Heat capacity pressure, relation 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 ... 

For the ideal-gas state there is an exact relation between the constant pressure heat capacity and the constant volume heat capacity, C, via the ideal-gas constant, R. 

The constant-volume and constant-pressure heat capacities of a solid substance are similar the same is true of a liquid but not of a gas. We can use the definition of enthalpy and the ideal gas law to find a simple quantitative relation between CP and Cv for an ideal gas. 

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. 

Relation between the constant-pressure and constant-volume molar heat capacities of an ideal gas ... 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

This relation is called Kirchhoff s law. To use it, we need to know ACP, the difference between the molar constant-pressure heat capacities of the products and reactants ... 

The effect of temperature on retention has been described experimentally,(4-8) but the functional dependence of k with temperature has only recently been described.W A thermodynamic model was outlined relating retention as a function of temperature at constant pressure to the volume expansivity of the fluid, the enthalpy of solute transfer between the mobile phase and the stationary phase and the change in the heat capacity of the fluid as a function of temperature.(9) The solubility of a solid solute in a supercritical fluid has been discussed by Gitterman and Procaccia (10) over a large range of pressures. The combination of solute solubility in a fluid with the equation for retention as a function of pressure derived by Van Wasen and Schneider allows one to examine the effect of solubility on solute retention. 

The methods used in predicting these thermodynamic properties employ (a) an equation of state, relating the pressure-volume-temperature characteristics of the fluids (b) ideal gas state heat capacities of the individual components and (c) binary interaction coefficients between the components. The development of these basic relationships is not within the scope of this paper. Technical literature sources of the thermodynamic equations and data are given in the references. 

Duration of a cycle of HHP operation is defined as time required for reaction hydrogenation/dehydrogenation in pair hydride system. This time determines heat capacity of HHP. Duration of a cycle depends on kinetics of hydrogenation reactions, a heat transfer between the heated up and cooling environment, heat conductivities of hydride beds. Rates of reactions are proportional to a difference of dynamic pressure of hydrogen in sorbers of HHP and to constants of chemical reaction of hydrogenation. The relation of dynamic pressure is adjusted by characteristics of a heat emission in beds of metal hydride particles (the heat emission of a hydride bed depends on its effective specific heat conductivity) and connected to total factor of a heat transfer of system a sorber-heat exchanger. The modified constant of speed, as function of temperature in isobaric process [1], can characterize kinetics of sorption reactions. In HHP it is not sense to use hydrides with a low kinetics of reactions. The basic condition of an acceptability of hydride for HHP is a condition of forward rate of chemical reactions in relation to rate of a heat transmission. 

This equation relates temperature and volume for a mechanically reversible adiabatic process involving an ideal gas with constant heat capacities. The analogous relationships between temperature and pressure and between pressure and volume can be obtained from Eq. (3.22) and the ideal-gas equation. Since P, V,/ T, = P2V2j T2, we may eliminate V,/ V2 from Eq. (3.22), obtaining ... 

Example 7.1 Consider the filling of an evacuated tank with a gas from a cons, pressure line. What is the relation between the enthalpy of the gas in the entras line and the internal energy of the gas in the tank Neglect heat transfer between 1 gas and the tank. If the gas is ideal and has constant heat capacities, how is ti temperature of the gas in the tank related to the temperature in the entrance line ... 

The relation of velocity to pressure in a nozzle can be given analytically the fluid behaves as an ideal gas. When an ideal gas with constant heat capaciti undergoes isentropic expansion, Eq. (3.24) provides a relation between P a V, that is, PVy = const. Integration of Eq. (7.20) then gives... 

There are two flow regimes corresponding to sonic (or choked) flow for liigher pressure drops and subsonic flow for lower pressure drops. The transition between the two flow regimes occurs at tlie dimensionless critical pressure ratio, rent, which is related to tlie gas heat capacity ratio y via... 

We turn to relations characterizing heat capacities. Experimentally, it is much easier to measure heat capacities at constant pressure than at constant volume the latter inevitably changes with temperature. However, the quantity relevant to theoretical interpretation is Cy rather than Cp. A relation between these may be established, beginning with the definition H — E + PV, so that... 

The coefficients of expansion and compressibility of condensed phases are important also because they are related to the difference between the heat capacity at constant volume and constant pressure. By writing equation (4.45) for one mole of substance, and substituting (12.1) and (12.2) we obtain the relation... 

It may also be observed that, because of the relation between heat capacities at constant volume and constant pressure (12.5), conditions (15.34) and (15.41) imply also that... 

For systems in which no change in composition (chemical reaction) occurs, things are even simpler to a very good approximation, the enthalpy depends only on the temperature. This means that the temperature of such a system can serve as a direct measure of its enthalpy. The functional relation between the internal energy and the temperature is given by the heat capacity measured at constant pressure ... 

Understand the relation between specific heat capacity and heat transferred in both constant-pressure (coffee-cup) and constant-volume (bomb) calorimeters ( 6.3) (SPs 6.3-6.5) (EPs 6.17-6.30)... 

Equation (2.28) establishes a relationship between pressure and specific volume (enthalpy is related via heat capacity and Eq. (2.29) with pressure and specific volume). It is called Hugoniot equation or Hugoniot shock adiabatic and consists of thermodynamic quantities only. 

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