Energy isothermal conditions

Under isothermal conditions where energy is not added or removed from the system, the second law of thermodynamics obtains, and... 

The change in total energy or free energy, under isothermal conditions, is then given by... 

A quantity of heat (q, say) is added during the expansion so as to maintain isothermal conditions. The change in the internal energy is therefore given by ... 

As the pressure in a pipe falls, the kinetic energy of the fluid increases at the expense of the internal energy and the temperature tends to fall. The maintenance of isothermal conditions therefore depends on the transfer of an adequate amount of heat from the surroundings. For a small change in the system, the energy balance is given in Chapter 2 as ... 

We have seen several examples of a technique for separation of gas mixtures which, in contrast with most commercial processes, requires no physical transfer of solvent, handling of solids, or cycling of temperature or pressure. The energy requirements can also be far lower The thermodynamic minimum work of separation is, under isothermal conditions, the free energy difference between the process stream and byproduct, or permeate, stream. When this difference is due only to the partial pressure difference of component 1, it becomes ... 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

Under isothermal conditions, we have seen that the apparent activation energy of the reaction is approximately one half the intrinsic value when rj is sufficiently low. When rj exceeds unity, an opposite effect occurs (i.e., the apparent activation energy will exceed the true activation energy). 

If we compare the work required to compress a given gas to a given compression ratio by isothermal and isentropic processes, we see that the isothermal work is always less than the isentropic work. That is, less energy would be required if compressors could be made to operate under isothermal conditions. However, in most cases a compressor operates under more nearly adiabatic conditions (isentropic, if frictionless) because of the relatively short residence time of the gas in the compressor, which allows very little time for heat generated by compression to be transferred away. The temperature rise during an isentropic compression is determined by eliminating p from Eqs. (8-17) and (8-19) ... 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

The measurement of an enthalpy change is based either on the law of conservation of energy or on the Newton and Stefan-Boltzmann laws for the rate of heat transfer. In the latter case, the heat flow between a sample and a heat sink maintained at isothermal conditions is measured. Most of these isoperibol heat flux calorimeters are of the twin type with two sample chambers, each surrounded by a thermopile linking it to a constant temperature metal block or another type of heat reservoir. A reaction is initiated in one sample chamber after obtaining a stable stationary state defining the baseline from the thermopiles. The other sample chamber acts as a reference. As the reaction proceeds, the thermopile measures the temperature difference between the sample chamber and the reference cell. The rate of heat flow between the calorimeter and its surroundings is proportional to the temperature difference between the sample and the heat sink and the total heat effect is proportional to the integrated area under the calorimetric peak. A calibration is thus... 

Measurements based on the law of conservation of energy are of two main types. In phase change calorimetry the enthalpy of the reaction is exactly balanced by the enthalpy of a phase change of a contained compound surrounded by a larger reservoir of the same compound used to maintain isothermal conditions in the calorimeter. The latter enthalpy, the measurand, is often displayed indirectly through the change in the volumetric properties of the heat reservoir compound, e.g. ice/water. 

Equation 6.38 is the basic form of the energy equation to be used for isothermal conditions, however it is instructive to write the equation in a slightly different form that allows easy comparison with incompressible flow. 

For an ideal gas under isothermal conditions, the enthalpy remains constant and hence it follows from equation 6.45 that the required heat leak into the pipe is equal to the increase in kinetic energy. This is usually a small quantity and therefore flow in long, uninsulated pipes will be virtually isothermal. 

In this case, the impact of g, and g2 is negligible, and we can write the equation for the excess of internal energy under isothermal conditions in the following differential form ... 

In the bomb process, reactants at the initial pressure pi and temperature 7 are converted to products at the final pressure pf and temperature Tf. The primary goal of a combustion calorimetric experiment, however, is to obtain the change of internal energy, Ac//°(7r), associated with the reaction under study, with all reactants and products in their standard states pi = pf = O.IMPa) and under isothermal conditions at a reference temperature 7r (usually 298.15 K). Once AC//°(298.15K) is known, it is possible to derive the standard enthalpy of combustion, AC77°(298.15K), and subsequently calculate the standard enthalpy of formation of the compound of interest from the known standard enthalpies of formation of the products and other reactants. 

The experiments are usually carried out at atmospheric pressure and the initial goal is the determination of the enthalpy change associated with the calorimetric process under isothermal conditions, AT/icp, usually at the reference temperature of 298.15 K. This involves (1) the determination of the corresponding adiabatic temperature change, ATad, from the temperature-time curve just mentioned, by using one of the methods discussed in section 7.1 (2) the determination of the energy equivalent of the calorimeter in a separate experiment. The obtained AT/icp value in conjunction with tabulated data or auxiliary calorimetric results is then used to calculate the enthalpy of an hypothetical reaction with all reactants and products in their standard states, Ar77°, at the chosen reference temperature. This is the equivalent of the Washburn corrections in combustion calorimetry... 

The affect of diffusion on catalyst selectivity in porous catalysts operating under non-isothermal conditions has been examined by a number of workers. The mathematical problem has been comprehensively stated in a paper [21] which also takes into account the affect of surface diffusion on selectivity. For consecutive first-order exothermic reactions, the selectivity increases with an increase in Thiele modulus when the parameter A (the difference between the activation energy for reaction... 

The reaction at this voltage is endothermic and hence at isothermal conditions heat energy (=TAS, where is S is entropy and T is the absolute temperature) must be absorbed from the surrounding environment for the increase in entropy associated with water... 

The least resolved measurement is determination of the isothermal rate constant k(T), where T is the isothermal temperature. Although conceptually simple, such measurements are often exceedingly difficult to perform for activated process without experimental artifact (contamination) because they require high pressures to achieve isothermal conditions. For dissociative adsorption, k(T) = kcol (T) [S (Tg = TS = T)), where kcol(T) is simply the collision rate with the surface and is readily obtainable from kinetic theory and Tg and T, are the gas and surface temperatures, respectively [107]. (S ) refers to thermal averaging. A simple Arrhenius treatment gives the effective activation energy Ea for the kinetic rate as... 

The gas is now reversibly compressed under the isothermal conditions (tc) of the low-temperature reservoir, with energy change... 

The conceptual mystery of the Gibbs free energy is largely associated with the entropic contribution. Sidebars 5.10-5.13 describe some alternative ways to think about entropy. Sidebar 5.10 evaluates the entropy change ASmix for the prototypical mixing of ideal gases A, B under isothermal conditions,... 

Let us recall the basic expression (5.53) for the P dependence of AG under isothermal conditions. For certain purposes, it is convenient to choose a standard state pressure, denoted P°, such that the free energy G = G(P) for any other pressure P is given by... 

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