Reversing heat capacity

Complex heat capacity Reversing heat capacity... 

Figure 2.114. Nonisothermal cure at 1 °C/min of a stoichiometric DGEBA + MDA mixture (a) nonreversing (NR) heat flow (b) glass transition as a function of temperature calculated from the total heat flow signal (c) heat capacity (reversing signal) for the first (1 ) and second (2 ) heating (l°C/60s modulation period) [reprinted from Swier et al. (2004) with permission of John Wiley Sons, Inc.].

Figure 2.117. Cure of the reactive blend DGEBA + aniline (r = l)/20wt% PES at 100°C (a) change in heat capacity (reversing signal) during isothermal cure at 100°C, with cloud point from OM (A), onset of heat flow phase relaxation (O) (b) glass transitions measured after different isothermal cure times at 100°C (heating rate = 2.5°C/min) numbers indicate cure times on (a) (Swier, unpublished results).

Simultaneous measurement of the cure exotherm and glass transition in nonisothermal conditions, separated into the nonreversing heat flow and heat capacity (reversing) signal, respectively... 

Example 3.8 Show that for the reversible adiabatic expansion of ideal gas with constant heat capacity... 

E3.13 A gas obeys the equation of state PVm RT + Bp and has a heat capacity Cy m that is independent of temperature. Derive an expression relating T and Vm in an adiabatic reversible expansion. 

This is the infinitesimal form of Eq. 5a of Chapter 6, which also applies to reversible changes.) If the change in temperature is carried out at constant volume, we use the constant-volume heat capacity, Cv. If the change is carried out at constant pressure, we... 

Thermod5mamics is a fundamental engineering science that has many applications to chemical reactor design. Here we give a summary of two important topics determination of heat capacities and heats of reaction for inclusion in energy balances, and determination of free energies of reaction to calculate equihbrium compositions and to aid in the determination of reverse reaction... 

REVTEMP - Reversible Reaction with Variable Heat Capacities... 

In most processes, a reversible absorption of heat is accompanied by a change in temperature, and a calculation of the corresponding entropy change requires an evaluation of the integral of q/T. The term q is related to the heat capacity of the system which is usually expressed as a function of temperature. In a constant volume process, for example, the entropy change is... 

For reversible exothermic reactions, the situation is more complex. Figure 6.5a shows the behavior of an exothermic reaction as a plot of equilibrium conversion against temperature. Again, the plot can be obtained from values of AG° over a range of temperatures and the equilibrium conversion calculated as discussed previously. If it is assumed that the reactor is operated adiabatically, and the mean molar heat capacity of the reactants and products is constant, then for a given starting temperature for the reaction Tin, the temperature of the reaction mixture will be proportional to the reactor conversion X for adiabatic operation, Figure 6.5a. 

Charge is a solution of pure A at the rate of 100 cuft/hr at 560 R with an inlet concentration Ca0 = 1.5 Ibmol/cuft. Heat of reaction is AHr = -4000 Btu/lbmol and independent of temperature. Heat capacity of the solution is 60 Btu/(cuft)(R). Specific rates of the forward and reverse reactions, ka = exp(9.26-6989/T), cuft/(lbmol)(min) (1)... 

Continuous Multicomponent Distillation Column 501 Gas Separation by Membrane Permeation 475 Transport of Heavy Metals in Water and Sediment 565 Residence Time Distribution Studies 381 Nitrification in a Fluidised Bed Reactor 547 Conversion of Nitrobenzene to Aniline 329 Non-Ideal Stirred-Tank Reactor 374 Oscillating Tank Reactor Behaviour 290 Oxidation Reaction in an Aerated Tank 250 Classic Streeter-Phelps Oxygen Sag Curves 569 Auto-Refrigerated Reactor 295 Batch Reactor of Luyben 253 Reversible Reaction with Temperature Effects 305 Reversible Reaction with Variable Heat Capacities 299 Reaction with Integrated Extraction of Inhibitory Product 280... 

The other extreme case is the adiabatic change, which occurs with no heat transfer between the gas and the surroundings. For a reversible adiabatic change, k = y where y = Cp/Cv, the ratio of the specific heat capacities at constant pressure (Cp) and at constant volume (C ). For a reversible adiabatic change of an ideal gas, equation 6.27 becomes... 

MDSC is particularly useful for the study of reversible (related to the heat capacity) thermal reactions, and is less useful for non-reversing (kinetically controlled) reactions. Examples of reversible thermal events include glass transitions, heat capacity, melting, and enantiotropic phase transitions. Examples of non-reversible events include vaporization,... 

Pyda and co-workers [49, 60] measured the reversible and irreversible PTT heat capacity, Cp, using adiabatic calorimetry, DSC and temperature-modulated DSC (TMDSC), and compared the experimental Cp values to those calculated from the Tarasov equation by using polymer chain skeletal vibration contributions (Figure 11.7). The measured and calculated heat capacities agreed with each other to within < 3 % standard deviation. The A Cp values for fully crystalline and amorphous PTT are 88.8 and 94J/Kmol, respectively. 

The reversible potential of a fuel cell at temperature T is calculated from AG for the cell reaction at that temperature. This potential can be computed from the heat capacities (Cp) of the species involved as a function of T and from values of both AS° and AH° at one particular temperature, usually 298K. Empirically, the heat capacity of a species, as a function of T, can be expressed as... 

For the reversible adiabatic expansion, a definite expression can be derived to relate the initial and final temperatures to the respective volumes or pressures if we assume that the heat capacity is independent of temperature. This assumption is exact at all temperatures for monatomic gases and above room temperature for diatomic gases. Again we start with Equation (5.39). Recognizing the restriction of reversibility, we obtain... 

We can calculate AH from thermal data alone, that is, from calorimetric measurements of enthalpies of reaction and heat capacities. It would be advantageous if we could also compute AS from thermal data alone, for then we could calculate AG or Ay without using equilibrium data. The requirement of measurements for an equilibrium state or the need for a reversible reaction thus could be avoided. The thermal-data method would be of particular advantage for reactions for which AG or AT is very large (either positive or negative) because equilibrium measurements are most difficult in these cases. 

In the above expression, the first term represents the accumulation and convective transport of enthalpy, where is the heat capacity of phase k. The second term is energy due to reversible work. For condensed phases this term is negligible, and an order-of-magnitude analysis for ideal gases with the expected pressure drop in a fuel cell demonstrates that this term is negligible compared to the others therefore, it is ignored in all of the models. 

As shown by Helgeson et al. (1978), satisfactory estimates of standard state molar entropy for crystalline solids can be obtained through reversible exchange reactions involving the compound of interest and an isostructural solid (as for heat capacity, but with a volume correction). Consider the generalized exchange reaction... 

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