Intensive state function

The internal energy per unit mass e is an intensive (state) function. Enthalpy h, a compound thermodynamic function defined by Equation 1.8, is also an intensive function. 

A lot of thermodynamics makes use of the important concept of state function, which is a property with a value that depends only on the current state of the system and is independent of the manner in which the state was prepared. For example, a beaker containing 100 g of water at 25°C has the same temperature as 100 g of water that has been heated to 100°C and then allowed to cool to 25°C. Internal energy is also a state function so the internal energy of the beaker of water at 25°C is the same no matter what its history of preparation. State functions may be either intensive or extensive temperature is an intensive state function internal energy is an extensive state function. 

Internal energy (through the enthalpy, defined in Sec. 2.5) is useful for the calculation of heat and work quantities for such equipment as heat exchangers, evaporators, distillation columns, pumps, compressors, turbines, engines, etc., because it is a state function. The tabulation of all possible Q s and W s for all possible processes is impossible. But the intensive state functions, such as specific volume and specific internal energy, are properties of matter, and they can be measured and their values tabulated as functions of temperature and pressure for a particular substance for future use in the calculation of Q or W for any process involving that substance. The measurement, correlation, and use of these state functions is treated in detail in later chapters. 

Loosely speaking, pressure of an ideal gas is the random translational kinetic energy per volume. Pressure is an intensive state function. Pressure and volume are discussed in depth in Chemistry Lecture 2 and Physics Lecture 5. 

The corresponding intensive state function, which enjoys only r - -1 degrees of freedom, is obtained via division by the total number of moles, V,- = iVtotai. Hence,... 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

That is, IjL is equal to the molar Gibbs energy of tbe substance at a given temperature and pressure. (Section 9.2.6 will introduce a more general definition of chemical potential that applies also to a constituent of a mixture.) The chemical potential is an intensive state function. 

This is the rate at which property X changes with the amount of species i added to the mixture as the temperature, the pressure, and the amounts of all other species are kept constant. A partial molar quantity is an intensive state function. Its value depends on the temperature, pressure, and composition of the mixture. 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

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. 

Equation 19.17 is not the original way the N, p(r) ambiguity was resolved [20]. As mentioned previously, the original paper on the shape function in the isomorphic representation performed the Legendre transform on the energy per particle. This gives an intensive, per electron, state function [20] ... 

It is important to distinguish between the intensive (state) properties (functions) and the extensive properties (functions). 

It should be emphasized that the criterion for macroscopic character is based on independent properties only. (The importance of properly enumerating the number of independent intensive properties will become apparent in the discussion of the Gibbs phase rule, Section 5.1). For example, from two independent extensive variables such as mass m and volume V, one can obviously form the ratio m/V (density p), which is neither extensive nor intensive, nor independent of m and V. (That density cannot fulfill the uniform value throughout criterion for intensive character will be apparent from consideration of any 2-phase system, where p certainly varies from one phase region to another.) Of course, for many thermodynamic purposes, we are free to choose a different set of independent properties (perhaps including, for example, p or other ratio-type properties), rather than the base set of intensive and extensive properties that are used to assess macroscopic character. But considerable conceptual and formal simplifications result from choosing properties of pure intensive (R() or extensive QQ character as independent arguments of thermodynamic state functions, and it is important to realize that this pure choice is always possible if (and only if) the system is macroscopic. 

Let us return to Figure 8-1 and ask about the nature of the steady state in a multi-component, multiphase system when we establish different (constant) intensive thermodynamic functions of state at the end reservoirs (R, and R2). Hereby, we generalize the situations which have been discussed so far. Without working out the solutions in any detail, let us nevertheless consider the necessary conditions and equations for a quantitative treatment and visualize the multiphase demixing with the help of reaction paths in the pertinent phase diagrams. The nomenclature is given in Figure 8-1. 

It is always convenient to use intensive thermodynamic variables for the formulation of changes in energetic state functions such as the Gibbs energy G. Since G is a first order homogeneous function in the extensive variables V, S, and rtk, it follows that [H. Schmalzried, A.D. Pelton (1973)]... 

The intensive variables T, P, and nt can be considered to be functions of S, V, and dj because U is a function of S, V, and ,. If U for a system can be determined experimentally as a function of S, V, and ,, then T, P, and /q can be calculated by taking the first partial derivatives of U. Equations 2.2-10 to 2.2-12 are referred to as equations of state because they give relations between state properties at equilibrium. In Section 2.4 we will see that these Ns + 2 equations of state are not independent of each other, but any Ns+ 1 of them provide a complete thermodynamic description of the system. In other words, if Ns + 1 equations of state are determined for a system, the remaining equation of state can be calculated from the Ns + 1 known equations of state. In the preceding section we concluded that the intensive state of a one-phase system can be described by specifying Ns + 1 intensive variables. Now we see that the determination of Ns + 1 equations of state can be used to calculate these Ns + 1 intensive properties. 

State functions which depend on the mass of material are called extensive properties (e.g., U, V). On the other hand some state functions are independent of the amount of materials. These are called intensive properties (e.g., P, T). 

Enthalpy plays a role in constant-pressure processes similar to that of internal energy in constant-volume processes. The heat added to a system in a constant-pressure process is the enthalpy increase of the system. Because U, P, and V (and /, and Lt) are all state functions, H is also a state function. It is extensive. The molar enthalpy, Hm = H/n, is intensive. Dividing Eq. (23) by dTP gives... 

In addition to being a function of T, the partition function is also a function of V, on which the quantum description of matter tells us that the molecular energy levels, , depend. Because, for single-component systems, all intensive state variables can be written as functions of two state variables, we can think of q(T, V) as a state function of the system. The partition function can be used as one of the independent variables to describe a single-component system, and with one other state function, such as T, it will completely define the system. All other properties of the system (in particular, the thermodynamic functions U, H, S, A, and G) can then be obtained from q and one other state function. 

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

The characteristic feature of all these processes is the coexistence of two phases. According to the phase rule, a two-phase system consisting of a single species is univariant, and its intensive state is determined by the specification of just one intensive property. Thus the latent heat accompanying a phase change is a function of temperature only, and is related to other system properties by an exact thermodynamic equation ... 

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