Heat capacity variation with pressure

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

The enthalpies and internal energies of steam and water also converge at the critical point. Tlie heat capacity at constant pressure. C,. is defined as the derivative of enthalpy with respect to temperature. The value of Cr, becomes very large in the vicinity of the critical point. The variation is much smaller for tlie heat capacity at constant volume, C,.. 

The magnitude of the variation of with pressure is thus determined by the variation of a with temperature. Since docjdT is always positive, the heat capacity at constant pressure decreases with increased pressure, but the effect becomes small at high temperatures since docjdT falls off more rapidly than T increases (c/. fig. 12.1). A similar formula is readily obtained for the effect of volume on c. ... 

It is of interest to note that (d P/dT )v is zero for a van der Waals gas, as well as for an ideal gas hence, Cv should also be independent of the volume (or pressure) in the former case. In this event, the effect of pressure on Cp is equal to the variation of Cp — Cv with pressure. Comparison of equations (21.4) and (21.13), both of which are based on the van der Waals equation, shows this to be true. For a gas obeying the Berthelot equation or the Beattie-Bridgeman equation (d P/dT )v would not be zero, and hence some variation (f Cv with pressure is to be expected. It is probable, however, that this variation is small, and so for most purposes the heat capacity of any gas at constant volume may be regarded as being independent of the volume or pressure. The maximum in the ratio y of the heat capacities at constant pressure and volume, respectively, i.e., Cp/Cv, referred to earlier ( lOe), should thus occur at about the same pressure as that for Cp, at any temperature. 

The heat capacity at constant pressure, Cp, is the derivative with respect to temperature of the enthalpy change induced by temperature variation (c.f. Eq. (4.6)). At high temperature, the methods used for Cp determination are based on the simultaneous measurement of the enthalpy temperature variation versus time at a programmed rate of heating. 

In order to assess the variation in AH with temperature, we apply KirchhofTs equation where Cp = molar heat capacity at constant pressure ... 

The time variation of enthalpy per m (left hand side of the equation) is caused by forced convection (first term on the right hand side), the effective heat transfer (second term on the right hand side) and the heat release caused by the chemical reaction (third term on the right hand side) )i is the effective coefficient of thermal conductivity of the mixture of substances in W/(m K), p is its density in kg/m, Cp the corresponding heat capacity at constant pressure in J/(kg K) and AHrj the enthalpy of reaction of reaction j in J/kg (with a negative sign for exothermic reactions). 

Water becomes less dense due to thermal expansion with increase in temperature. The density of water is 1.0 g/cm at room temperature, which changes to 0.7 g/cm at 306 C. At critical point, the densities of the two phases become identical and they become a single fluid, which is called supercritical fluid. The density of water at this point is 0.3 g/cm In the supercritical region, most of the properties of water vary widely. The most important of these is the heat capacity at constant pressure, which approach infinity at the critical point. Also, the dielectric constant of dense, supercritical water ranges from 5 to 20 on variation of applied pressure. 

Figure 1.4 Variation of thermodynamic quantities and order parameter with temperature for (a) a first-order phase transition and, (b) a second-order phase transition. The notation is as follows /x, chemical potential H, enthalpy S, entropy V, volume Cp, heat capacity (at constant pressure) order parameter. The phase transition occurs at a temperature T = T ...

The variations of the heat capacities at constant pressure with the absolute temperature T are given in the following form ... 

On the experimental side, one may expect most progress from thermodynamic measurements designed to elucidate the non-configurational aspects of solution. The determination of the change in heat capacity and the change in thermal expansion coefficient, both as a function of temperature, will aid in the distinction between changes in the harmonic and the anharmonic characteristics of the vibrations. Measurement of the variation of heat capacity and of compressibility with pressure of both pure metals and their solutions should give some information on the... 

R.L. Bohon, AnalChem 35 (12), 1845-52 (1963) CA 60,1527 (1964) Approx heats of expin, Qv were detd on mg amounts of propints and expls by differential thermal analysis (DTA). Small-screw-cap metal cupsi sealed with a Cu washer served as constant vol sample containers the initial cup pressure could be controlled from 0 to approximately lOOOpsia. The calibration constant was calcd for each run from the total heat capacity of the cup and the relaxation curve, thereby compensating for equipment variations. 

Because the feed is primarily air and because substantial amounts of N2 and 02 are present in the effluent stream, we will assume that the heat capacity of the reactant mixture may be taken as that of air at 630° K (0.255 cal/g-°K). The variations of the heat capacity with temperature and pressure may be neglected. 

By differentiating Eq.(6) with respect to time, considering that the variations of the volume and total pressure are negligible, and using the enthalpy definition. Hi = CpiT, where Cp is the heat capacity of i-reactant (kJ/mol °C), Eq.(5) can be written as follows ... 

We also account for density, heat capacity, and molecular weight variations due to temperature, pressure, and mole changes, along with temperature-induced variations in equilibrium constants, reaction rate constants, and heats of reaction. Axial variations of the fluid velocity arising from axial temperature changes and the change in the number of moles due to the reaction are accounted for by using the overall mass conservation or continuity equation. 

Tenperature and conposition affect physical properties, but the effect of pressure is generally small and we can neglect it. One exception is gas density. A well known example of the effect of temperature is the variation of heat capacity of a gas with temperature, which is generally curve fitted in the form of a polynomial. 

Heat capacities are also useful in determining the variation of temperature with pressure or volume in isentropic processes. For this purpose introduce Eq. (1.3.8) to write... 

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