The ideal gas

Siirce the terms B/V, C/V, etc., of the virial expairsion [Eq. (3.12)] arise on account of molecular interactions, the virial coefficients B, C, etc., would be zero if iro such iirteractions existed. The virial expatrsiotr would tlren reduce to  

For a real gas, molecular interactions do exist, and exert an itrfluetrce otr the observed behaviorof the gas. As the pressure of a real gas is reduced at constant temperature, F increases and the contributions of the terms B/V, C/ V, etc., decrease. For a pressure approacliingzero, Z approachesunity, not because of any change in the virial coefficients, but because F becomes 

The definition of heat capacity at constant volume, Eq. (2.16), leads for an ideal gas to the conclusion that Cy is a function of temperature only  

Since pressure represents the force per area, the unit of [N/m ] is most straightforward however, in the context of the energy balances we will be addressing in this text, the unit of [j/m ] is often more useful. 

An equation that relates the measured properties T, P, and v is called an equation of state. The simplest equation of state is given by the ideal gas model  

Applying Equation (1.1), the ideal gas model can be written in terms of extensive volume, V, and number of moles, n, as follows  

Values for the gas constant, R, in different units are given in Table 1.1. The ideal gas model was empirically developed largely through the work of the chemists Boyle, Gay-Lussac, and Charles. It is valid for gases in the limits of low pressure and high temperature. In practice, the behavior of most gases at atmospheric pressure is well approximated 

From a molecular viewpoint, we can develop the ideal gas relation based on the assumption that the gas consists of molecules that are infinitesimally small, hard, round spheres that occupy negligible volume and exert forces on each other only through collisions. Thus, there are no potential energy interactions between the molecules. When a gas takes up a significant part of the system s volume or exerts other intermolecular forces, alternative equation of state should be used. Such Equations will be addressed in Chapter 4. 

For an ideal gas the general equation of state or overall relationship connecting the possible variables P, pressure, T, 

The combined gas law above is compounded from three individual laws those of  

There is no change in (internal) energy of the gas as its volume is increased (i.e. gas is expanded) whilst the temperature is kept constant. This provides a convenient mathematical definition (in thermodynamic terms) of an ideal gas. 

The general form of a curve of y plotted versus x for the function is  

When a variable approaches infinity the curve at this limit is often called an asymptote. 

An ideal gas is defined as one whose properties are given by the two equations (Sect. 2.8) 

These two equations are applicable to mixtures of ideal gases as well as to pure gases, provided n is taken to be the total number of moles of gas. However, we must consider how the properties of the gas mixture depend upon the composition of the gas mixture and upon the properties of the pure gases. In particular, we must define the Dalton s pressures, the partial pressures, and the Amagat volumes. Dalton s law states that each individual gas in a mixture of ideal gases at a given temperature and volume acts as if it were alone in the same volume and at the same temperature. Thus, from Equation (7.1) we have 

the total pressure of a mixture of ideal gases is the sum of the Dalton pressures. 

The partial pressure of a gas in a gas mixture, whether such a mixture is ideal or real, is defined by 

Amagat s law is very similar to Dalton s law, but deals with the additivity of the volumes of the individual components or species of the gas mixture when mixed at constant temperature and pressure. The Amagat volume may be defined as the volume that n, moles of the pure /th gas occupies when 

This law is closely obeyed by real gases under conditions where the actual volume of the molecules is small compared with the total volume, and where the molecules exert only a very small attractive force on one another. These conditions are met at very low pressures when the distance apart of the individual molecules is large. TTie value of R is then the same for all gases and in SI units has the value of 8314 J/kmol K. 

When the only external force on a gas is the fluid pressure, the equation of state is  

Any property may be expressed in terms of any three other properties. Considering the dependence of the internal energy on temperature and volume, then  

This relation applies to any fluid. For the particular case of an ideal gas, since Pv = RT/M (equation 2.16)  

Thus the internal energy of an ideal gas is a function of temperature only. The variation of internal energy and enthalpy with temperature will now be calculated. 

We begin by considering the case of the simplest possible substance, the ideal gas. 

Equations of state are commonly described as equations which relate the P, V and r of a substance. The simplest example is the ideal gas equation. 

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. 

We can show this analytically without much effort too. In Equation (5.42) we indicated that the effect of pressure on the enthalpy of an ideal gas is zero. That this is also true for the effect of pressure on internal energy, we note that 

In the gaseous phase, we can usually think about gas molecules being on average very far apart from each other and moving very quickly. The behavior of a gas can be described using the ideal gas model. In an ideal gas, we assume that 

The molecules (or atoms) are very small point masses. 

They do not experience any intermolecular forces (apart from behaving like hard spheres). 

FIGURE 1.4 The Maxwell-Boltzmann distribution for different temperature gases. At three different temperatures, the plot represents the probability of a gas molecule in the system having a particular speed. 

This model works very well for gases that are far from a phase transition and not under high pressure. That is, for the model to work, we make sure that the molecules are not, on average, sufficiently close to each other to experience any significant intermolecular forces, or that their physical size needs to be taken into account when describing their behavior. 

Taking into account that the (average) kinetic energy of the enclosed gas particles is proportional to the absolute temperature, T, leads to the equation known as the ideal gas law 

Since the ideal gas state does not depend on the chemical nature of the gas particles, Eq. (3.1.1) applies not only to pure gases but also to mixtures of gases. Each component i contributes with its partial pressure pt to the total pressure p of the gas mixture 

The correlation between the average kinetic energy of the molecules and the absolute temperature is contained by the expression 

The impinging particles may either remain, that is, stick, on the surface, or be reflected back into the gas. The surface coverage, 9, is a relative measure of how many particles NAd remain adsorbed on the surface. 9 can either be given in relationship to the number Ng of atoms per cm of the bare surface (for most solid surfaces. Ns 10 cm ), that is, the substrate, 0s = Nm/Ns, or in relationship to the maximal possible number NAd,max of adsorbed particles in direct contact with the substrate, that is, a full monolayer of adsorbed particles, 0ml = NAd/NAd,max-The difference between both descriptions obviously corresponds to the specific adsorbate configuration, that is, to how many surface atoms on average one adsorbate particle is bound. 

The sticking probability or the so-caUed sticking coefficient, that is, the fraction s of the impinging particles that stick, can be a very complicated function of a number of parameters, such as the kinetic energy of the impinging particles the number of already adsorbed particles, that is, the precoverage the surface temperature the 

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Unfortunately, the ideal-gas assumption can sometimes lead to serious error. While errors in the Lewis rule are often less, that rule has inherent in it the problem of evaluating the fugacity of a fictitious substance since at least one of the condensable components cannot, in general, exist as pure vapor at the temperature and pressure of the mixture. 

The computation of pure-component and mixture enthalpies is implemented by FORTRAN IV subroutine ENTH, which evaluates the liquid- or vapor-phase molar enthalpy for a system of up to 20 components at specified temperature, pressure, and composition. The enthalpies calculated are in J/mol referred to the ideal gas at 300°K. Liquid enthalpies can be determined either with... 

Appendix C-3 gives constants for the ideal-gas, heat-capacity equation... 

The properties of real gases and liquids under pressure are calculated by adding a pressure correction to the properties determined for the ideal gas or the saturated liquid. 

Next the properties of each component must be determined at the temperature being considered in the ideal gas state and, if possible, in the saturated liquid state. 

Generally the properties of mixtures in the ideal gas state and saturated liquids are calculated by weighting the properties of components at the same temperature and in the same state. Weighting in these cases is most often linear with respect to composition ( ), ... 

Properties of mixtures as a real gas or as a liquid under pressure are determined starting from the properties of mixtures in the ideal gas state or saturated liquid after applying a pressure correction determined as a function of a property or a variable depending on pressure )... 

The other method is to employ the principle of corresponding states and calculate the Cp/ of the mixture in the liquid phase starting from the mixture in the ideal gas state and applying an appropriate correction ... 

Viscosity can be expressed as a function of reduced density to which the viscosity of the ideal gas must by added. We will use the formulation proposed by Dean and Stiel in 1965 ... 

This relation is easily transformed to express the ideal gas density ... 

The specific heat of gases at constant pressure is calculated using the principle of corresponding states. The for a mixture in the gaseous state is equal to the sum of the C g of the ideal gas and a pressure correction term ... 

The enthalpy of pure hydrocarbons In the ideal gas state has been fitted to a fifth order polynomial equation of temperature. The corresponding is a polynomial of the fourth order ... 

Hgp = specific enthalpy of the ideal gas Cpgp = specific heat of the ideal gas... 

For petroleum fractions, there is a problem of coherence between the expression for liquid enthalpy and that of an ideal gas. When the reduced temperature is greater than 0.8, the liquid enthalpy is calculated starting with the enthalpy of the ideal gas. On the contrary, when the reduced temperature is less than 0.8, it is preferable to calculate the enthalpy of the ideal gas starting with the enthalpy of the liquid (... 

Hgp = enthalpie of the component i in the ideal gas state Xw = weight fraction of the component i... 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

It is necessary to determine first the properties of each component in the ideal gas state, next to weight these values in order to obtain the property of the mixture in the ideal gas state. 

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