The thermodynamic equation of state

The equations of state discussed so far, the ideal gas law, the van der Waals equation, and others, were relations between p, V, and T obtained from empirical data on the behavior of gases or from speculation about the effects of molecular size and attractive forces on the behavior of the gas. The equation of state for a liquid or solid was simply expressed in terms of the experimentally determined coefficients of thermal expansion and compressibility. These relations applied to systems at equilibrium, but there is a more general condition of equilibrium. The second law of thermodynamics requires the relation, Eq. (10.19), 

By restricting the second fundamental equation, Eq. (10.20), to constant temperature and dividing by (dp)j- we obtain 

If we knew the value of either (dU/dV) or (dH/dp)T for a substance, we would know its equation of state immediately from Eqs. (10.28) or (10.30). More commonly we do not know the values of these derivatives, so we arrange Eq. (10.28) in the form 

From the empirical equation of state, the right-hand side of Eq. (10.31) can be evaluated to yield a value of the derivative dU dV)j. For example, for the ideal gas, p = nRT/V, so (dp/dT)v = nR/V. Using these values in Eq. (10.31), we obtain (dU/dV)T = nRT/V — p = p — p = 0. We have used this result. Joule s law, before this demonstration proves its validity for the ideal gas. 

It is now possible, using Eqs. (10.32) and (10.33), to write the total differentials of 17 and if in a form containing only quantities that are easily measurable  

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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... 

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). 

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. 

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. 

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... 

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)... 

There seems to be a law of nature that, in an equilibrium system, the chemical hardness and the physical hardness have maximum values, compared with nearby non-equilibrium states. However, it must not be inferred that these maximum principles are being proposed to take the place of estabished criteria for equilibrium. Instead, they are necessary consequences of these fundamental laws. It is very clear that the Principle of Maximum Hardness for electrons is a result of the quantum mechanical criterion of minimum energy. Similarly, Sanchez has recently derived the relationship (dB/dP) = 5 by a straightforward manipulation of the thermodynamic equation of state.The PMPH is a result of the laws of thermodynamics. 

Figure 2.29- - An analysis of the thermodynamic equation of state [Eq. (2.69)] for rubber elasticity using a general experimental curve of force versus temperature at constant length. The tangent to the curve at T is extended back to 0°K. For an ideal elastomer, the quantity (dU/df)r is zero, and the tangent goes through the origin. The experimental line is, however, straight in the ideal case. (After Flory, 1953.)...

This equation is sometimes called the thermodynamic equation of state for rubber elasticity. For an ideal elastomer dUjdl )t = 0 Equations (2.63) and (2.69) then reduce to... 

Clarke has examined the thermodynamic equation of state and the specific heat for a Lennard-Jones liquid cooled through 7 at zero pressure. He found that drops with decreasing temperature near where the selfdiffusion becomes very small. Wendt and Abraham have found that the ratio of the values of the radial distribution function at the first peak and first valley shows behavior on cooling much like that observed for the volume of real glasses (Fig. 6), with a clearly defined 7. Stillinger and Weber have studied a Gaussian core model and find a self-diffusion constant that drops essentially to zero at a finite temperature. They also find that the ratio of the first peak to the first valley in the radial distribution function showed behavior similar to that found by Wendt and Abraham" for Lennard-Jones liquids. However, the first such evidence for a nonequilibrium (i.e. kinetic) nature of the transition in a numerical simulation was obtained by Gordon et al., who observed breakaways in the equation of state and the entropy of a hard-sphere fluid similar to those in real materials. 

Finally, we resolve the long-standing problem of behavior of this equation of state in the limit of low densities, which arises in the integration of the thermodynamic equation of state, to obtain the change of internal energy as a function of density along isotherms. 

The function (p,T) of Equation 6 approaches infinity as p- 0, (Ta(p) —> 0). This brings in the question of its validity for use in the thermodynamic equation of state. At very low densities, however, the coexistence temperature diminishes roughly only as the logarithm of density, l/T p) ln(l/p), and hence this behavior must yield a finite integral from the thermodynamic equation of state. 

Placing Equation 4 in the thermodynamic equation of state yields a leading term as follows... 

Using the van der Waals equation with the thermodynamic equation of state, evaluate (dU/dV)j-for the van der Waals gas. 

From this expression and the thermodynamic equation of state show that (dH/dp)T = h-ila/RT). 

Using the thermodynamic equation of state p = N kT, we can replace the density Ab in (6.54) by the pressure p and obtain the Stern-Vollmer equation ... 

This particular equation gives a useful relationship between the variables Uy Vy T and p and is known as the thermodynamic equation of state. ... 

Figure 9.6 An analysis of the thermodynamic equations of state for rubber elasticity (18b).

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