V, internal energy

Any characteristic of a system is called a property. The essential feature of a property is that it has a unique value when a system is in a particular state. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the size of a system, such as temperature T and pressure p. Extensive properties are those that are dependent on the size of a system, such as volume V, internal energy U, and entropy S. Extensive properties per unit mass are called specific properties such as specific volume v, specific internal energy u, and specific entropy. s. Properties can be either measurable such as temperature T, volume V, pressure p, specific heat at constant pressure process Cp, and specific heat at constant volume process c, or non-measurable such as internal energy U and entropy S. A relatively small number of independent properties suffice to fix all other properties and thus the state of the system. If the system is composed of a single phase, free from magnetic, electrical, chemical, and surface effects, the state is fixed when any two independent intensive properties are fixed. 

The system of our choice will usually prevail in a certain macroscopic state, which is not under the influence of external forces. In equilibrium, the state can be characterized by state properties such as pressure (P) and temperature (T), which are called "intensive properties." Equally, the state can be characterized by extensive properties such as volume (V), internal energy (U), enthalpy (H), entropy (S), Gibbs energy (G), and Helmholtz energy (A). 

TEMPERATURE ABS ECIPIC VOLUME V INTERNAL ENERGY U ENTHALPY H ENTROPY S ... 

The composition of a solution is obviously one of its important properties. In the preceding section various ways of describing the composition of a two-component system were described. Other properties include its volume, V, internal energy, U, and entropy, S. In order to specify any one of these, one must specify not only the amounts of each of the components but also the temperature, T, and pressure, P. These quantities are known as the independent variables of the system. They are... 

V. Internal energy distribution measurements using PFI-PE spectroscopy, 85... 

State defined by (3.222) (or by (3.220), (3.221)) the persistence of which is achieved by the zero body heating (3.231), the zero inertial and body forces (i = o, b = o) and the zero velocity v = o (3.223) everywhere. The body is in the uniform equilibrium state mentioned above and as may be seen, such a state may be realized in the isolatedhody in which no exchange of heat, work and mass with environment exists and the boundary of which is fixed. Denoting constant (throughout the body and time) equilibrium values of temperature T° density p° and therefore also specific volume v°, internal energy and entropy s° (cf. (3.191), (3.192), (3.199)) we can express the volume V°, energy E° and entropy S° of the body in such equilibrium by... 

Mowry, C. D. Johnston, M. V. Internal energy of neutral molecules ejected by matrix-assisted laser-desorption. J. Phys. Chem. 1994, 98, 1904—1909. 

One can trivially obtain the other thennodynamic potentials U, H and G from the above. It is also interesting to note that the internal energy U and the heat capacity Cy can be obtained directly from the partition fiinction. Since V) = 11 exp(-p , ), one has... 

Now let us write down explicit expressions for p Q), -Pr(v,) and g-j-. Denoting the internal energy for a given state as e. and the relative translational energy as = I we have (in tluee dimensions)... 

Enthalpy. Enthalpy is the thermodynamic property of a substance defined as the sum of its internal energy plus the quantity Pv//, where P = pressure of the substance, v = its specific volume, and J = the mechanical equivalent of heat. Enthalpy is also known as total heat and heat content. 

The systems of interest in chemical technology are usually comprised of fluids not appreciably influenced by surface, gravitational, electrical, or magnetic effects. For such homogeneous fluids, molar or specific volume, V, is observed to be a function of temperature, T, pressure, P, and composition. This observation leads to the basic postulate that macroscopic properties of homogeneous PPIT systems at internal equiUbrium can be expressed as functions of temperature, pressure, and composition only. Thus the internal energy and the entropy are functions of temperature, pressure, and composition. These molar or unit mass properties, represented by the symbols U, and S, are independent of system size and are intensive. Total system properties, J and S do depend on system size and are extensive. Thus, if the system contains n moles of fluid, = nAf, where Af is a molar property. Temperature... 

Cp = specific heat e = specific internal energy h = enthalpy k =therm conductivity p = pressure, s = specific entropy t = temperature T = absolute temperature u = specific internal energy [L = viscosity V = specific volume f = subscript denoting saturated hquid g = subscript denoting saturated vapor... 

The same manipulations can be done for internal energy as a function of T and V. 

Because V is related to T and P through an equation of state, V rather than P can serve as an independent variable. In this case the internal energy and entropy are the properties of choice whence... 

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 equation that expresses conservation of energy can also be determined by considering Fig. 2.3. Since the piston moves a distance u At, the work done by the piston on the fluid during this time interval is Pu At. The mass of material accelerated by the shock wave to a velocity u is PqU At. The kinetic energy acquired by this mass element is therefore (pqUu ) At/2. If the specific internal energies of the undisturbed and shocked material are denoted by Eq and E, respectively, the increase in internal energy is ( — o)Po V At per unit mass. The work performed on the system is equal to the sum of kinetic and... 

These equations can be combined to eliminate the velocities, yielding the Rankine Hugoniot equation for internal energy jump in terms of pressures and specific volumes (V s 1/p)... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Figure 2.5. Relationship of the P-V Hugoniot to the Rayleigh line and a graphical illustration of kinetic and internal energy increase.

V is the material velocity. a is the stress tensor. g is the acceleration of gravity. e is the internal energy per unit mass. h is the energy flux. 

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