Heat capacity under constant pressure

Dividing all terms by the air parcel mass and recognizing the heat capacity under constant pressure term, we obtain... 

Here, Cp is the heat capacity under constant pressure Aqa is the enthalpy of composition Molar fraction q is the heat fraction vector frozen point. 

Heat capacity under constant pressure (Cp) is defined as the heat quantity which is required to increase the temperature of the unit mass of a material by 1 K or 1°C under constant pressure. It is given by the following equation ... 

Another heat capacity, i.e., the heat capacity under constant pressure Cp, can be defined, which is calculated by using enthalpy h, as... 

FIGURE 7.11 The experimental determination of entropy, (a) The heat capacity at constant pressure in this instance) of the substance is determined from close to absolute zero up to the temperature of interest, (b) The area under the plot of CP/T against T is determined up to the temperature of interest. 

For purposes of this calculation, latent heats at constant volume and at constant pressure are assumed equal, heat capacities at constant pressure and at constant volume are assumed equal for solids and liquids [See also Calculation of Temperature of Detonation (and Explosion) 1 and Experimental Determination of Temperature of Detonation [and Explosion) , under Detonation (and Explosion) Temperature Developed On in Vol 4 of Encycl, pp D589 L to D601-R]... 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

This table lists standard enthalpies of formation AH°, standard third-law entropies S°, standard free energies of formation AG°, and molar heat capacities at constant pressure, Cp, for a variety of substances, all at 25 C (298.15 K) and 1 atm. The table proceeds from the left side to the right side of the periodic table. Binary compounds are listed under the element that occurs to the left in the periodic table, except that binary oxides and hydrides are listed with the other element. Thus, KCl is listed with potassium and its compounds, but CIO2 is listed with chlorine and its compounds. 

From temperature dependence data and other environmental factors, it is possible to evaluate the standard free energy change, AF , the molal changes in heat content (A// ), entropy (AS ) and heat capacity at constant pressure (ACp) which occur when the dissociation reaction occurs under standard conditions. 

The subscript p on Cp indicates that this is the heat capacity at constant pressure. Under other conditions, such as constant volume, the value of the heat capacity may differ slightly.)... 

The calculation of C, the integrated heat capacity of the electron, is based on treating the free electrons as an ideal gas and computing the heat capacity according to Boltzmann statistics. At 298 K, the heat capacity at constant pressure is that of an ideal gas, C = 5/2 RT, or 6.197 kJ/mol. This computation allows data tabulated under the Electron Convention to be rationalized with Ion Convention data as follows ... 

The heat capacity at constant pressure, Cp, is the ratio dq/dT for a process in a closed system with a constant, uniform pressure and with expansion work only. Under these conditions, the heat dq is equal to the enthalpy change dH (Eq. 5.3.7), and we obtain a relation analogous to Eq. 5.6.1 ... 

Many density-dependent properties of H2O, such as viscosity, polarity (dielectric constant s changes from 74 to 2), heat capacity at constant pressure (which is infinite at the critical point), ion product and solvent power can be tuned for specific requirements by setting the correct temperature and pressure, and they show significant changes near the critical point (Figure 25.2). Several studies have demonstrated that the transition from sub- to supercritical conditions also affects the elementary steps in reaction mechanisms, and radical intermediates are favoured over ionic species. Another consequence is that subcritical water shows potential for acid catalysis. Reactions can be run either under non-polar/aprotic or polar/pH controlled conditions (water can take part in these reactions). Consequently, non-polar compounds like aromatics become soluble whereas inorganic salts precipitate. Therefore, the properties of water as a solvent are tunable over much wider parameter ranges than for most other compounds. 

For pure substances, the heat capacity is often expressed per mole or per gram of substance. The molar heat capacity at constant pressure (symbol j ) is the quantity of heat required to raise the temperature of one mole of a substance by one degree under the condition of constant pressure. The constant-pressure specific heat capacity (symbol, Cp), sometimes called the specific heat is the quantity of heat required to change the temperature of one gram of a substance by one degree at constant pressure. Here are the values for water at 25 °C. 

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. 

Enthalpy-Temperature Relation and Heat Capacity When heal is adsorbed by a substance, under conditions such that no chemical reaction or slate transition occur and only pressure-volume work is done, the temperature. T, rises and the ratio of the heat adsorbed, over the differential temperature increase, is by definition the heat capacity. For a process at constant pressure (following Equation (2)). this ratio is equal to the partial derivative of the enthalpy, and it is called the hear capacity at constant pressure. C,. (usually in calories/degree-mole) ... 

Q, now measures the heal absorbed under constant pressure, and c, is the constant pressure specific heat. Since the right side of Equation (2) contains three quantities, a mere choice of a mass unit and a degree unit is insufficient to establish a unit of heal. It is necessary to select some substance as a standard reference body and assign an arbitrary value of, say cp equal to unity for it. Water is the universal choice for this standard body due not only to its cheapness and ease nf purification, hut also to its large heat capacity. 

The actual change of temperature depends on the heat capacity C which is again dependent on whether the process is done under constant pressure (Cp) or at constant volume (Cv). Therefore,... 

Heat capacity Without a change in phase and under constant pressure, the amount of heat (usually in joules or calories) required to raise the temperature of a given quantity (usually a mole) of a substance by 10 K. 

Most distillation columns are operated under constant pressure, because at constant pressure temperature measurement is an indirect indication of composition. When the column pressure is allowed to float, the composition must be measured by analyzers or by pressure-compensated thermometers. The primary advantage of floating pressure control is that one can operate at minimum pressure, and this reduces the required heat input needed at the reboiler. Other advantages of operating at lower temperatures include increased reboiler capacity and reduced reboiler fouling. 

When a system is heated, its temperature generally increases. This increase in temperature is dependent on the heat capacity of the system under constant volume or constant pressure. Therefore, the heat capacity is defined as the ratio of heat added to a system to its corresponding temperature change. If the system is under constant volume, the molar heat capacity is Cv, whereas the molar capacity is Cp for a system under constant pressure. Then,... 

In (1.18.12) the symbols Cv and CP represent the heat capacities at constant volume and constant pressure, respectively. As will be seen in Section 1.19, these quantities represent the heat absorbed by a system per unit increase in its temperature under the indicated constraints. Such a quantity may be measured experimentally by use of a calorimeter. Equations (1.18.12) thus furnish a basis for determining E, H, or S from experimental measurements. 

The enthalpy change, dH = T dS + V dp, can be described as dH = dq - -V dp, and for a constant-pressure process, c/p = 0, we have dH = dqp. For a finite state change at constant pressure, qp = AH, that is, the heat transferred is equal to the enthalpy change of the system. This relation is the basis of constant pressure calorimetry, the constant-pressure heat capacity being Cp = dqldT)p. The relationship qp = AH is valid only in the absence of external work, w. When the system does external work, the first law must include dw. Then, the heat transferred to the system under constant-pressure conditions is qp = AH -f w. Thus, if a given chemical reaction has an enthalpy change of -50 kJ mol and does 100 kJ mol" of electrical work, the heat transferred to the system is —50 + 100 = 50 kJ mol". ... 

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