Ideal gas relationships

As a first exarhple, we will consider the reversible compression of an ideal gas at constant pressure. Suppose that we have 1 mol of an ideal gas confined in a cylinder with a piston, at a pressure of Pu a volume of and a temperature (in kelvins) of 

Pressure-volume isotherms for an ideal gas at a lower temperature T2 and a higher temperature A reversible compression occurs at constant pressure from state A to state B. 

Our problem is to calculate how much work is done on the system during this process, how much heat is lost, and what are the changes in energy and enthalpy. 

Note that as far as work and heat are concerned there would have been no definite answdr to this question unless we had specified the path taken, since work and heat are not state functions. In this case we have specified that the compression is reversible and is occurring at constant pressureJThe work done on the system is 

The heat absorbed by the system during the process A B is given by 

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Initial velocities needed to kickstart the simulation is taken as a random distribution. Temperature is calculated from atomic velocities using the ideal-gas relationship... 

Solution From the general identity (3.52) and the ideal gas relationship (3.58), we obtain... 

Eq. (22) can be recast using vapor phase pressure as the driving force behind the mass transfer, using the ideal gas relationship. 

It will normally be sufficient to use the ideal-gas relationship for the partial pressure function, fj, for component j on plate i ... 

Assume that, to a first approximation, the vapor of the organic material analyzed in Problem 1.1 can be considered an ideal gas that is, it obeys the ideal gas relationship PV = nRT. If the pressure on the day the experiment was run was 728torr and a volume of air corresponding to 15.2cc was displaced from the hot tube at 373 K (but collected at 300 K) by introducing and vaporizing 34.3 mg of the material from Problem 1.1, what must be its molecular weight (R = 0.083Latmmor K )... 

Substituting the ideal gas relationship into this last expression, we can also obtain the work requirement in terms of the pressure change ... 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

The pressure and the density of a gas are related by an equation of state. If the maximum pressure permitted within the centrifuge bowl is not too high, the equation of state for an ideal gas will suffice. The relationship between the pressure and density of an ideal gas is given by the weU-known equation ... 

To understand the flow in turbomachines, an understanding of the basic relationships of pressure, temperature, and type of flow must be acquired. Ideal flow in turbomachines exists when there is no transfer of heat between the gas and its surroundings, and the entropy of the gas remains unchanged. This type of flow is characterized as a rever.sible adiabatic flow. To describe this flow, the total and static conditions of pressure, temperature, and the concept of an ideal gas must be understood. 

The traditional unipolar diffusion charging model is based on the kinetic theory of gases i.e., ions are assumed to behave as an ideal gas, the properties of which can described by the kinetic gas theory. According to this theory, the particle-charging rate is a function of the square of the particle size dp, particle charge numbers and mean thermal velocity of tons c,. The relationship between particle charge and time according White s... 

Many process components do not conform to the ideal gas laws for pressure, volume and temperature relationships. Therefore, when ideal concepts are applied by calculation, erroneous results are obtained—some not serious when the deviation from ideal is not significant, but some can be quite serious. Therefore, when data are available to confirm the ideality or non-ideality of a system, then the choice of approach is much more straightforward and can proceed with a high degree of confidence. 

The solution of the work compression part of the compressor selection problem is quite accurate and easy when a pressure-enthalpy or Mollier diagram of the gas is available (see Figures 12-24A-H). These charts present the actual relationship of the gas properties under all conditions of the diagram and recognize the deviation from the ideal gas laws. In the range in which compressibility of the gas becomes significant, the use of the charts is most helpful and convenient. Because this information is not available for many gas mixtures, it is limited to those rather common or perhaps extremely important gases (or mixtures) where this information has been prepared in chart form. The procedure is as follows ... 

The ideal gas law is readily applied to problems of this type. A relationship between the variables involved is derived from this law. In this case, pressure and temperature change, while n and V remain constant. 

The law of combining volumes, like so many relationships involving gases, is readily explained by the ideal gas law. At constant temperature and pressure, volume is directly proportional to number of moles (V = kin). It follows that for gaseous species involved in reactions, the volume ratio must be the same as the mole ratio given by the coefficients of the balanced equation. 

The Absolute Temperature Scale The absolute temperature scale is based on the (p, V, T) relationships for an ideal gas as given by equation (1.7)... 

If a relationship is known between the pressure and volume of the fluid, the work can be calculated. For example, if the fluid is the ideal gas, then pV = nRT and equation (2.14) for the isothermal reversible expansion of ideal gas becomes... 

Thus, for the ideal gas the molar heat capacity at constant pressure is greater than the molar heat capacity at constant volume by the gas constant R. In Chapter 3 we will derive a more general relationship between Cp m and CV m that applies to all gases, liquids, and solids. 

In an adiabatic expansion or compression, the system is thermally isolated from the surroundings so that q = 0. If the change is reversible, we can derive a general relationship between p, V, and T, that can then be applied to a fluid (such as an ideal gas) by knowing the equation of state relating p, V, and T. 

We used the system (.vic-Q,H 1CH3 +. vic-CeH ) as an example of a system that closely approximates ideal behavior. Figure 6.5 showed the linear relationship between vapor pressure and mole fraction for this system. In this Figure, vapor pressure could be substituted for vapor fugacity, since at the low pressure involved, the approximation of ideal gas behavior is a good one, and... 

Alternative forms of the equilibrium constant can be obtained as we express the relationship between activities, and pressures or concentrations. For example, for a gas phase reaction, the standard state we almost always choose is the ideal gas at a pressure of 1 bar (or 105 Pa). Thus... 

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... 

Stirling s approximation is a mathematical relationship that becomes more accurate as N becomes larger, and becomes very precise for the large number of units we will consider as we work with moles of ideal gas. Substituting equations (10.17) and (10.18) into equation (10.16) gives... 

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