Heating baths constant temperature

The first factor on the right-hand side of the above equation, p(z0), is the distribution of initial conditions zo, which, in many cases, will just be the equilibrium distribution of the system. For a system at constant volume in contact with a heat bath at temperature T, for instance, the equilibrium distribution is the canonical one... 

Here k is Boltzmann s constant, S the entropy of the heat bath, and T its temperature. Hence detailed balance for a system in contact with a heat bath at temperature T has the form... 

A weaker formulation of this approach is the Berendsen thermostat where to keep temperature constant, system is coupled to an external heat bath of temperature To. The velocities are scaled in such a way that the rate of change of temperature is proportional to the difference in temperature between system and bath, that is. 

All calorimeters consist of the calorimeter proper and its surround. This surround, which may be a jacket or a batii, is used to control tlie temperature of the calorimeter and the rate of heat leak to the environment. For temperatures not too far removed from room temperature, the jacket or bath usually contains a stirred liquid at a controlled temperature. For measurements at extreme temperatures, the jacket usually consists of a metal block containing a heater to control the temperature. With non-isothemial calorimeters (calorimeters where the temperature either increases or decreases as the reaction proceeds), if the jacket is kept at a constant temperature there will be some heat leak to the jacket when the temperature of the calorimeter changes. 

In a molecular dynamics calculation, you can add a term to adjust the velocities, keeping the molecular system near a desired temperature. During a constant temperature simulation, velocities are scaled at each time step. This couples the system to a simulated heat bath at Tq, with a temperature relaxation time of "r. The velocities arc scaled bv a factor X. where... 

One of the disadvantages of oil and metal baths is that the reaction mixture cannot be observed easily also for really constant temperatures, frequent adjustment of the source of heat is necessary. These difficulties are overcome, when comparatively small quantities of reactants are involved, in the apparatus shown in Fig. II, 5,4 (not drawn to scale). A... 

For a constant temperature simulation, a molecular system is coupled to a heat bath via a Bath relaxation constant (see Temperature Control on page 72). When setting this constant, remember that a small number results in tight coupling and holds the temperature closer to the chosen temperature. A larger number corresponds to weaker coupling, allowing more fluctuation in temper-... 

The simplest method that keeps the temperature of a system constant during an MD simulation is to rescale the velocities at each time step by a factor of (To/T) -, where T is the current instantaneous temperature [defined in Eq. (24)] and Tq is the desired temperamre. This method is commonly used in the equilibration phase of many MD simulations and has also been suggested as a means of performing constant temperature molecular dynamics [22]. A further refinement of the velocity-rescaling approach was proposed by Berendsen et al. [24], who used velocity rescaling to couple the system to a heat bath at a temperature Tq. Since heat coupling has a characteristic relaxation time, each velocity V is scaled by a factor X, defined as... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

With the known temperatures on each end of the coil, the heat duty for each coil can be calculated from the heat transfer theoiy in Chapter 2. Since the bath is at a constant temperature, LMTD can be calculated as ... 

At definite times samples are withdrawn and titrated by an appropriate method. The temperature is controlled either with a constant temperature oil bath or a heating jacket and a P.I.D. regulation with a captor plunged in the reaction medium. 

If a constant-temperature bath is not available, a bucket of water, initially at 25°, serves to dissipate the heat of reaction. At higher temperatures the potassium permanganate is rapidly consumed, presumably by reaction with the acetone. 

In a 300-mL round-bottom flask, a 5% sodium hydroxide solution (250 mL) was heated to 80° C in a constant-temperature bath. The catalysts were added in the following amounts in separate experiments trioctylmethy-lammonium chloride (TOMAC) (0.04 g, 0.0001 mol) trioctylmethylammo-nium bromide (TOMAB) (0.045 g, 0.0001 mol) hexadecyltrimethylammo-nium bromide (HTMAB) (0.045 g, 0.0001 mol) tetraethylammonium hydroxide (TEAOH) (0.015 g, 0.0001 mol) and phenyltrimethylammonium chloride (PTMAC) (0.02 g, 0.0001 mol). PET fibers (1.98 g, 0.01 mol) were added to the mixture and allowed to react for 30, 60, 90, 150, and 240 min. Upon filtration, any remaining fibers were washed several times with water, dried in an oven at 130-150°C, and weighed. The results are shown in Table 10.1. 

A 500 mL round-bottom flask containing aqueous sodium hydroxide (25 or 50 wt%) and benzyltrimethylammonium bromide as the phase transfer agent (0.2 g, 0.0008 mole) was placed in a constant-temperature oil bath and heated to the reflux temperature of 165°C. Chopped nylon-4,6 fibers (6 g, 0.030 mol) were placed in the reaction flask. The reaction mixture was constantly stirred with a magnetic stirrer and the reaction was carried out at atmospheric pressure for a period of 24 or 36 h (see Table 10.3 for results). The aqueous sodium hydroxide solution containing products of depolymerization was concentrated by evaporation and 30 mL of 35% aqueous hydrochloric acid was added to the concentrate. The precipitate was filtered and washed with water. The product was dried in a vacuum oven at 120°C for 24 h. 

The high pressure continuous reactor consists of five Kenics type in-line static mixers, that were connected in series [3]. Each reactor unit has 27 Kenics elements and dimensions of 19 cm tube length and 3.3 mm inner diameter. Acetonylacetone and 1 % NaOH aqueous solution were pumped into the in-line static mixer reactor using two independent HPLC pumps. The in-line static mixer reactors were immersed in a constant temperature controlled oil bath at 200 °C so that the reaction mixture was heated to the reaction temperature. When the reaction was completed, the fluid was cooled down rapidly in a constant temperature cold bath at 0 °C. At the end of the cooling line, a backpressure regulator was placed to allow experiments to be run at 34 bar. 

Heat added to an ice-water mixture melts some of the ice, but the mixture remains at 0 °C. Similarly, when an ice-water mixture in a freezer loses heat to the surroundings, the energy comes from some liquid water freezing, but the mixture remains at 0 °C until all the water has frozen. This behavior can be used to hold a chemical system at a fixed temperature. A temperature of 100 °C can be maintained by a boiling water bath, and an ice bath holds a system at 0 °C. Lower temperatures can be achieved with other substances. Dry ice maintains a temperature of -78 °C a bath of liquid nitrogen has a constant temperature of-196 °C (77 K) and liquid helium, which boils at 4.2 K, is used for research requiring ultracold temperatures. 

The surroundings may be so large that they can absorb or release significant amounts of heat before the temperature changes by a measurable amount. An experiment performed in a constant-temperature water bath is a common example of this category. 

The experiments were conducted batchwise in small stainless-steel pipe-bombs immersed in a molten-salt bath that was maintained at a desired, constant temperature. Pipe-bomb heat-up and quench times, on the order of 1 min each, were negligible compared with reaction times, which were on the order of 1 hr. The reagents used were obtained commercially all were of purity > 98% except for the A2-dialin which had a composition of ( 0)> 0 0 9 Q)) = 9 20, 64) mol%. The proportions of sub-... 

The two ends of the system are put into contact with thermal baths at temperature Tl and Tr for left and right bath, respectively. In fact, Eq. (5) is the Hamiltonian of the Frenkel-Kontorova (FK) model which is known to have normal heat conduction(Hu Li Zhao, 1998). For simplicity we set the mass of the particles and the lattice constant m = a = 1. Thus the adjustable parameters are (ki, hnt, kR, Vl, Vr, Tr, Tr), where the letter L/R indicates the left/right segment. In order to reduce the number of adjustable parameters, we set Vr = XVr, kR = Xki, Tl = T0(l + A),Tr = To(l — A) and, unless otherwise stated, we fix Vl = 5, ki = 1 so that the adjustable parameters are reduced to four, (A, A, kint, To)- Notice that when A > 0, the left bath is at higher temperature and vice versa when A < 0. 

Sz(i) being the total angular momentum of the i spins. The linear combination 5 (l)/a1 + Sz(2)jx2 is now coupled to the dipole-dipole heat bath, whereas the difference Sz(l)/x1 — Sz(2)jx2 remains constant. It may be shown that the chemical equilibrium condition (31) is then equivalent to an equalization of the temperatures of the collective coordinate Sz(l)jx1 + Sz(2)/x2 and of the dipole-dipole heat bath. 

Viscosity depends on temperature. The higher the temperature, the lower the viscosity Pancake syrup, for example, flows more freely when heated. For reasonable accuracy when measuring viscosity, the temperature must be very carefully controlled. This means that the viscometer and sample must be immersed in a constant temperature bath and the temperature given time to equilibrate before the measurement is recorded. A calibrated thermometer must be used to measure the temperature. 

Modern heat flow microcalorimeters employ a diversity of heat sinks and cells, depending on the applications for which they were designed. The heat sinks can be water baths, kept at a constant temperature ( 5 x 10-4 K) and typically operating in the range of 20-80 °C, or metal blocks, allowing much wider temperature ranges (e.g., -196°C to 200°C, 20°C to 1000°C). In some cases it is possible to scan the temperature at a predetermined rate (see chapter 12). 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

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