Internal equations of state

Table 3.4 The International Equation of State of Seawater, 1980, Definition.

I. The One Atmosphere International Equation of State of Seawater formulated to compute p(S, t, 0). This can be converted into o using Eq. 3.5... 

Note that p and a are thus functions of properties, so they too are properties. The properties s, v and all functions thereof are called internal properties (thus u and b are called internal energy and internal availability respectively). The functions in Eqs. 27 through 30 are called thermostatic constitutive relations or internal "equations" of state — or better, internal functions of state. 

The Practical Salinity Scale 1978 and the International Equation of State of Seawater 1980, Unesco Technical Papers in Marine Science No. 36, Unesco, Paris, 1981 sections No. 37, 38, 39, and 40 in this series give background papers and detailed tables. 

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... 

Because V is related to T and P through an equation of state, V rather than P can serve as an independent variable. In this case the internal energy and entropy are the properties of choice whence... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Chirat, R. and Baute, J., An Extensive Application of WCA4 Equation of State for Explosives, in Eighth Symposium (International) on Detonation, NSWC MP 86-194 (edited by Short, J.M.), Naval Surface Weapons Center, White Oak, Silver Spring, MD, 1986, pp. 751-761. 

Gas thermometers that employ equation (1.10) can be constructed to measure either pressure while holding the volume constant (the most common procedure) or volume while holding the pressure constant. The (pV) product can be extrapolated to zero p. but this is an involved procedure. More often, an equation of state or experimental gas imperfection data are used to correct to ideal behavior. Helium is the usual choice of gas for a gas thermometer, since gas imperfection is small, although other gases such as hydrogen have also been used. In any event, measurement of absolute temperature with a gas thermometer is a difficult procedure. Instead, temperatures are usually referred to a secondary scale known as the International Temperature Scale or ITS-90. 

The major difficulty in applying this hydrodynamic theory of detonation to practical cases lies in the calculation of E2, the specific internal energy of the explosion products immediately behind the detonation front, without which the Rankine-Hugoniot curve cannot be drawn. The calculations require a knowledge of the equation of state of the detonation products and also a full knowledge of the chemical equilibria involved, both at very high temperatures and pressures. The first equation of state used was the Abel equation... 

An infinitesimal change in internal energy is an exact differential and is a unique function of temperature and pressure (for a given composition). Since the density of a given material is also uniquely determined by temperature and pressure (e.g., by an equation of state for the material), the internal energy may be expressed as a function of any two of the three terms T, P, or p (or v = 1/p). Hence, we may write ... 

The equation of state (7.2) can now be applied in estimating the maximum pressure in the firework body, assuming the internal volume to be 4cm. ... 

Johannes van der Waals developed his famous equation of state by the introduction of both the attractive and the repulsive forces between the molecules. First he postulated that the gas behaves as if there is an additional internal pressure to augment the external applied pressure, which is based on the mutual attraction of molecules since the density of molecules is proportional to 1/V, the intensity of the binary attractive force would be proportional to 1/V. Then he postulated that when the measured total volume begins to approach the volume occupied by the real gaseous molecules, the free volume is obtained by subtracting the molecular volume from the measured volume. Then he introduced the parameter a, which represents an attractive force responsible for the internal pressure, and the parameter b, which represents the volume taken by the molecules. He arrived at... 

While Eqs. (9)-(16) provide a recipe for evaluating Sc T) for constant volume (V) systems (i.e., constant cj)), they can easily be applied to constant pressure systems by computing cj) for a given pressure P and temperature T from the equation of state. For internal consistency, this equation of state must be derived from the free energy T expression appropriate to the /-ensemble. 

It is std practice in foe formulation of interior ballistic theory to assume an equation of state of the simple covolume type. For a gas obeying the Abel equation of state, the internal energy depends only on temp and not on density. It is expressed as... 

Making use the equation of state in Eq. (28), the work, heat and internal energy change at V, T = const, can be derived... 

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