State of a gas

As noted earlier, the standard state of a gas is the hypothetical ideal gas at 1 atmosphere and the specified temperature T. 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

The fact that gases have a simple equation of state makes possible the use of absorptiometry with polychromatic beams to give information about the state of a gas under conditions (in detonation waves,16 boundary layers,17 or supersonic flow18) transient or difficult of access. Temperature measurements19 have also been made. The technique is a unique method for studying the fluidization of a finely divided solid by a gas. Bed density profiles, which reveal the character and effectiveness of fluidization, have been readily determined20 without disturbing the system as probes would inevitably do. 

Standard State of a Gas The standard state for a gas is usually chosen as the... 

In general the conditions under which a change in state of a gas takes place are neither isothermal nor adiabatic and the relation between pressure and volume is approximately of the form Pvk = constant for a reversible process, where k is a numerical quantity whose value depends on the heat transfer between the gas and its surroundings, k usually lies between 1 and y though it may, under certain circumstances, lie outside these limits it will have the same value for a reversible compression as for a reversible expansion under similar conditions. Under these conditions therefore, equation 2.70 becomes ... 

The total entropy of a substance in a state defined as standard. Thus, the standard states of a solid or a liquid are regarded as those of the pure solid or Ihe pure liquid, respectively, and at a stated temperature. The standard state of a gas is at 1 atmosphere pressure and specified temperature, and its standard entropy is the change of entropy accompanying its expansion to zero pressure, or its compression from zero pressure to 1 atmosphere. The standard entropy of an ion is defined in a solution of unit activity, by assuming that the standard entropy of the hydrogen ion is zero. 

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... 

The work of the empirical gas laws made it possible to formulate a state function to completely describe the state of a gas under a given set of conditions. This is called the ideal gas law, which has the form ... 

The ideal gas law is most useful when the problem describes only a single state (temperature, pressure, and/or volume). The empirical gas laws are useful for changing the state of a gas (F, n, V, and/or T change). 

Show that the Joule coefficient can he obtained from the equation of state of a gas... 

The normal state of a gas is determined by the normal temperature (Tn) and normal pressure (/ ) ... 

Equation n. B. 19. gives the difference in free energy from the standard state of a gas at any temperature and pressure. The relationship of the entropy to the pressure is given by ... 

VIII) The actually observed sequence of states of a gas quantum from time tA on is identical with that created by the overwhelming majority144 of the motions discussed in (VI). 

Gibbs If one affects all the members of an initially canonical ensemble in a manner corresponding to the reversible change of the state of a gas,198 then it is permissible to assume (cf. XVIII) that the ensemble will always pass through canonical distributions only. Under these assumptions we can represent the average value over the ensemble of the 5Q defined above by199... 

The state of a gas at the limiting condition where P - 0 deserves some discussion. As the pressure on a gas is decreased, the individual molecules become more and more widely separated. The volume of the molecules themselves becomes a smaller and smaller fraction of the total volume occupied by the gas. Furthermore, the forces of attraction between molecules become ever smaller because of the increasing distances between them. In the limit, as the pressure approaches zero, the molecules are separated by infinite distances. Their volumes become negligible compared with the total volume of the gas, and the inter-molecular forces approach zero. A gas which meets these conditions is said to be ideal, and the temperature scale established by Eq. (3.9) is known as the ideal-gas temperature scale. 

Integration at constant temperature from the state of a gas at a low pressure to the state at higher pressure P gives... 

Water has an activity of 1 when Nw (see Eq. 2.8) is 1. The concentration of water on a molality basis (number of moles of a substance per kilogram of water for aqueous solutions) is then 1/(0.018016 kg mol-1) or 55.5 molal (m). The accepted convention for a solute, on the other hand, is that aj is 1 when yfj equals 1 m. For example, if yj equals 1, a solution with a 1 -m concentration of solute j has an activity of 1 m for that solute. Thus the standard state for an ideal solute is when its concentration is 1 m, in which case RT In a - is zero.2 A special convention is used for the standard state of a gas such as CO2 or O2 in an aqueous solution—namely, the activity is 1 when the solution is in equilibrium with a gas phase containing that gas at a pressure of 1 atm. (At other pressures, the activity is proportional to the partial pressure of that gas in the gas phase.)... 

The ideal gas law is an equation of state for a gas, where the state of the gas is its condition at a given time. A particular state of a gas is described by its pressure, volume, temperature, and number of moles. Knowledge of any three of these properties is enough to completely define the state of a gas, since the fourth property can then be determined from the equation for the ideal gas law. 

The state of a gas is completely determined by specifying the states of all molecules composing the gas. In view of the large number of molecules involved (and the continual external perturbations present in any real system), one cannot hope to determine the state of a gas the most information that can be obtained is a specification of some sort of average state. This average state is the thermodynamic state (that is, only the thermodynamic properties are specified), which might be expected to depend on all possible states of the molecules composing the gas. 

The state of a gas at the limiting condition where P 0 deserves some discussion. The... 

A pure compound may be in the state of a gas, sohd or liquid (or even multiphase) depending on the pressure and temperature, the interrelationship being shown in Figure 23.6. Following the vapour pressure curve which separates the gas and liquid states in the direction of increased pressure and temperature leads to an area in which the densities of both phases are identical. A phase that is neither gas nor liquid follows on from the critical point P (shaded area) and this is known as the fluid or supercritical... 

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