Isochoric temperature derivative

J. C. Dore, M. A. M. Sufi, and M. Bellissent-Funel, Structural change in D2O water as a function of temperature the isochoric temperature derivative function for neutron diffraction. Phys. Chem. Chem. Phys. 2, 1599-1602 (2000). 

The excess energy of solvation at infinite dilution can be obtained by taking the isochoric temperature derivative [135, 136, 129],... 

Fig. 4.16. DC electrical conductivity

Fig. 4.18. Thermopower Q of fluid mercury as a function of density at three constant temperatures. Inset, Isochoric temperature derivative (< In Q/dT)y for the liquid at intermediate densities (Gotzlaff et al., 1988).

Taking the isochoric P-derivative, and recalling that B depends only on temperature,... 

From eq 10.46 it follows that the isochoric specific heat capacity in the two-phase region is related to the temperature derivates of and P by ... 

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 strength of acid sites can be approximated by determining the heat of adsorption using the Van t Hoff isochore. In this method the pressures P required to give a fixed surface coverage at a series of temperatures T is determined. The derivative of In P with respect to T is related to the isosteric heat of adsorption. This technique is not used routinely. Nor is that of IR spectroscopy, where it is reported that Lewis and Bronsted sites can be differentiated by using substituted compounds. 

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. 

In eqs Al-1—Al-3, k is the Boltzmann constant, T is the absolute temperature, Np is the number of particles of species / in the volume v, is the chemical potential per molecule of species a, Va is the partial molar volume per molecule of species a, kf is the isothermal compressibility, and Ca is the bulk molecular concentration of component a (ca = Na/v). The derivative (dfiJdNp)T,v y f is taken rmder isothermal—isochoric conditions and with Ny = constant for any y B o/3 represents the cofactor... 

Using eq. (3) the temperatures are determined that will yield a value of the quantity TV at the higher P equal to values for the atmospherie pressure data. The resulting curves are shown in Figure 2, along with the corresponding derivative plots. From the intersection of the linear segments, we obtain the Ta for each pressure and hence ta(P). The latter, listed in Table 1, is invariant to pressure. We also include in Fig. 2 an isochoric curve at an arbitrary V. Likewise, the transition to Arrhenius behavior under constant volume is associated with the same value of xa- A similar analysis was applied to... 

In this chapter, we shall derive one further equation, the reaction isochore (sometimes called isobar ), and apply this and other equations to a variety of experiments. We shall see that relatively simple measurements of equilibria, taken over a range of temperature, provide information on both enthalpy changes, AH, and entropy changes, AS, and therefore on changes of free energy, AG. 

While the heat capacities at constant pressme are useful for calciflation of various thermodynamic quantities, related to Gibbs free energy for aqueous species, and chemical equilibria involving aqueous species, the isochoric heat capacity measurements are useftil in developing equations of state (EOS) because they yield valuable information about the second derivative of the pressure and of the Helmholtz energy with respect to temperature. Detailed comparisons of experimental Cv data with available EOS are needed to establish their accuracy. Cv experiments contain direct information on the curvature behavior of the p-T isochores, which are extremely important in the development of a reliable EOS. Furthermore, the temperature dependence of Cv at fixed density serves as a sensitive indicator of the phase transition boundary. Thus C measurements provide a tool for investigation of temperatures T and densities d on the coexistence ciuve, especially near the critical point. 

Pressure at the critical point Density at the critical point Temperature at the critical point Isochoric derivative at the critical point First derivative of pressure with respect to density at the critical point Second and subsequent derivatives of pressure with respect to density at the critical point Pc Pc Tc (dp/dT) =dp/dT f dp/dp)T = 0... 

At a given pressure and temperature, the density must first be calculated before the isothermal compressibility or the heat ctq>acities can be obtained. The isothermal compressibility may then be calculated directly fi m equation (8.3) or (8.5) by using the derivatives given by equation (8.4) or (8.6). The isobaric heat capacity Cp at any point is related to the isochoric heat capacity Cy by the expression... 

The values for Tc and Pc are those given by Stewart Jacobsen (1989), whereas the value for Pc is derived from isochoric specific-heat-capacity data of Anisimov et al. (1978). Since the global equation of state is based on temperatures in toms of IPTS-68, all temperatures are consistently expressed in this temperature scale. 

Thus, by using thermodynamic relations, in the same way as in the case of volumetric and thermal properties of solntions (see Eq. (2.57)), it is possible to correlate the compressibility and thermal properties. By differentiation of the isochoric thermal pressure coefficient y T m) with regard to T, the change of isochoric heat capacity with volume at constant temperature can be evaluated. Its value for pure water and citric acid solutions increases with increasing volume because the second derivative of the pressure with respect to temperature is positive, g T m) = T d P / dT )y > 0. 

Looking at the isothermal-isobaric or isothermal—isochoric reaction, it can be concluded that under either constant pressure or constant volume, the thermal effect depends on the reaction temperature, as is described by Kirchhoff equations. In order to derivate them, the expressions for Qp and Qy must be differentiated, e.g. the equation for Qy is as follows ... 

Equations 2.12 and 2.15 are called Kirchhoffs equations, whereas derivatives QQy/Qt)y and (dQp/dt) are called the temperature coefihcients of reaction heat for the isochoric or isobaric reactions. 

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