Isochoric heat capacity

Helmholtz energy Gibbs energy Isobaric heat capacity Isochoric heat capacity Isobaric expansivity Isothermal compressibility Isentropic compressibility... 

A more recent compilation includes tables giving temperature and PV as a function of entropies from 0.573 to 0.973 (2ero entropy at 0°C, 101 kPa (1 atm) and pressures from 5 to 140 MPa (50—1400 atm) (15). Joule-Thorns on coefficients, heat capacity differences (C —C ), and isochoric heat capacities (C) are given for temperatures from 373 to 1273 K at pressures from 5 to 140 MPa. 

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... 

In the Kieffer model (Kieflfer 1979a, b, and c cf section 3.3), the isochoric heat capacity is given by... 

Figure 11.5 compares the fluid entropy vectors, whose lengths range from about 0.25 (ideal gas) to about 0.75 (ether). As expected, the entropy vectors exhibit an approximate inverted or complementary (conjugate) relationship to the corresponding T vectors of Fig. 11.3. The length of each S vector reflects resistance to attempted temperature change (under isobaric conditions), i.e., the capacity to absorb heat with little temperature response. The lack of strict inversion order with respect to the T lengths of Table 11.3 reflects subtle heat-capacity variations between isochoric and isobaric conditions, as quantified in the heat-capacity or compressibility ratio...

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. 

While there are conflicting views about the performances and benefits of all these versions, there are some criteria that may serve for a critical assessment. These include (a) Onsager s well-known lower bound for the mean electrostatic energy per ion [253] in its reformulation by Totsuji [254] and (b) Gillan s upper bound for the free energy [255]. Moreover, the condition for thermal stability requires the configurational isochoric heat capacity to be positive. 

Isochoric heat capacity of crystals can be presented as a sum of contributions from lattice vibrations (6 degrees of freedom) and intramolecular vibrations (3n-6 degrees of freedom, where n is a number of atoms in a molecule)... 

A certain gas obeys the equation of state P(V -nb) - nRT and has a constant volume heat capacity, Cv, which is independent of temperature. The parameter b is a constant. For 1 mol, find W, AE, Q, and AH for the following processes (a) Isothermal reversible expansion. (b) Isobaric reversible expansion. (c) Isochoric reversible process, (d) Adiabatic reversible expansion in terms of Tlf Vlt V2, Cp, and Cv subscripts of 1 and 2 denote initial and final states, respectively. (c) Adiabatic irreversible expansion against a constant external pressure P2, in terms of Plf P2, Tj, and 7 = (Cp/Cy). 

What Gucker and Rubin called the apparent molal isochoric heat capacity of an electrolyte is (P(Co2), and the corresponding isopiestic heat capacity is 0(Cp2), where ... 

We now specialize to isochoric, isentropic conditions, dV = dS = 0 and we replace the first partial derivative with Cg/T, where Cg = Cy is the corresponding heat capacity. We use lower case letters to denote the densities and we set Cg — fy Cg d r. Then the above reduces to... 

On the basis of this brief summary of RPM criticality, one might be tempted to conclude that the problem has been solved all finite-size scaling analysis point towards the Ising universality class. There is, however, one critical phenomenon which does not seem to have been demonstrated unambiguously in the RPM. This is the critical divergence of the constant-volume heat capacity, Cy. Recall that on the critical isochore and close to the critical temperature where the parameter t = (T — Tc)/Tc is small,... 

Figure 3. Constant-volume heat capacity, Cy, for the CHD fluid on the critical isochore as obtained from NVT MC simulations fluctuation formula (6) (points) from a [5,5] Pade approximant (9) fitted to the energy (solid line). The kinetic contribution is not included.

We now turn to measurements of the constant-volume heat capacity, Cy, along the critical isochore. In Fig. 5 we show Cy as measured in NVT and pVT simulations of a system with a = 6 and L = lOer. The bulk critical... 

Figure 5. Constant-volume heat capacity, Cy, of the AHS fluid with a = 6 as obtained from NVT (filled circles and solid line) and pVT (open circles and dashed line) MC simulations of systems with L = 10a along the bulk critical isochore. The kinetic contribution (equal to SNIcb/Z) is included.

Typical uncertainties in density are 0.02% in the liquid phase, 0.05% in the vapor phase and at supercritical temperatures, and 0.1% in the critical region, except very near the critical point, where the uncertainty in pressure is 0.1%. For vapor pressures, the uncertainty is 0.02% above 180 K, 0.05% above 1 Pa (115 K), and dropping to 0.001 mPa at the triple point. The uncertainty in heat capacity (isobaric, isochoric, and saturated) is 0.5% at temperatures above 125 K and 2% at temperatures below 125 K for the liquid, and is 0.5% for all vapor states. The uncertainty in the liquid-phase speed of sound is 0.5%, and that for the vapor phase is 0.05%. The uncertainties are higher for all properties very near the critical point except pressure (saturated vapor/liquid and single pliase). The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to about 2.5% from 100 to 475 K up to about 30 MPa, and to about 4% outside this range. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%. 

Gases Methods for estimating low-pressure gas thermal conductivities are based on kinetic theory and generally correlate the dimensionless group kM/r C (M = molecular weight, T] = viscosity, C = isochoric heat capacity) known as the Eucken factor. The method of Stiel and Thodos is recommended for pure nonpolar compounds, and the method of Chung is recommended for pure polar compounds. 

It is known that incompressible fluids represent a useful model for real fluids in fluid mechanics and heat and mass transfer. Their thermal equation of state is v = v0 = const. For pure substances and also for mixtures, isobaric and isochoric specific heat capacities agree with each other, cp = cv = c. 

Here 2 is the exponent for the heat capacity measured along the critical isochore (i.e. in the two-phase region) below the critical temperature, while is the exponent for the isothermal compressibility measured in the one-phase region at the edge of the coexistence curve. These inequalities say nothing about the exponents a and y in the one-phase region above the critical temperature. 

---