Heat capacity pressure, difference between

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

Liquid Heat Capacity The two commonly used liqmd heat capacities are either at constant pressure or at saturated conditions. There is negligible difference between them for most compounds up to a reduced temperature (temperature/critical temperature) of 0.7. Liquid heat capacity increases with increasing temperature, although a minimum occurs near the triple point for many compounds. 

Thermocycle capacity is a function of the temperature difference between the chilled-water outlet temperature leaving the cooler and the inlet condenser water. The cycle finally stops when these two temperatures approach each other and there is not sufficient vapor pressure difference to permit flow between the heat exchangers. 

The heat capacity of a subshince is defined as the quantity of heat required to raise tlie temperature of tliat substance by 1° the specific heat capacity is the heat capacity on a unit mass basis. The term specific heat is frequently used in place of specific heat capacity. This is not strictly correct because traditionally, specific heal luis been defined as tlie ratio of the heat capacity of a substance to the heat capacity of water. However, since the specific heat of water is approxinuitely 1 cal/g-°C or 1 Btiiyib-°F, the term specific heal luis come to imply heat capacity per unit mass. For gases, tlie addition of heat to cause tlie 1° tempcniture rise m iy be accomplished either at constant pressure or at constant volume. Since the mnounts of heat necessary are different for tlie two cases, subscripts are used to identify which heat capacity is being used - Cp for constant pressure or Cv for constant volume. Tliis distinction does not have to be made for liquids and solids since tliere is little difference between tlie two. Values of heat capacity arc available in the literature. ... 

The specific heat of a substance must always be defined relatively to a particular set of conditions under which heat is imparted, and it is here that the fluid analogy is very liable to lead to error. The number of heat units required to produce unit rise of temperature in a body depends in fact on the manner in which the heat is communicated. In particular, it is different according as the volume or the pressure is kept constant during the rise of temperature, and we have to distinguish between specific heats (and also heat capacities) at constant volume and those at constant pressure, as well as other kinds to be considered later. 

To use this expression, we need to know ACP, the difference between the constant-pressure heat capacities of the products and reactants ... 

The temperature variation of the standard reaction enthalpy is given by Kirchhoff s law, Eq. 23, in terms of the difference in molar heat capacities at constant pressure between the products and the reactants. 

Kirchhoff s law The relation between the standard reaction enthalpies at two temperatures in terms of the temperature difference and the difference in heat capacities (at constant pressure) of the products and reactants. 

Typical values of the isobaric expansivity and the isothermal compressibility are given in Table 1.2. The difference between the heat capacities at constant volume and constant pressure is generally negligible for solids at low temperatures where the thermal expansivity becomes very small, but the difference increases with temperature see for example the data for AI2O3 in Figure 1.2. 

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

The thermal expansivity of a solid is in general low at low temperatures and the anharmonic contribution to the heat capacity is therefore small in this temperature region and Cv m Cpm. At high temperatures the difference between the heat capacity at constant pressure and at constant volume must be taken into consideration. 

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

ACp (T) is the difference between the heat capacities of the products and the reactants at temperature, T. The heat capacity, Cp, is the rate of change of enthalpy with temperature at constant pressure. The dependence of Cp on T is given by. 

Hence, for temperatures very close to the boiling point, we integrate Eq. 4-7 by assuming that Avap//,(T) = AvapH,(Tb) = constant (see Section 4.2). However, in most cases, one would like to estimate the vapor pressure at temperatures (e.g., 25°C) that are well below the boiling point of the compound. Therefore, one has to account for the temperature dependence of Avap// below the boiling point. A first approximation is to assume a linear temperature dependence of Avap/7, over the temperature range considered, that is, to assume a constant heat capacity of vaporization, A Cpi (the difference between the vapor and liquid heat capacities). Thus, if the heat capacity of vaporization, AvapCpi(Tb), at the boiling point is known, Avap/7,(7) can be expressed by (e.g., Atkins, 1998) ... 

These two expressions differ only by the leading constant terms. The simple thermal conductivity expression derived here is roughly 40% the size of the rigorous result. It captures the functional dependence on temperature, molecular mass, heat capacity, and pressure (independent of pressure) of the exact result. Experimentally the thermal conductivity is generally found to be independent of pressure, except at very low pressures. The thermal conductivity is predicted to increase as the square root of temperature, which somewhat underestimates the actual temperature dependence. Consideration of interactions between molecules, as in the next section, brings the temperature dependence into better accord with observation. 

In the CSM laboratory, Rueff et al. (1988) used a Perkin-Elmer differential scanning calorimeter (DSC-2), with sample containers modified for high pressure, to obtain methane hydrate heat capacity (245-259 K) and heat of dissociation (285 K), which were accurate to within 20%. Rueff (1985) was able to analyze his data to account for the portion of the sample that was ice, in an extension of work done earlier (Rueff and Sloan, 1985) to measure the thermal properties of hydrates in sediments. At Rice University, Lievois (1987) developed a twin-cell heat flux calorimeter and made AH measurements at 278.15 and 283.15 K to within 2.6%. More recently, at CSM a method was developed using the Setaram high pressure (heat-flux) micro-DSC VII (Gupta, 2007) to determine the heat capacity and heats of dissociation of methane hydrate at 277-283 K and at pressures of 5-20 MPa to within 2%. See Section 6.3.2 for gas hydrate heat capacity and heats of dissociation data. Figure 6.6 shows a schematic of the heat flux DSC system. In heat flux DSC, the heat flow necessary to achieve a zero temperature difference between the reference and sample cells is measured through the thermocouples linked to each of the cells. For more details on the principles of calorimetry the reader is referred to Hohne et al. (2003) and Brown (1998). 

However, AH, the difference between the molar enthalpy of the gas and the condensed phase, depends in general on both the temperature and the pressure. The enthalpy for an ideal gas is independent of pressure and, fortunately, the enthalpy for the condensed phase is only a slowly varying function of the pressure. It is therefore possible to assume that AH is independent of the pressure and a function of the temperature alone, provided that the limits of integration do not cover too large an interval. With this final assumption, the integration can be carried out. When the molar heat capacities of the two phases are known as functions of the temperature, AR is obtained by integration. If ACP, the difference in the molar heat capacities of the two phases, is expressed as... 

---