Properties of the ideal gas

If arbitrary values are assigned to any two of the three variables p, V, and T, the value of the third variable can be calculated from the ideal gas law. Hence, any set of two variables is a set of independent variables the remaining variable is a dependent variable. The fact that the state of a gas is completely described if the values of any two intensive variables are specified allows a very neat geometric representation of the states of a system. 

In Fig. 2.4, p and V have been chosen as independent variables. Any point, such as A, determines a pair of values of p and V this is sufficient to describe the state of the system. Therefore every point in the p- V quadrant (both p and V must be positive to make physical sense) describes a diff erent state of the gas. F urthermore, every state of the gas is represented by some point in the p-V diagram. 

It is frequently useful to pick out all of the points that correspond to a certain restriction on the state of the gas, as, for example, the points that correspond to the same temperature. In Fig. 2.4 the curves labeled Ti, T2, and T3 collect all the points that represent states of the ideal gas at the temperatures T, T2, and T3, respectively. These curves are called isotherms. The isotherms of the ideal gas are rectangular hyperbolas determined by the relation 

In Fig. 2.5 every point corresponds to a set of values for the coordinates V and T again each point represents a state of the gas, just as in Fig. 2.4. In Fig. 2.5 points corresponding to the same pressure are collected on the lines, which are called isobars. The isobars of the ideal gas are described by the equation 

As in the other figures, every point in Fig. 2.6 represents a state of the gas, because it determines values of p and T. The lines of constant molar volume, isometrics, are described by the equation 

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We will more thoroughly define the properties of the ideal gas later. 

JThis calculation is one of the most satisfying in science. The values of the thermodynamic properties of the ideal gas calculated from molecular parameters are usually more accurate than the same thermodynamic results obtained from experimental measurements. 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation... 

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

We are now ready to relate the thermodynamic properties of the ideal gas to Z. We start with U - Uq. where Uq is the energy when all the molecules are at zero... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

Clusius and Frank 61) find 83.78 K. for the melting point with 280.8 cal./gram atom for the heat of melting as well as 87.29 K. for the normal boiling point and 1558 cal./gram atom for the associated heat of vaporization. These vapor pressure data are substantiated by the more recent work of Clark, Din, Robb, Michels, Wassenaar, and Zwietering 57). Thermodynamic properties of the ideal gas have been calculated at the National Bureau of Standards 296). Kobe and Lynn 193) select 151 K. for the critical temperature and 48.0 atmospheres for the critical pressure. 

Here we have used the superscripts IG and IGM to indicate properties of the ideal gas and the ideal gas mixture, respectively, and taken pressure and temperature to be the independent variables. From Eq. 8.1-12 it then follows that for the ideal gas mixture... 

In this section, we will show how to derive a fundamental form from the state equations. Further, we will show how to get from certain assumptions on the fundamental from the state equations. The latter approach allows getting information on the thermodynamic properties of the ideal gas. 

Here it has been shown that the property of the ideal gas mixture follows from the model of a simple fluid mixture of gases. In fact both these models are equivalent, because from (4.430), (4.431), (4.419) Eqs. (4.413), (4.412), (4.414) follow [61]. 

The properties of the ideal-gas state will be discussed in detail in Chapter g . but some important results will be introduced here on the basis of qualitative molecular arguments. In the ideal-gas state, intermolecular interactions are unimportant because distances between molecules are large, beyond the... 

The partial molar volume and partial molar enthalpy are equal to the corresponding properties of the pure component. The partial molar entropy is higher than the entropy of the pure component by -R In x. a component in the mixture contribute more entropy per mole than the original entropy of the pure component before mixing. The properties of the ideal-gas state are summarized in Table o-i. 

Other properties of the ideal-gas mixture can be obtained easily. For example, for the internal energy we write... 

The thermodynamic properties of the hypothetical ideal gas are of practical interest because, as we will see ( 13.2.3, 13.6.1) it is common practice in developing an equation for real systems to first subtract the properties of the ideal gas, which are known, and then deal only with the deviations from these properties. The properties of the ideal gas are in many cases not quite as simple as you might suppose. 

It is fairly intuitive that many properties of the ideal gas should be independent of pressure, but not independent of temperature. If there is no interaction whatsoever between molecules, which have zero volume, then it should not matter how close together they are (the effect of P). But if we add heat to the gas (we raise the T), that energy cannot disappear, but must be reflected in the thermodynamic properties of the ideal gas. 

The macroscopic properties of the ideal gas can be described very well in terms of pressure P, temperature T, and volume Vusing the ideal gas equation. 

In the books of classical statistical mechanics, the properties of the ideal gas have been evaluated, so we can focus on the configurational contribution. With the Monte Carlo method [17-19], we explore the configuration phase subspace according to the canonical or grand canonical distributions. The corresponding trajectory is essentially the projection of the phase space onto the subspace of coordinates, becoming independent of the subspace of moments. 

By combining Boyle s law with Charles law, we can develop a relationship that describes the properties of the ideal gas through changes in the pressure (p), volume (V), and temperature (T) ... 

This combined relationship serves to relate the properties of a fluid in a closed system, but does not permit the absolute calculation of any specific property, since the amount of the substance present is not known. For that, we require Avogadro s law, developed in 1811 by Amedeo Avogadro, which states that the number of molecules (n) in a specific volume of fluid (V) at a given pressure (P) and temperature (7) is always the same. Together, this provides the ideal gas law, which can be used to relate the properties of the ideal gas under any conditions. 

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