Application to Perfect Gases

The oldest and most important experimental proof of the correctness of equation (2) was given by Gay-Lussac. If we allow a gas to expand without absorption or loss of heat, and so that no external pressure has to be overcome in the expansion, then dQ and p dv are both zero. From equation (2) it follows that dT = 0 also, i.e. there should be no change in temperature. In accordance with this Gay-Lussac found that when a gas Digitized by Microsoft  

Isothermal processes. If we conduct the compression or expansion of a gas so that its temperature remains constant, the change in volume is said to take place isothermally. This condition can be realised experimentally by keeping the gas during expansion in a thermostat of large capacity for heat, and by causing the change in volume to take place so slowly that the gas always maintains the constant temperature of its surroundings. In this case we have dT = 0 and dQ=pdv. The heat absorbed by the gas is completely converted into work. 

The integration can only be carried out when p is known as. a function of v in the whole interval from to Vg, i.e. when we know the pressure which has to be overcome during expansion for every value of v between and i g. 

We shall assume that this pressure is equal throughout to the pressure in the interior of the gas. We have then 

This is an expression for the maximum amount of work which can be done in the expansion of a gas from volume to volume 2 at constant temperature, for the greatest pressure against which expansion can take place is that which is equal to the internal pressure of the gas. If the external pressure is smaller than the internal pressure, the work done will also be smaller than in the limiting case, in which both are equal. On the other hand, when a gas is compressed, the work done in the compression is a minimum when the external and internal pressures are equal to one another. In the limiting case, which is characterised by the condition 

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We can show that the relations obtained are valid if the polycomponent phases exhibit perfect behavior in the sense of the solutions. In the case of gases, those relations are applicable to real gases, provided the mixture of gases is perfect in the sense of the solution. Thus, the relations are applicable when the Lewis approximation is satisfactory (see relation [3.5]). In the case of solutions with non-perfect behavior, to the difference between the fundamental energies we must add the energy (or enthalpy) of nuxing, which can be obtained either on the basis of the measurements or by modeling the interactions between the molecules in the solution. 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

Van der Waals equation is an attempt to modify the general gas law so that it will be applicable to non-perfect gases. The equation for one mole of a single, pure gas is written... 

Mole per cent or mole fraction, weight per cent or weight fraction, and volume per cent or volume fraction may be employed to designate the composition of a solution. Avogadro s Law is not applicable to liquids, and equal volumes of different liquids do not contain the same number of molecules. Consequently, mole per cent and volume per cent are not equivalent in liquid solutions as they were in perfect-gas mixtures. To convert mole per cent to weight per cent the procedure is identical with that previously described for gases. To calculate the volume per cent of a liquid solution from the mole per cent or weight per cent the densities of the pure components must be known. 

Deviations from the simple laws. The exact proportionality between the osmotic pressure and the concentration can only hold in dilute solutions. No matter how we account for the osmotic pressure laws, whether by an attraction between the solvent and the solute, or by the impacts of the dissolved molecules, or whether we deduce them from the lowering of the vapour pressure of the solution, we are always forced to restrict the applicability of the simple laws of van t Holf to the region of very dilute solutions. Similarly, the laws of perfect gases can only be regarded as valid in the limiting case of very great... 

Since then further progress has extended the field of applicability of Gibbs chemical thermodynamics. Thus the introduction of the ideas of fugacity and activity by G. N. Lewis enabled the thermodynamic description of imperfect gases and of real solutions to be expressed with the same formal simplicity as that of perfect gases and ideal solutions. These results were completed when N. Bjerrum and E. A. Guggenheim introduced osmotic coefficients. 

Enthalpy is used as the measure of the thermal driving potential to broaden the application of Eq. 6.1 to thermally perfect gases with temperature-dependent specific heats where... 

Most introductory accounts use the combination (la), (26) and (3a), which is one of the easiest to grasp. However, the disadvantage of averaging over the quantum states of molectdes as in (la), is that the statistical formulae which are obtained are applicable only to qrstems in which the particles are independent, as in perfect gases. In many elementary accounts of this method it is also necessary to adopt a subterfuge in order to introduce an important term, n , into the formulae. The method (26) may also lead to an erroneous impression of the nature of equilibrium, for it might be taken to imply that the system stays always in the particular distribution known as the most probable distribution. Tliis would be to preclude fluctuation phenomena. 

The great value of the third law in physical chemistry is this application to chemical reactions. In the case of simple molecules, which exist as perfect or almost perfect gases, the standard free energy can be evaluated from the band spectrum, as discussed in 12 7. This method is not applicable to the more complex organic substances, and it is for this reason that the third law is extremely valuable. 

These equations are essentially incomplete until expressions are given for the Kg and the various fluxes. For simplicity, we restrict the discussion of these expressions to forms applicable to mixtures of perfect gases. The chemical kinetics of a reacting mixture may be described in terms of a set of chemical reactions, which may be written symbolically in the form... 

Figures 2-38A and 2-38B are based on the perfect gas laws and for sonic conditions at the outlet end of a pipe. For gases/vapors that deviate from these laws, such as steam, the same application will yield about 5% greater flow rate. For improved accuracy, use the charts in Figures 2-38A and 2-38B to determine the dowmstream pressure when sonic velocity occurs. Then use the fluid properties at this condition of pressure and temperature in ...

All gas particles have some volume. All gas particles have some degree of interparticle attraction or repulsion. No collision of gas particles is perfectly elastic. But imperfection is no reason to remain unemployed or lonely. Neither is it a reason to abandon the kinetic molecular theory of ideal gases. In this chapter, you re introduced to a wide variety of applications of kinetic theory, which come in the form of the so-called gas laws. ... 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

Kinetic theory of gases was given by Kronig, Clausius, Maxwell etc. to explain the behaviour of gases theoretically. This theory is applicable only to a perfect or an ideal gas. The main postulates or assumptions of the kinetic theory are ... 

These characteristics led to the ideal gas law. They are accurate enough for most applications. They are not perfect, however. In fact, most gases fall short of these characteristics in many ways ... 

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