Calculation of As for an Ideal Gas

Finally, rearranging Equation (E3.6A) and plugging in values gives  

The entropy increases, as we would expect for this spontaneous process. 

The change in internal energy for an ideal gas is, by definition, zero thus, the differential energy balance is  

Applying this expression for Sq ev to the definition of entropy. Equation (3.2), gives  

we want to solve for the reversible heat transfer, q ev, and use it in the definition of entropy. Applying the first law gives  

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The Ideal-gas equation, PV = iiRT, is the equation of state for an Ideal gas. The term R in this equation is the gas constant We can use the ideal-gas equation to calculate variations in one variable when one or more of the others are changed. Most gases at pressures less than 10 atm and temperatures near 273 K and above obey the ideal-gas equation reasonably well The conditions of 273 K (0 °C) and I atm are known as the standard temperature and pressure (STP). In all applications of the ideal-gas equation we must remember to convert temperatures to the absolute-temperature scale (the Kelvin scale). 

Calculation of AS for the Reversible Isothermal Expansion of an Ideal Gas Integration of equation (2.38) gives... 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

A typical use for this model would be to solve for the number of moles of a gas, given its identity, pressure, volume, and temperature. The iterative solver is used for this purpose. You must decide which variable to choose for iteration and what a reasonable initial guess is. Real gases approach ideal behavior at low pressure and moderate temperatures. Since the compressibility factor z is 1 for an ideal gas, and since knowing z along with P, V, and T allows a calculation of n, we choose z as the iteration variable and 1.0 as the initial guess. 

A noteworthy aspect of Eqs. (10.16) and (10.18) is that they are altogether free of any requirement to carry out an electronic structure calculation. Equation (10.16) is well known for an ideal gas and is entirely independent of the molecule in question, and Eq. (10.18) can be computed trivially as soon as the molecular weight is specified. Note, however, that the units chosen for the various quantities must be such that the argument of the logarithm in Eq. (10.18) (i.e., the partition function), is unitless. 

The f, fg, f , and f,are determined for the pure gas at the pressure of the mixture and depend on the pressure and the temperature. In gaseous mixtures, the quantity Kp as defined by Equation 2-38 is used. For an ideal gas reaction mixture, Kf = Kp. For a non-ideal system, Equation 2-39 can be used to calculate Kp from the measured equilibrium compositions Ky using Equation 2-42. The composition... 

It is possible, however, to determine a relationship exists between the fluid pressure and its density, thus allowing indirect measurement and control of the density. For an ideal gas, the relationship is simply, PV/RT= 1, where Fis the molar volume (reciprocal of the molar density). From the molecular weight of the gas (M), the mass density, p, can be calculated as M/V. The simple equation breaks down at the high densities characteristic of supercritical fluids, but, the work of Pitzer et allows the ideal gas law to be extended... 

It should be noted that the solvation energy is defined as the energy of transfer of a solute from vacuum, or an ideal gas phase, to a solvent [24]. Therefore, if the dielectric constant of the solute is different from 1 (the dielectric constant of vacuum or an ideal gas phase), two calculations are needed for the estimation of the solvation energy, one to find the energy of the solute in vacuum and another to find the energy of the solute in the solvent ... 

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

In gas systems, the flow becomes choked when the exit velocity through the orifice plate reaches sonic velocity. The mass flow rate is essentially independent of downstream conditions but can be increased by increasing the upstream pressure or decreasing the temperature. For an ideal gas and isentropic flow, the pressure ratio to calculate the onset of sonic conditions depends on the ratio of the specific heats, g, and is often known as the isentropic expansion factor ... 

Example 7.2 illustrates the use of Equation 7.7 for calculating the work done in the isothermal, reversible expansion of an ideal gas. Because q - -w for an ideal gas. Equation 7.7 can also be written as follows ... 

For a fixed sample of ideal gases, both U and PV depend only on temperature. As a result, the enthalpy of the gas (calculated using Equation 7.10) will also depend only upon temperature. Like changes in internal energy (Equation 7.5), the change in enthalpy for an ideal gas undergoing an isothermal process is identically zero ... 

Figure 4.9 A family of reversible adiabatic curves (two-dimensional reversible adiabatic surfaces) for an ideal gas with V and T as independent variables. A reversible adiabatic process moves the state of the system along a curve, whereas a reversible process with positive heat moves the state from one curve to another above and to the right. The curves are calculated for n = 1 mol and Cv,m - (3/2)/ . Adjacent curves differ in entropy by 1J K. ...

Many thermodynamic quantities, such as total internal energy, entropy, chemical potential, etc., for an ideal gas have been derived in the preceeding chapters as examples. In this section we will bring all these results together and list the thermodynamic properties of gases in the ideal gas approximation. In the following section we will see how these quantities can be calculated for real gases, for which we take into account molecular size and molecular forces. 

For an ideal gas, the partial pressure of species i is given by pt = yi P, where y,- is the mole fraction and P is the total pressure. Finally, the mole fractions of CO, CI2, and COCI2 can be written as functions of the quantity of CO that has reacted. This permits the amount of CO reacted, and therefore the factional conversion of CO, to be calculated fiom the equilibrium expression. 

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