Two-step expansion

Two-Step Expansion. Next, we will expand the gas in two steps by using two different weights on the pan. First, we put a weight with mass Mj/2 on the pan. In this case... 

The total quantity of work in this two-step expansion is... 

In the initial state there are four weights of mass M/2 holding the gas back. In the intermediate state one of these weights has been removed and in the final state a second weight of mass M/2 has been removed. The initial and final states in this figure are the same as in Figure 7-8. This two-step expansion helps us to establish that the work of expansion depends on the path taken. 

In comparing the quantity of work done in the two different expansions (Figs. 7-8 and 7-11), we found them to be different, thereby proving that work is not a state function. Additionally, the quantity of work performed is greater in the two-step expansion (Fig. 7-11) than in the single-step expansion (Fig. 7-8). We leave it to the interested student to demonstrate, through Feature Problem 125, that the maximum possible work is that done in a reversible expansion (Fig. 7-12). 

Ring expansion of haloalkyloxiranes provides a simple two-step procedure for the preparation of azetidin-3-ols (Section 5.09.2.3.2(f)) which can be extended to include 3-substituted ethers and O-esters (79CRV331 p. 341). The availability of 3-hydroxyazetidines provides access to a variety of 3-substituted azetidines, including halogeno, amino and alkylthio derivatives, by further substitution reactions (Section 5.09.2.2.4). Photolysis of phenylacylamines has also found application in the formation of azetidin-3-ols (33). Not surprisingly, few 2-0-substituted azetidines are known. The 2-methoxyazetidine (57) has been produced by an internal displacement, where the internal amide ion is generated by nucleophilic addition to an imine. 

To reeap, a turboexpander is a radial inflow turbine. The expansion proeess is aeeomplished in two steps primary and seeondary expansion. Primary expansion oeeurs in the inlet guide vanes and seeondary expansion oeeurs in the radial wheel. The proeess is isentropie and thermally effieient with reeoverable eold energy. Turboexpanders used in dew point eontrol require the following ... 

Earlier ehapters explained tliat air separation expanders are mrbines that expand gases in two steps, using primary and seeondary gas expansion deviees. Inlet guide vanes (or inlet nozzles) are the primary expansion deviee. Their funetion is to eonvert almost half of... 

A reversible cycle with turbine expansion split into two steps (high pressure, HP, and low pressure, LP) is illustrated in the T, s diagram of Fig. 4.3. The mass flow through the heater is still unity and the temperature rises from T2 to Tt, = Tq hence the heat supplied (3b is unchanged, as is the overall isentropic temperature ratio (x). But cooling air of mass flow i//H is used at entry to the first HP turbine (of isentropic temperature ratio. xh) and additional cooling of mass flow is introduced subsequently into the LP turbine (of isentropic temperature ratio Xl)- The total cooling flow is then i/( = i/ h + >h.-... 

For two step cooling, now with irreversible compression and expansion, Fig. 4.7 shows that the turbine entry temperature is reduced from Ti. to by mixing with the cooling air i/ H taken from the compressor exit, at state 2, pressure p2, temperature T2 (Fig. 4.7a). After expansion to temperature Tg, the turbine gas flow (1 + lp ) is mixed with compressor air at state 7 (mass flow i/h.) at temperature Tg. This gas is then expanded to temperature T g. 

The two step cooling example given above can in theory be extended to multi-step cooling of the turbine. It is more convenient to treat the turbine expansion as a modification of normal polytropic expansion the analysis is essentially an adaptation of that given in Section 4.2.1.3 for the multi-step cooled turbine cycle. 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

Path A is an isothermal, reversible expansion. Path B has two steps. In the first step, the gas is cooled at constant volume to 1.19 atm. In the second step, the gas is heated and allowed to expand against a constant external pressure of 1.19 atm until the final volume is 7.39 L. Calculate the work for each path. 

A system undergoes a two-step process. In step 1, it absorbs 50. J of heat at constant volume. In step 2, it releases 5 J of heat at 1.00 atm as it is returned to its original internal energy. Find the change in the volume of the system during the second step and identify it as an expansion or compression. 

To get analytic results for the convergence behaviour of an expansion in a Gaussian basis we shall proceed in two steps. 

Despite the extensive utility of osmylation reactions, the mechanism remains controversial - generally focusing on either a concerted [3 + 2] cycloaddition to directly yield the cyclic osmate ester or a two-step mechanism where [2 + 2] cycloaddition forms a 4-membered osmaoxetane which then undergoes a ligand-assisted ring expansion to yield the cyclic osmate ester216 (Scheme 21). 

One of the central problems in the chemistry of the singlet state of la was the mechanism of the ring expansion reaction that it undergoes.111 In particular, chemists sought to determine whether the ring expansion takes place in a single step, or in two steps via a bicyclic cyclopropene intermediate (Scheme 5).17... 

All three levels of theory predict the ring expansion of singlet phenylcarbene ( A -la) to cycloheptatetraene (3a) to occur in two steps, via bicy-clo[4.1.0]hepta-2,4,6-triene (2a) as an intermediate. The first step is addition of the carbene carbon to an adjacent 7t bond of the ring. The second step involves a six-electron, disrotatory, electrocyclic ring opening, which is allowed by orbital symmetry67 and thus proceeds by a highly delocalized transition state. Fig. 4... 

Our calculations on the ring expansion of the lowest singlet state of phenylnitrene ( A2-lb) to azacycloheptatetraene (3b) predict a two-step mechanism that is analogous to that for the rearrangement of la to 3a and which involves the bicyclic azirine intermediate 2b.61 The CASPT2 energetics are depicted in Fig. 5, and the CASSCF optimized geometries of the stationary points are shown in Fig. 6. 

Here G(vj, v2, v3) is the level energy in wave number units (as far as possible we follow the notation of Herzberg, 1950) and the constants in Equation (0.1) are given in Table 0.1. As usual the vs are the vibrational quantum numbers of S02 and rather high (above 10) values can be reached using the SEP technique. Equation (0.1) provides a fit to the observed levels to within an error below 10 cm 1, which is almost the experimental accuracy. We need, however, to be able to relate the parameters in this expansion directly to a Hamiltonian. The familiar way of doing this proceeds in two steps. First, the electronic problem is solved in the Bom-Oppenheimer approximation, leading to the potential for the... 

The drop formation is considered to proceed exactly in the same fashion as the bubble formation under constant flow conditions, viz. the two step (the expansion and detachment) mechanism. The tensile force does not arise in the expansion stage because there is no neck formation. 

The first consists of two steps (1) an isothermal reversible expansion at the temperature Ta until the volume V is reached, and (2) an adiabatic reversible expansion from V to Vj,. The entropy change for the gas is given by the sum of the entropy changes for the two steps ... 

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