Thermodynamics equation of state

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids which equates the thermal expansion coefficient with the quantity where yis the Griineisen dimensionless ratio, C, is the specific heat of the solid, p is the density of the material, and B is the bulk modulus. For fee metals the average value of the Griineisen constant is near 2.3. However, there is a tendency for this constant to increase with atomic number. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

D.L. Hicks, von Neumann Stability of the Wondy Wavecode for Thermodynamic Equations of State, SAND77-0934, Sandia National Laboratories, Albuquerque, NM, 1977. 

For the purpose of illustrating the application of the thermodynamic equation of state to experimental data, consider the plot given in Fig. 84 for the retractive force, measured at fixed length, against the absolute temperature for a hypothetical elastic substance. The slope at any temperature T gives the important quantity —(dS/dL)T,p according to Eq. (12) an increase in / with T at constant L shows immediately, therefore, that the entropy decreases with increase in length... 

This version of the thermodynamic equation of state for elasticity is most useful for interpretation of the experimental data discussed below. By measuring the force as a function of temperature at constant pressure and elongation a, one may readily derive (dE/dL)T,v from Eq. (22) and dS/dL)T,v from Eq. (20). 

Thermodynamic Equation of State for Classical Fluid Mixtures of Molecules Interacting with Alpha-exponential-six Pair Potentials up to High Densities. 

See Activity Coefficients Additivity Principle Biochemical Thermodynamics Chemical Potential Equilibrium Constants Hess s Law Innate Thermodynamic Quantities Molecular Crowding Thermodynamics, Laws of Thermodynamic Cycle Thermodynamic Equations of State... 

THERMODYNAMIC EQUATIONS OF STATE Thermodynamic isotope effect, EQUILIBRIUM ISOTOPE EFFECT THERMODYNAMIC pK, THERMODYNAMICS, LAWS OF ENTROPY ENTHALPY... 

Single molecule pulUng experiments can be described with the formalism developed in Section lll.C.l. In the simplest setting the configurational variable C corresponds to the molecular extension of the complex (handles plus inserted molecule) and the control parameter X is either the force/measured in the bead or the molecular extension of the system, x. For small enough systems the thermodynamic equation of state is dependent on what is the variable that is externally controlled [87]. In the actual experiments, the assumption that either the force or the extension is controlled is just an approximation. Neither the molecular extension nor the force can be really controlled in optical tweezers [88]. For example, in order to control the force a feedback mechanism must operate at aU times. This feedback mechanism has a time delay response so the force is never really constant [89, 90]. By assuming that the force is constant. 

In the realni of classical thermodynamics, equations of state arc assumed given. They can be derived from first principles only by the methods of statistical mechanics and quantum mechanics These rely on the adoption of suitable molecular models for substances, and so far no universal, generally applicable model has heen discovered even for narrow classes of subslunces such as gases. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

Because the right-hand sides of Eqs. (30) and (31) can be evaluated from equations of state, we see that such equations plus heat capacity data allow us to completely calculate changes of U and H. Equations (30) and (31) are known as the thermodynamic equations of state. 

This equation 5.20, called the Gibbs-Duhem equation, is unique among a variety of the thermodynamic equations of state in that the characteristic variables are all intensive quantities, each multiplied by its conjugate extensive quantity. 

Graphical Gradients and Intercepts Differentiation in Thermodynamics Equation of State for an Ideal Gas... 

Maxwell first noted the cross relations based on a property of the total differentials of the state functions. The cross differentiations of a total differential of the state function are equal to each other. Table 1.14 summarizes the total differentials and the corresponding Maxwell relations. The Maxwell relations may be used to construct important thermodynamic equations of states. 

Equation (5.8.8a) specifies the heat capacity C at fixed volume and electromagnetic fields Eq. (5.8.8b) represents the generalization of Eq. (1.18.13a), i.e., the thermodynamic equation of state. The two remaining differential equations are more immediately relevant to the topic taken up in the present section they show how U varies with Ea or H0. 

In most cases, the growth of polymeric chains is accompanied by volume contraction. Therefore external pressure tends to shift the monomer polymer equilibrium in favour of the polymer or, in other words, it increases the ceiling temperature of polymerization (lowers 7 ). This analysis can be refined by means of the known thermodynamic relations. The change in enthalpy with pressure is described by the thermodynamic equation of state... 

Bina C. R. and Helflfich G. R. (1992) Calculation of elastic properties from thermodynamic equation of state principles. Ann. Rev. Earth Planet. Set 20, 527—552. 

Atmospheric GCMs simulate the time evolution of various atmospheric fields (wind speed, temperature, surface pressure, and specific humidity), discretized over the globe, through the integration of the basic physical equations the hydrostatic equation of motion, the thermodynamic equation of state, the mass continuity equation, and a water vapor transport equation. To reproduce the... 

Precisely similar methods are applied to determine the thermodynamic equations of state for the enthalpy. One obtains... 

Different concentration types are used for different reaction systems. For gas-phase reactions, volumetric concentration or partial pressures are equally useful and can be related by the thermodynamic equation of state. For instance, for ideal gases (approximation valid for gases at very low pressure)... 

Sengers, J. V., and International Union of Pure and Applied Chemistry Commission on Thermodynamics. Equations of State for Fluids and Fluid Mixtures. Vol. 5 of Experimental Thermodynamics. New York Elsevier, 2000. 

Another thermodynamic equation of state is obtained by the combination of equations (20.14) and (20.15) thus,... 

The first term on the left-hand side describes the variation of the fluid momentum in time and the second term describes the transport of the momentum in the flow (convective transport). The first term on the right-hand side describes the effect of gradients in the pressure p the second term, the transport of momentum due to the molecular viscosity p (diffusive transport) the third term, the effect of gravity g and in the last term, F lumps together all the other forces acting on the fluid. Techniques for solving the set of four equations (one continuity and three momentum equations) are discussed in a later section of this entry. When the flow is compressible, it is usually necessary to close the system of equations listed above using a thermodynamic equation of state (such as the ideal gas law) that calculates the density as a function of temperature and pressure. 

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