amounts (like mole fractions) of the components, C — 1 values must be specified. (The amount of the final component can be determined by subtraction.) Because the phase of each component must be specified, we need to know (C — 1) F values. Finally, if temperature and pressure need to be specified, we have a total of (C — 1) P -F 2 values that we need to know in order to describe our system. 

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