Variable volume thermodynamically determined

We have considered volume changes resulting from density changes in liquid and gaseous systems. These volume changes were thermodynamically determined using an equation of state for the fluid that specifies volume or density as a function of composition, pressure, temperature, and any other state variable that may be important. This is the usual case in chemical engineering problems. In Example 2.10, the density depended only on the composition. In Example 2.11, the density depended on composition and pressure, but the pressure was specified. 

These minimum number of variables that determine the state of a system are called the independent variables, and all other variables which can be functions of the independent variables are dependent variables or thermodynamic functions. For a system where no external force fields exists such as an electric field, a magnetic field and a gravitational field, we normally choose as independent variables the combination of pressure-temperature-composition or volume-temperature-composition. 

How do we know we don t need all three variables - temperature, pressure, and volume - to determine the phase of the system Because Gibbs s Phase Rule, the zeroth law of thermodynamics, guarantees that you only need two for a pure substance. 

Mathematical functions play an important role in thermodynamics, classical mechanics, and quantum mechanics. A mathematical function is a rule that delivers a value of a dependent variable when the values of one or more independent variables are specified. We can choose the values of the independent variables, but once we have done that, the function delivers the value of the dependent variable. In both thermodynamics and classical mechanics, mathematical functions are used to represent measurable properties of a system, providing values of such properties when values of independent variables are specified. For example, if our system is a macroscopic sample of a gas at equilibrium, the value of n, the amount of the gas, the value of T, the temperature, and the value of V, the volume of the gas, can be used to specify the state of the system. Once values for these variables are specified, the pressure, P, and other macroscopic variables are dependent variables that are determined by the state of the system. We say that P is a state function. The situation is somewhat similar in classical mechanics. For example, the kinetic energy or the angular momentum of a system is a state function of the coordinates and momentum components of all particles in the system. We will find in quantum mechanics that the principal use of mathematical functions is to represent quantitites that are not physically measurable. 

Certain thermodynamic relations exist between the state variables. In general for a binary alloy we choose p, T and Xg (the at% of component B) as the independent variables - though presently we shall drop p. The volume 1/ and the composition Xa (= 1 - Xg) are then determined they are the dependent variables. Of course, the weight percentages Wa and Wg can be used instead. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

Considering local equilibrium, the liquid structure in a small portion of space around a point r is determined at time t by the local thermodynamical variables T(r,t) and P(r, t). If further the variation of temperature ST(r,t) and pressure SP(r, t) with respect to the reservoir temperature To and pressure Po is small, the x-ray scattering signal of a small volume tfir at r, which contains dn r) molecules, can be written to first order ... 

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

Gibbs considered the statistical mechanics of a system containing one type of molecule in contact with a large reservoir of the same type of molecules through a permeable membrane. If the system has a specified volume and temperature and is in equilibrium with the resevoir, the chemical potential of the species in the system is determined by the chemical potential of the species in the reservoir. The natural variables of this system are T, V, and //. We saw in equation 2.6-12 that the thermodynamic potential with these natural variables is U[T, //] using Callen s nomenclature. The integration of the fundamental equation for yields... 

In thermodynamics, the observer is outside the system and properties are measured in the surroundings. For example, pressure is measured by an external observer reading a pressure gauge on the system. Volume can be determined by measuring the dimensions of the system and calculating the volume or, in the case of complex shapes, by using the system to displace a liquid from a filled container. Important thermodynamic properties have low information content (i.e., they can be expressed by relatively few numbers). The details of the shape of a system are usually not important in thermodynamics, except, sometimes, a characteristic of the shape, such as the surface-to-volume ratio, or radii of particles, may also be considered. Information only accessible to an observer within the system, such as the positions and velocities of the molecules, is not considered in thermodynamics. However, in Chapter 5 on statistical mechanics, we will learn how suitable averages of such microscopic properties determine the variables we study in thermodynamics. 

Equations (54) and (56) suggest that measurements of the force required to keep a rubber at constant length as a function of temperature would determine the thermodynamic properties of the rubber. However, due to thermal expansion, very large changes in pressure would be required to keep the polymer volume constant as its temperature is varied. Thus, measurements of (df /dT)l are usually made at constant pressure. Extending Eq. (10) of Appendix A to an additional variable gives... 

All observable quantities of the macroscopic property of a thermodynamic system, such as the volume V, the pressure p, the temperature T, and the mass m of the system, are called variables of the state, or thermodynamic variables. In a state of the system all observable variables have their specified values. In principle, once a certain number of variables of the state are specified, all the other variables can be derived from the specified variables. The state of a pure oxygen gas, for example, is determined if we specify two freely chosen variables such as temperature and pressure. 

The thermodynamic state of a pure component is determined by three variables the pressure P, the volume V, and the temperature T. The relationship between these three variables is known as the state equation and is represented by a surface in the three-dimensional plotting of P, V, and T. Any pure component, following the value of these three parameters, will be either a solid (S), a liquid (L), or a gas (G). A plot of P against T, or P against V is generally preferred because of its easier application. 

Variables involved in the study of the relationship of heat and energy are called thermodynamic variables. Examples of these variables are temperature, pressure, free energy, enthalpy, entropy, and volume. In our short discussion of thermodynamics, we will address enthalpy, entropy, and free energy. As mentioned, whether or not a particular reaction, such as a biological reaction, is possible can be determined by the free energy change between products and reactants. Free energy, in turn, is a function of the enthalpy and entropy of the reactants and products. 

Partial Molar Quantities. — The thermodynamic functions, such as heat content, free energy, etc., encountered in electrochemistry have the property of depending on the temperature, pressure and volume, i.e., the state of the system, and on the amounts of the various constituents present. For a given mass, the temperature, pressure and volume are not independent variables, and so it is, in general, sufficient to express the function in terms of two of these factors, e.g., temperature and pressure. If X represents any such extensive property, i.e., one whose magnitude is determined by the state of the system and the amounts, e.g., number of moles, of the constituents, then the partial molar value of that property, for any constituent i of the system, is defined by... 

However, Dubinin and co-workers do not accept the concept of monolayer formation in micropores and propose determining the microporous volume, Fq, on the basis of the thermodynamic theory of Polanyi adsorption. However, one can observe that the monolayer volume, Vm, when expressed in liquid nitrogen volume per unit mass, is very close to the Dubinin volume, Vo. The proportionality of the BET monolayer volume, Vm, and the so-called micropore volume, Va, (Vo 11 Vm) has been observed for many materials, as shown in different studies [2, 3]. This means that both variables are correlated, so determining one is equivalent to the determining the other. The discussion on the physicochemical meaning of these parameters may be interesting from a theoretical point of view but as far as practical characterization of porous materials is concerned, both methods can often be considered as equivalent. 

In traditional equilibrium MD an isolated system of fixed volume V and a fixed number of molecules N is studied. Because the system is isolated its total energy E is constant and thus the variables N, V and E determine the thermodynamic state. The molecules of the system interact through model potentials. The positions of the molecules are obtained by solving the equations of motion for each molecule. 

Nowadays, for a thermodynainicist, /pVT-calorimetry (further referred to as scanning transitiometry, its patented and commercial name ) is the most accomplished experimental concept. It allows direct determinations of the most important thermodynamic derivatives it shows how, in practice, the Maxwell relations can be used to fully satisfy the thermodynamic consistency of those derivatives. Of particular interest is the use of pressure as an independent variable this is typically illustrated by the relatively newly established pressure-controlled scanning calorimeters (PCSC). - Basically, the isobaric expansibility Op(p,T) =il/v)(dv/dT)p can be considered as the key quantity from which the molar volume, v, can be obtained and therefore all subsequent molar thermodynamic derivatives with respect to pressure. Knowing the molar volume as a function of p at the reference temperature, Tg, the determination of the foregoing pressure derivatives only requires the measurement of the isobaric expansibilities as... 

As seen above, a thermodynamic property of a homogeneous system of constant composition is completely determined by the three thermodynamic variables, pressure, volume and temperature. Since only two of these three variables are independent, it is possible to write... 

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