Equations for a closed system

We begin with the application of the first law of thennodynanucs first to a dosed system and then to an open system. A system is any bounded portion of the universe, moving or stationary, which is chosen for the application of the various thermodynamic equations. For a closed system, in wMch no mass... 

The ideal systems described are difficult to achieve in practice. In a real closed system adsorption occurs on the walls and gas is pumped by the pressure gauge. The waUs may degass as the pressure falls, presenting a difficult problem. Therefore the equation for a closed system is really represented by the equation for a flow system, where Fj and now denote gas flow or pumping of any type other than that due to adsorption on the filament. Closed systems are also hmited because for a given initial pressure, only a small range of surface concentrations can be measured. Fig. 6 shows a typical apparatus for adsorption measurements. The flow into the adsorption cell is controlled by valve and the pumping rate can be... 

Closed system A closed system is one which does not have flows in or out of the substance. Almost always, when one refers to a close system it is implied to be closed to mass flow but not to other flows such as energy or momentum. The equation for a closed system is ... 

Equation (1 19) is the basic equation for a closed system, i.e. one which does not exchange matter with its environment. It may be derived more directly as foDows. From the first law... 

In working with an open system, the problan is somewhat more complex. First, let s assume that the system is operating at steady state. Then, we can derive an equation nearly identical to the above equation for a closed system, except that we need to account for all of the flowing streams. This gives... 

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that ia such a system... 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

Since the mass is fixed for a closed system, the equations in this discussion will be valid for the entire mass (M) or on a unit mass basis. 

Example 9.3 Equation (9.20) was derived for a closed system. Repeat the derivation for an open system with i),>, > 0 and > 0 shown in Figure 9.9. 

Momentum space equations for a closed-shell system... 

Thus in the absence of non-expansion work for a closed system, the following important equation... 

Recall that the equilibrium condition for a closed system at constant T and p was given by eq. (1.41). For an open system the corresponding equation is... 

For a closed system of fixed composition, the extensive thermodynamic properties such as y, U, S, A, Y, and G are functions of any pair of convenient independent variables. For example. Equation (7.38) suggests that G is a natural function of T and P. That is G =f(T, P). The total differential of G would be... 

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that in such a system there are at most r mole-number-related quantities, S , capable of independent variation. Note the absence of implied restrictions with respect to chemical reaction equilibria the reaction coordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. 

Two cases must be considered one in which the state of aggregation is the same in the initial and final state, and the other in which the state of aggregation is different in the two states. In the first case the enthalpy is a continuous function of the temperature and pressure in the interval between (Th P,) and (T, Pj). Equation (4.86) can be used for a closed system and the integration of this equation is discussed in Section 8.1, where the emphasis is on standard states of pure substances. The result of the integration is valid in the present instance with change of the limits of integration and limitation to molar quantities. Equations (8.10) and (8.11) then become... 

This form of the fundamental equation applies at each stage of the reaction. The rate of change of G with extent of reaction for a closed system with a single reaction at constant T and P is given by... 

Another approach for estimating am is based on the pseudothermodynamic properties of the mixture, as suggested by Rudinger (1980). The equation for the isentropic changes of state of a gas-solid mixture is given by Eq. (6.53). Note that for a closed system the material density of particles and the mass fraction of particles can be treated as constant. Hence, in terms of the case for a single-phase fluid, the speed of sound in a gas-solid mixture can be expressed as... 

Thus it can be generalized that the integrated equation for a reaction of any order in a flow system with no mixing is the same as for a closed system except for the time variable replaced by V/u. 

For a closed system, the number of variables will include the p number of extensive variables of the masses of the phases 2+ n — l)p+p. The number of independent phase equilibrium equations will include the material balance equations for each n species, and we have (p )n + n. Therefore, the degrees of freedom of a closed system is... 

Comparing this equation with the first law of thermodynamics dH 8q - VdP (for a closed system and for - dP), we obtain... 

Some crucial aspects of studying the GHF wave functions are connected with the relationship between the GHF and UHF methods. First of all, it is evident that the RHF and UHF wave functions are particular solutions also to the GHF problem In this case the components p " and p" of the Fock-Dlrac density matrix are zero, and the GHF equations separate into two sets of equations for the orbitals of spins a and p, respectively. The system of equations obtained in this way is identical to that of the ordinary UHF scheme. We note that the two sets of equations are still coupled through the components p++ and p". The situation is in some way analogous to the case of the UHF equations for a closed-shell system, for which the RHF functions always provide a particular solution. Similarly to the RHF versus UHF case, the UHF (or RHF) solution can, in principle, represent either a true (local) minimum or a saddle point for the GHF problem. 

In the vast majority of experimental studies, the backmixing characteristics of a flowing phase are examined using a -pulse tracer input. For the fixed-bed systems shown in Fig. 3-2, if a perfect pulse input is used, then, as shown by Levenspiel,5 6 the axial dispersion coefficient or the Peclet number can be obtained from the variance of the RTD curve. For example, for a closed system and large extent of dispersion, the variance, it, is related to the Peclet number by the equation... 

Define the terras closed process system, open process system, isothermal process, and adiabatic process. Write the first law of thermodynamics (the energy balance equation) for a closed process system and state the conditions under which each of the five terms in the balance can be neglected. Given a description of a closed process system, simplify the energy balance and solve it for whichever term is not specified in the process description. 

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

Equation 7.3-4 is the basic form of the first law of thermodynamics for a closed system. 

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