Enthalpy constant-pressure heat capacity

The enthalpy change AH for a temperature change from to T2 can be obtained by integration of the constant pressure heat capacity... 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

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. 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

C = heat capacity at constant pressure = heat capacity at constant voiume = gravitationai constant (numericai vaiues in Tabie Ai) h = individuai heat transfer coefficient H = enthalpy k = thermai conductivity... 

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". ... 

The measurement of heat capacity and related quantities is known as calorimetry. Most often the constant-pressure heat capacity is measured some instruments measure the constant-volume heat capacity Cy. Often, what is actually measured is not the derivatives Cp and Cy but an energy change divided by a small but finite temperature change. In some cases, the original enthalpy increment data may be more useful than the approximate heat capacities derived from them. In addition to the lUPAC books referenced in Section 1.8.1, the monograph of Flemminger and Flohne... 

The quantity Cv is talled the constant-volume heat capacity, and Cp is the constant-pressure heat capacity both appear frequently throughout this book. Partial derivatives have been used in Eqs. 3.3-2 and 3.3-3 to indicate that although the internal energy is a function of temperature and density or. -pecific (or molari volume, Cv ha.s been measured along a path of constant volume and although the enthalpy is a function of temperature and pressure, Cp has been evaluated in an experiment in which the pressure was held constant. 

This example demonstrates that reliable PVT correlation and constant-pressure heat capacity of an ideal gas are sufficient to determine a variety of thermodynamic properties, as enthalpy, entropy, Gibbs free energy, etc., and built comprehensive charts. This approach will be extended by means of departure functions. 

The glass transition temperature of a polymer marks a second-order phase transition in which there is continuity of the free energy function and its first partial derivatives with respect to state variables such as temperature or pressure, but there is a discontinuity in the second partial derivatives of free energy. There is, therefore, continuity in enthalpy, entropy, or volume at the transition temperature, but not in the constant-pressure heat capacity, (42). Hence, measurements of changes in with increasing temperature yield information about, as well as about the magnitude of... 

Problem 5.5 Estimate the constant-pressure heat capacity of oxygen at - 50 °C, 38 bar, using the truncated virial equation. Hint Use the truncated virial to calculate the enthalpy at two pressures near -50 °C, 38 bar, then obtain the heat capacity by numerical differentiation. 

On the other hand, thermodynamic quantities that pertain to the formation of isolated ions from the elements in their standard states are well defined. The standard molar Gibbs energy and the enthalpy of formation, AfG°(F= =, g) and g), of many ions have been reported. The standard molar entropy and constant-pressure heat capacity, 5°(F, g) and Cp(F= =, g), of isolated ions are also well defined quantities and have been reported. Such data are generally available for the standard temperature T° = 298.15 K and pressure P° = 100 kPa, and suitable sources are the NBS tables (Wagman et al. 1982) and the book by Marcus (1997). The standard molar volume of an isolated ion is trivial, being the same for all ions V°(F , g)= RT /P = 0.02479m mor , where/ = 8.31451 J K mor is the gas constant. 

Similarly, the heat energy absorbed by a system in a constant-pressure process is equal to the change in enthalpy—that is, AH = qp (Equation 7.13). Hence, the constant-pressure heat capacity is given by... 

Equation 7.49 is known as Kirchhoff s law, after the German physicist Gustav-Robert Kirchhoff. According to Kirchhoff s law, the difference between the enthalpies of a reaction at two different temperatures is just the difference in the enthalpies of heating the products and reactants from Ti to T2. Note that in deriving Equation 7.48 we have assumed that the constant-pressure heat capacities are independent of temperature. Otherwise, they must be expressed as functions of T, and the integral in Equation 7.48 must be done explicitly. 

Since depends on and is completely independent of T, it is at once apparent that there is no electrostatic contribution to the enthalpy, and that the constant-pressure heat capacity is just 3Nk, or exactly what would have been obtained for an ideal one-di-... 

The thermodynamic properties of liquids that could be of interest for solvents of electrolytes are the vapor pressure, p, the molar enthalpy of vaporization, A/T, the molar constant pressure heat capacity, Cp, and the surface tension, [Pg.69]

TABLE 3 A Some Thermodynamic Properties of Solvents for Electrofytes The Vapor Pressure,/ , the Molar Enthalpy of Vaporization, Afl, the Molar Constant Pressure Heat Capacity, Cp, the Surface Tension a, and the Hildebrand Soluhility Parameter, at 25°C... 

Just as we saw that the constant-volume heat capacity tells us about the temperature-dependence of the internal energy at constant voliune, so the constant-pressure heat capacity tells us how the enthalpy of a system changes as its temperature is raised at constant pressure. To derive the relation, we combine the definition of heat capacity in eqn 1.5 (C=q/AT) with eqn 1.13 and obtain... 

That is, the constant-pressure heat capacity is the slope of a plot of enthalpy against temperature of a system kept at constant pressure. Because the plot might not be a straight fine, in general we interpret Cp as the slope of the tangent to the curve at the temperature of interest (Fig. 1.16, Table 1.1). That is, the constant-pressure heat capacity is the derivative of the function H with respect to the variable T at a specified pressure or... 

Figure 1.26 illustrates the technique. As we have seen, the enthalpy of a substance increases with temperature therefore the total enthalpy of the reactants and the total enthalpy of the products increase, as shown in the illustration. Provided the two total enthalpy increases are different, the standard reaction enthalpy (their difference) will change as the temperature is changed. The change in the enthalpy of a substance depends on the slope of the graph and therefore on the constant-pressure heat capacities of the substances (recall Fig. 1.16). We can therefore expect the temperature dependence of the reaction enthalpy to be related to the difference in heat capacities of the products and the reactants. We show in the following Justification that this is indeed the case and that, when the heat capacities do not vary with temperature, the standard reaction enthalpy at a temperature T is related to the value at a different temperature Tby a special formulation of KirchholF slaw ...

Jt2 Estimate the enthalpy of vaporization of water at lOO C from its value at 25°C (+44.01 kl mol ) given the constant-pressure heat capacities of75.291K moE and 33.581K" mol" for liquid and gas, respectively. 

The protein lysozyme unfolds at a transition temperature of 75.5°C, and the standard enthalpy of transition is 509 kj mol". Calculate the entropy of unfolding of lysozyme at 25.0°C. given that the difference in the constant-pressure heat capacities on unfolding is 6.28 kJ K" mol" and can be assumed to be independent of temperature. Hint Imagine that the transition at 25.0°C occurs... 

Calculate the partition function and the molar energy, enthalpy, and entropy and constant-pressure heat capacity 1.000 mol of Cl2 at 298.15 K and the standard-state pressure P°. Assume that the uncorrected harmonic oscillator-rigid rotor energy levels can be used. 

Our analysis using standard state values implies dP = 0, and under that constraint, the second term in Equation 6.44 is zero. The remaining partial derivative is the constant-pressure heat capacity of the fth substance, Cp. To evaluate the enthalpy (or other thermodynamic state functions) at a temperature, call it T2, different from a temperature Ti for which there are known enthalpies, we invoke Hess s law, treating the temperature changes of reactants and products as reaction steps. Here is an example ... 

The entropy of binding is thus available by subtraction. The partial differential of enthalpy with respect to temperature defines the partial change in constant pressure heat capacity. Over short temperature ranges this derivative is approximated as... 

The enthalpies of reactants and products increase with temperature. If the total enthalpy of the reactants increases more than that of the products, then the reaction enthalpy will decrease as the temperature is raised (Fig. 6.34). On the other hand, if the enthalpy of the products increases more with an increase in temperature than that of the reactants, then the reaction enthalpy will increase. The increase in enthalpy of a substance when the temperature is raised depends on its heat capacity at constant pressure (Eq. 20), so we ought to be able to predict the change in reaction enthalpy from the heat capacities of all the reactants and products. 

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