Equation isochoric

Finally, we consider the isothennal compressibility = hi V/dp)y = d hi p/5p) j, along tlie coexistence curve. A consideration of Figure A2.5.6 shows that the compressibility is finite and positive at every point in the one-phase region except at tlie critical point. Differentiation of equation (A2.5.2) yields the compressibility along the critical isochore ... 

Besides shear-induced phase transitions, Uquid-gas equilibria in confined phases have been extensively studied in recent years, both experimentally [149-155] and theoretically [156-163]. For example, using a volumetric technique, Thommes et al. [149,150] have measured the excess coverage T of SF in controlled pore glasses (CPG) as a function of T along subcritical isochoric paths in bulk SF. The experimental apparatus, fully described in Ref. 149, consists of a reference cell filled with pure SF and a sorption cell containing the adsorbent in thermodynamic equilibrium with bulk SF gas at a given initial temperature T,- of the fluid in both cells. The pressure P in the reference cell and the pressure difference AP between sorption and reference cell are measured. The density of (pure) SF at T, is calculated from P via an equation of state. 

This model was applied by Mukherjee et al. [20] for various natural fibers. By considering diverse mechanisms of deformation they arrived at different calculation possibilities for the stiffness of the fiber. According to Eq. (1), the calculation of Young s modulus of the fibers is based on an isochoric deformation. This equation sufficiently describes the behavior for small angles of fibrils (<45°) [19]. 

Two isotherms, isochores, adiabatics, or generally any two thermal lines of the same kind, never cut each other in a surface in space representing the states of a fluid with respect to the three variables of the characteristic equation taken as co-ordinates, for a point of intersection would imply that two identical states had some property in a different degree (e.g., two different pressures, or temperatures). Two such curves may, however,... 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

We have seen how to calculate q for the isochoric and isobaric processes. We indicated in Chapter 1 that q = 0 for an adiabatic process (by definition). For an isothermal process, the calculation of q requires the application of other thermodynamic equations. For example, q can be obtained from equation (2.3) if AC and w can be calculated. The result is... 

This equation is sometimes called the van t Hoff isochore, to distinguish it from van t Hoff s osmotic pressure equation (Section 8.17). An isochore is the plot of an equation for a constant-volume process. 

Again using Euler s chain rule, the value in brackets equals the negative of the isochoric derivative of temperature with respect to pressure. Thus, Equation 24.16 simplifies to... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

Strategy. For convenience, we will call the higher temperature T> and the lower temperature T. (1) The van t Hoff isochore, Equation (4.78), is written in terms of a ratio, so we do not need the absolute values. In other words, in this example, we can employ the solubilities s without further manipulation. We can dispense with the units of. v for the same reason. (2) We convert the two temperatures to kelvin, for the van t Hoff isochore requires thermodynamic temperatures, so T2 = 343.9 K and 7) = 312.0 K. (3) We insert values into the van t Hoff isochore (Equation (4.78)) ... 

The isochore, Equation (4.81), was derived from the integrated form of the Gibbs-Helmholtz equation. It is readily shown that the van t Hoff isochore can be rewritten in a slightly different form, as ... 

Figure 8.6B shows a wider P-T portion with the location of the critical region for H2O, bound by the 421.85 °C isotherm and the p = 0.20 and 0.42 glcvci isochores. The PVT properties of H2O within the critical region are accurately described by the nonclassical (asymptotic scaling) equation of state of Levelt Sengers et al. (1983). Outside the critical region and up to 1000 °C and 15 kbar, PVT properties of H2O are accurately reproduced by the classical equation of state of Haar et al. (1984). An appropriate description of the two equations of state is beyond the purposes of this textbook, and we refer readers to the excellent revision of Johnson and Norton (1991) for an appropriate treatment. 

Compression in a Roots pump is performed by way of external compression and is termed as isochoric compression. Experience shows that the following equation holds approximately ... 

Accdg to von Stein Alster (Ref 41), accurate determination of isochoric adiabatic flame temp of an expl often involves a series of tedious calcns of the equilibrium of compn of the expln products at several temps. Calcg the expln product compn at equilibrium is a tedious process for it.requires the soln of a number of non-linear simultaneous equations by a laborious iterative procedure. Damkoehler Edse (Ref 11) developed a graphical procedure and Wintemitz (Ref 26a) improved it. by transforming it into its algebraic equivalent. Unfortunately both.methods proved less useful with.hetorogeneous equilibria which. contain solid carbon... 

As shown above, the most important quantity in making foamed plastics (16) is the closing pressure. In the discussion below, the pressure is understood to be that which must be applied to keep the volume of the specimen constant as the temperature is raised. In this way the closing pressure is expressed as the isochore of an equation of state. 

Equation (11.163) shows how the isochoric heat capacity of a heterogeneous two-phase system can be evaluated from known isobaric properties (CP, aP) of the individual phases and the direction y(T of the coexistence coordinate cr. 

Of course, the vapour pressure is very temperature dependent, and reaches P° = 101.325 kPa at the normal boiling point, Tb. The isochoric thermal pressure coefficient, dp/dT)v = otp/KT, can be obtained from the two quantities on the right hand side listed in Table 3.1. Except at T it does not equal the coefficient along the saturation line, (dp/dT)a, which is the normal vapour pressure curve. The latter temperature dependence is often described by means of the Antoine equation ... 

For some purposes, a graphical presentation of information is preferable to an equation. As shown in Eq. (6), the ideal gas law is a relation among three variables. In order to represent this equation in a two-dimensional plot, one variable must be held constant. The plots are called isotherms, isobars, or isochores, depending on whether temperature, pressure, or volume, respectively, is held constant. The plots for an ideal gas are shown in Fig. 1. 

Problem 5 Give the thermodynamic derivation of van t Hoff isochore or van t Hoff equation. Mention its applications also. 

Equation (5) represents the variation of equilibrium constant with temperature at constant pressure. This equation is referred to as van t Hoff reaction isochore (Greek isochore = equal space), as it was first derived by van t Hoff for a constant volume system. Since AH is the heat of reaction at constant pressure, the name isochore is thus misleading. Therefore, equation (S) is also called as van s Hoff equation. 

The van t Hoff Isochore or van t Hoff Equation (Equation (46.11), Frame 46) expressed the variation of the In Kp/po term with change in temperature, T in the form ... 

Some texts refer to this equation (46.12) as the van t Hoff Isochore (Frame 1) (but there is no constancy of volume involved). 

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