Variables, thermodynamic

Thermodynamics is concerned only with the macroscopic properties of a body and not with its atomic properties, such as the distance between the atoms in a particular crystal. These macroscopic properties form a large class and include the volume, pressure, surface 

The extensive properties, such as volume and mass, are those which are additive, in the sense that the value of the property for the whole of a body is the sum of the values for all of its constituent parts. 

The intensive properties, such as pressure, density, etc., are those whose values can be specified at each point in a system and which may vary from point to point, when there is an absence of equilibrium. Such properties are not additive and do not require any specification of the quantity of the sample to which they refer. 

Consider the latter claes and let it be supposed that the S3mtem under discussion is closed and consists of a single phase which is in a state of equilibrium, and is not significantly affected by external fields. For such a system it is usually found that the specification of any two of the intensive variables will determine the values of the rest. For example, if /j, /2,//. / are the intensive properties then the fixing of, say, and will give the values of all the others. 

For example, if the viscosity of a sample of water is chosen as 0.506 X 10 N s m- and its refractive index as 1.328 9, then its density is 0.988 1 g cm 8, its hotness is 50 C, etc. In the next section, instead of choosing viscosity and refractive index, we shall take as our reference variables the pressure and density, which are a more convenient choice. On this basis we shall discuss what is meant by hotness or temperature (which it is part of the business of thermodynamics to define), and thereafter we shall take pressure and temperature as the independent intensive variables, as is always done in practice. 

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. 

A state function is a property of a system that has a value that depends on the conditions (state) of the system and not on how the system has arrived at those conditions (the thermal history of the system). For example, the temperature in a room at a given time does not depend on whether the room was heated up to that temperature or cooled down to it. The difference in any state function is identical for every process that takes the system from the same given initial state to the same given final state it is independent of the path or process connecting the two states. Whereas the internal energy of a system is a state function, work and heat are not. Work and heat are not associated with one given state of the system, but are defined only in a transformation of the system. Hence the work performed and the heat 

The thermodynamic state of unit mass of a homogeneous fluid is definite when fixed values are assigned to any two of the following three variables pressure, p, temperature, T, and specific volume, v. These variables will be connected by an equation of state of the form  

In particular, it is useful to emphasize that the thermodynamic state is defined completely if the pressure and the temperature of the fluid under consideration are known. 

For a gas or a vapour, we may express the equation of state with good accuracy by 

R is the universal gas constant = 8314J/(kmolK), T is the absolute temperature (K), w is the molecular weight of the gas, and R = R/w is the characteristic gas constant, which applies only to the gas or gas mixture in question (J/kgK). 

Pressure, temperature and specific volume have a claim to be regarded as the most basic of the thermodynamic variables because of the ease with which they can be sensed and measured, and hence their familiarity to the practising physicist or engineer. However, there are a number of other thermodynamic variables necessary for the simulation of industrial processes that will be considered here. 

DSC data are recorded on a thermogram delineated by ACp v vs AT or At. Inasmuch as it is the energy consumption that is recorded, the thermogram shows + AE proportional to the change in property. DSC is therefore a method of directly quantifying ACp v, AH, AS [Eq. (3.16)], and ACp v/g (the specific heat). Exothermic reactions are characterized by positive peaks on the thermogram, and endothermic reactions are characterized by negative peaks. 

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A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

The second, third, and fourth corrections to [MPd/b-Jl lG(d,p)] are analogous to A (- -). The zero point energy has been discussed in detail (scale factor 0.8929 see Scott and Radom, 1996), leaving only HLC, called the higher level correction, a purely empirical correction added to make up for the practical necessity of basis set and Cl truncation. In effect, thermodynamic variables are calculated by methods described immediately below and HLC is adjusted to give the best fit to a selected group of experimental results presumed to be reliable. 

Since entropy plays the determining role in the elasticity of an ideal elastomer, let us review a couple of ideas about this important thermodynamic variable ... 

Figure 4.3 Behavior of thermodynamic variables at T for an idealized phase transition (a) Gibbs free energy and (b) entropy and volume.

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

Relationships which exist between ordinary thermodynamic variables also apply to the corresponding partial molar quantities. Two such relationships are... 

To extract a desired component A from a homogeneous liquid solution, one can introduce another liquid phase which is insoluble with the one containing A. In theory, component A is present in low concentrations, and hence, we have a system consisting of two mutually insoluble carrier solutions between which the solute A is distributed. The solution rich in A is referred to as the extract phase, E (usually the solvent layer) the treated solution, lean in A, is called the raffinate, R. In practice, there will be some mutual solubility between the two solvents. Following the definitions provided by Henley and Staffin (1963) (see reference Section C), designating two solvents as B and S, the thermodynamic variables for the system are T, P, x g, x r, Xrr (where P is system pressure, T is temperature, and the a s denote mole fractions).. The concentration of solvent S is not considered to be a variable at any given temperature, T, and pressure, P. As such, we note the following ... 

The work on iron-nickel alloys has described shock-compression measurements of the compressibility of fee 28.5-at. % Ni Fe that show a well defined, pressure-induced, second-order ferromagnetic to paramagnetic transition. From these measurements, a complete description is obtained of the thermodynamic variables that change at the transition. The results provide a more complete description of the thermodynamic effects of the change in the magnetic interactions with pressure than has been previously available. The work demonstrates how shock compression can be used as an explicit, quantitative tool for the study of pressure sensitive magnetic interactions. 

The properties of a system at equilibrium do not change with time, and time therefore is not a thermodynamic variable. An unconstrained system not in its equilibrium state spontaneously changes with time, so experimental and theoretical studies of these changes involve time as a variable. The presence of time as a factor in chemical kinetics adds both interest and difficulty to this branch of chemistry. 

The second law of thermodynamics also consists of two parts. The first part is used to define a new thermodynamic variable called entropy, denoted by S. Entropy is the measure of a system s energy that is unavailable for work.The first part of the second law says that if a reversible process i f takes place in a system, then the entropy change of the system can be found by adding up the heat added to the system divided by the absolute temperature of the system when each small amount of heat is added ... 

The temperature dependence of the equilibrium cell voltage forms the basis for determining the thermodynamic variables AG, A//, and AS. The values of the equilibrium cell voltage A%, and the temperature coefficient dA< 00/d7 which are necessary for the calculation, can be measured exactly in experiments. 

Most of the 50,000,000 equations have little use. However, a significant number are invaluable in describing and predicting the properties of chemical systems in terms of thermodynamic variables. They serve as the basis for deriving equations that apply under experimental conditions, some of which may be difficult to achieve in the laboratory. Their applications will form the focus of several chapters. 

But just what are the thermodynamic variables that we use to describe a system And what is a system What are Energie (energy) and Entropie (entropy) as described by Clausius We will soon describe the thermodynamic variables of interest. But first we need to be conversant in the language of thermodynamics. 

Before describing these thermodynamic variables, we must talk about their properties. The variables are classified as intensive or extensive. Extensive variables depend upon the amount while intensive variables do not. Density is an example of an intensive variable. The density of an ice crystal in an iceberg is the same as the density of the entire iceberg. Volume, on the other hand, is an extensive variable. The volume of the ocean is very different from the volume of a drop of sea water. When we talk about an extensive thermodynamic variable Z we must be careful to specify the amount. This is usually done in terms of the molar property Zm, defined as... 

The second thing to note about the thermodynamic variables is that, since they are properties of the system, they are state functions. A state function Z is one in which AZ = Zi — Z that is, a change in Z going from state (l) to state (2), is independent of the path. If we add together all of the changes AZ, in going from state (1) to state (2), the sum must be the same no matter how many steps are involved and what path we take. Mathematically, the condition of being a state function is expressed by the relationship... 

In addition to the fundamental variables p, V, T, U, and S that we have described so far, three other thermodynamic variables are commonly encountered enthalpy Helmholtz free energy and Gibbs free energy. They are extensive variables that do not represent fundamental properties of the... 

We have now described the thermodynamic variables n, p, T, V, U, S, H, A, and G and expressed expectations for their usefulness. We will be ready to start deriving relationships between these variables after we review briefly the mathematical operations we will employ. 

One of the pleasant aspects of the study of thermodynamics is to find that the mathematical operations leading to the derivation and manipulation of the equations relating the thermodynamic variables we have just described are relatively simple. In most instances basic operations from the calculus are all that are required. Appendix 1 reviews these relationships. 

Of special interest are the properties of the exact differential. We have seen that our thermodynamic variables are state functions. That is, for a thermodynamic variable Z... 

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. 

We start by noting that any dependent thermodynamic variable Z is completely specified by two — and only two — independent variables X and Y (if n held constant). As an example, the molar volume of the ideal gas depends upon the pressure and temperature. Setting p and T fixes the value of Vm through the equation... 

Similar types of relationships can be found between the other thermodynamic variables. In general, specifying two variables fixes the state of the third.y Thus specifying Vm and T fixes the value of Sm, specifying Hm and Gm fixes Um, and so on. As another example, Figure 1.4 shows the (Sm, p, T) surface for an ideal monatomic gas.z The entropy, Sm is restricted to values of p and T on the surface. 

By using relationships for an exact differential, equations that relate thermodynamic variables in useful ways can be derived. The following are examples. 

We have seen that a thermodynamic variable, Z, can be completely specified by two other thermodynamic variables, X and F, in which case, we can write... 

Pl.l Use the properties of the exact differential and the defining equations for the derived thermodynamic variables as needed to prove the following relationships ... 

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... 

The substitutions can be made because the extensive thermodynamic variables in the equations are homogeneous of degree one.d Thus, dividing the equation by n converts the extensive variable to the corresponding molar intensive variable. For example, to prove that equation (3.48) follows from equation... 

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