Internal energy reversible

Consider the free expansion of a gas shown in Fig. 3.8 on page 79. The system is the gas. Assume that the vessel walls are rigid and adiabatic, so that the system is isolated. When the stopcock between the two vessels is opened, the gas expands irreversibly into the vacuum without heat or work and at constant internal energy. To carry out the same change of state reversibly, we confine the gas at its initial volume and temperature in a cylinder-and-piston device and use the piston to expand the gas adiabatically with negative work. Positive heat is then needed to return the internal energy reversibly to its initial value. Because the reversible path has positive heat, the entropy change is positive. 

Compressible Vlow. The flow of easily compressible fluids, ie, gases, exhibits features not evident in the flow of substantially incompressible fluid, ie, Hquids. These differences arise because of the ease with which gas velocities can be brought to or beyond the speed of sound and the substantial reversible exchange possible between kinetic energy and internal energy. The Mach number, the ratio of the gas velocity to the local speed of sound, plays a central role in describing such flows. 

Note that under choked conditions, the exit velocity is V = V = c = V/cKTVM not V/cKT(/M, . Sonic velocity must be evaluated at the exit temperature. For air, with k = 1.4, the critical pressure ratio p /vo is 0.5285 and the critical temperature ratio T /Tq = 0.8333. Thus, for air discharging from 300 K, the temperature drops by 50 K (90 R). This large temperature decrease results from the conversion of internal energy into kinetic energy and is reversible. As the discharged jet decelerates in the external stagant gas, it recovers its initial enthalpy. 

The adiabatic expansion of a gas is an example of (b). In the reversible adiabatic expansion of one mole of an ideal monatomic gas, initially at 298.15 K, from a volume of 25 dm3 to a final volume of 50 dm3, 2343 J of energy are added into the surroundings from the work done in the expansion. Since no heat can be exchanged (in an adiabatic process, q = 0), the internal energy of the gas must decrease by 2343 J. As a result, the temperature of the gas falls to 188 K. 

In this expression consistent units must be used. In the SI system each of the terms in equation 2.1 is expressed in Joules per kilogram (J/kg). In other systems either heat units (e g. cal/g) or mechanical energy units (e.g. erg/g) may be used, dU is a small change in the internal energy which is a property of the system it is therefore a perfect differential. On the other hand, Sq and SW are small quantities of heat and work they are not properties of the system and their values depend on the manner in which the change is effected they are, therefore, not perfect differentials. For a reversible process, however, both Sq and SW can be expressed in terms of properties of the system. For convenience, reference will be made to systems of unit mass and the effects on the surroundings will be disregarded. 

The change in the internal energy may be expressed in terms of properties of the system itself. For a reversible process ... 

FIGURE 6.16 (a) On the reversible path, the work done in Example 6.5 is relatively large (w = -2.12 kj) because the change in internal energy is zero, heal flows in to maintain constant temperature and constant internal energy. Therefore, q =... 

FIGURE 7.21 The changes in entropy and internal energy when an ideal gas undergoes (a) reversible and (b) irreversible changes between the same two states, as described in Example 7.12. 

In order to define the surface tension we will consider the change in internal energy connected with a reversible change in the system. For an open system dU is given by eq. (1.79) as... 

While the chemical substance involved dictates the magnitude of A U (i.e. the amount of it), its sign derives from the direction of the thermodynamic process. We can go further if the same mass of substance is converted from state A to state B, then the change in internal energy is equal and opposite to the same process occurring in the reverse direction, from B to A. This essential truth is depicted schematically in Figure 3.3. 

Figure 3.3 The change in internal energy when converting a material from state A to state B is equal and opposite to the change in U obtained when performing the same process in reverse, from B to A...

On the other hand, A C/°(0) can be related with the same parameter for the reverse reaction and with the standard reaction internal energy at 0 K ... 

Hvistendahl, G. Williams, D.H. Partitioning of Reverse Activation Energy Between Kinetic and Internal Energy in Reactions of Organic Ions. J. Chem. Soc., Perkin Trans. 2 1975, 881-885. 

However, if there is no other exothermic pathway available, all the intermediate can do is revert to reactants. In such a situation, the more favorable the addition process is, the more internal energy is in the intermediate and the faster the reverse dissociation will occur. The better the addition is thermochemically, the worse it is kinetically. For the proton transfer pathway (6b), the neutral methanol product can carry off the excess energy as translational energy (and capture some of it in the newly formed OH bond) and the reaction proceeds. 

A corollary to this law is that all reversible Carnot cycles, operating between the same two temperatures, must have the same efficiency. For a perfectly reversible cycle in which the only pressure, P, is uniform and external, then dU = TdS - PdU where df/ is the differential change in internal energy, T is the absolute temperature, dS is the differential change in entropy, and dV is the differential change in volume. This relation, which is a... 

Fig 6 The Rankine-Hugoniot curve defines states that can be induced in substance by shock compression in terms of pressure (p), specific volume (V), and internal energy (E). Shock compression from initial state B to shocked state C follows the R-H curve and dissipates energy shown by the hatched area. Thus shock compression is not a reversible process—unlike adiabatic compression, which is, at least ideally reversible... 

Rubber elasticity, which is a unique characteristic of polymers, is due to the presence of long chains existing in a temperature range between the Tg and the Tm. The requirements for rubbery elasticity are (1) a network polymer with low cross-link density, (2) flexible segments which can rotate freely in the polymer chain, and (3) no volume or internal energy change during reversible deformation. 

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