THE PERFECT GAS LAW

It was found experimentally by Boyle (1662) that the volume of a given gas varies inversely with absolute pressure (P), and by Charles (1800) that volume varies directly with absolute temperature (T). It was subsequently found that these two results could be combined as follows  

All dimensions in biaekets are expressed in terms of the fundamental set foree (F), length (Z), and temperature (6). 

It was later found that for all gases, R equals a universal constant where  

Equation (11.9) is known as the ideal gas law, and is found to hold over a wide range of conditions. For example, for air Eq. (11.9) is good to an accuracy of 2% for pressures below 3,200 psia (100 atmospheres), and for temperatures from 0°F to 750°K. 

---

The deviation from the perfect gas law is not great at ordinary pressures and temperatures. At the highest pressure normally encountered commercially, 41 MPa (6000 psig), the compressibiUty factor of nitrogen is 1.3629 at 25°C (12). 

The gas usually deviates considerably from the perfect-gas laws, and in many cases temperature or other limitations necessitate a thor-... 

Compressibility of Natural Gas All gases deviate from the perfect gas law at some combinations of temperature and pressure, the extent depending on the gas. This behavior is described by a dimensionless compressibility factor Z that corrects the perfect gas law for real-gas behavior, FV = ZRT. Any consistent units may be used. Z is unity for an ideal gas, but for a real gas, Z has values ranging from less than 1 to greater than 1, depending on temperature and pressure. The compressibihty faclor is described further in Secs. 2 and 4 of this handbook. 

Ideal gas obeys the equation of state PV = MRT or P/p = MRT, where P denotes the pressure, V the volume, p the density, M the mass, T the temperature of the gas, and R the gas constant per unit mass independent of pressure and temperature. In most cases the ideal gas laws are sufficient to describe the flow within 5% of actual conditions. When the perfect gas laws do not apply, the gas compressibility factor Z can be introduced ... 

Charles and Gay-Lussac, working independently, found that gas pressure varied with the absolute temperature. If the volume was maintained constant, the pressure would vary in proportion to the absolute temperature [I j. Using a proportionality constant R, the relationships can be combined to form the equation of state for a perfect gas, otherwi.se known as the perfect gas law. 

The use of a gas mixture presents a two-part problem. If the state of the mixture is such that it may be considered a mixture of perfect gases, classical thermodynamic methods can be applied to determine the state of each gas constituent. If, however, the state of the mixture is such that the mixture and constituents deviate from the perfect gas laws, other methods must be used that recognize this deviation. In any case, it is important that accurate thermodynamic data for the gases are used. 

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

It is convenient to calculate a TNT equivalent of a physical explosion to use the military results of Figures 9.1-4 and 5. Baker et al. (1983) give a recipe for the rupture of a gas filled container assuming expansion occurs isothermally and the perfect gas laws apply (equation 9.1-25), where W is... 

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 ...

Compressibility is expressed as the multiplier for the perfect gas law to account for deviation from the ideal. At a given set of conditions of temperature and pressure ... 

Ideal (or perfect) gas behavior is approached by most vapors and gases in the limit of low pressures and elevated temperatures. Two special forms of restricted utility known as the Boyle s law and the Charles law preceded the development of the perfect gas law. 

The mole is particularly useful when working with gas mixtures. It is based on Avogdro s law that equal volumes of gases at given pressure and temperature (pT) conditions contain equal number of molecules. Since this is so, then the weight of these equal volumes will be proportional to their molecular weights. The volume of one mole at any desired condition can be found by the use of the perfect gas law. 

Volatility is the weight of vapor present in a unit volume of air, under equilibrium conditions, at a specified temperature. It is a measure of how much material (agent) evaporates under given conditions. The volatility depends on vapor pressure. It varies directly with temperature. We express volatility as milligrams of vapor per cubic meter (mg/m3). Calculate it numerically by an equation derived from the perfect gas law. This equation follows ... 

The vapor density pc was expressed in terms of pG and T0 through the perfect gas law ... 

The role of approximate mechanisms in organic chemistry is somewhat like that of the perfect gas laws in physical chemistry. The fact that an approximate mechanism has some value does not of course mean that precise mechanisms are not still better. 

For methane at 25 °C or 298 K, cp = 2.24 J/gK. Note that on substituting for the temperatures in this steady state example it makes no difference whether K or °C units are used. This follows from the conservation of mass. However, for unsteady applications of Equation (3.40), since we have used the perfect gas law in which T is in K, we should be consistent and use it through the equations. When in doubt, use K without error. Substituting ... 

The conditions that apply for the saturated liquid-vapor states can be illustrated with a typical p-v, or (1 /p), diagram for the liquid-vapor phase of a pure substance, as shown in Figure 6.5. The saturated liquid states and vapor states are given by the locus of the f and g curves respectively, with the critical point at the peak. A line of constant temperature T is sketched, and shows that the saturation temperature is a function of pressure only, Tsm (p) or psat(T). In the vapor regime, at near normal atmospheric pressures the perfect gas laws can be used as an acceptable approximation, pv = (R/M)T, where R/M is the specific gas constant for the gas of molecular weight M. Furthermore, for a mixture of perfect gases in equilibrium with the liquid fuel, the following holds for the partial pressure of the fuel vapor in the mixture ... 

The introduction of the perfect gas law to the Clausius-Clapeyron equation (Equation (6.14)) allows us to obtain a more direct approximation to p p(T) in the saturation region. We use the following ... 

Purely phenomenological as well as physically based equations of state are used to represent real gases. The deviation from perfect gas behaviour is often small, and the perfect gas law is a natural choice for the first term in a serial expression of the properties of real gases. The most common representation is the virial equation of state ... 

The equilibrium constant based on concentration (in moles per cubic centimeter) is sometimes used, particularly in chemical kinetic analyses (to be discussed in the next chapter). This constant is found by recalling the perfect gas law, which states that... 

Representing p in terms of the perfect gas law and using the logarithmic form, one obtains... 

The pressure dependence, as before, is derived not only from the perfect gas law for p, but from the density-pressure relationship in Z as well. Also, the effect of the stoichiometry of a reacting gas mixture would be in Z. But the mole fraction terms would be in the logarithm, and therefore have only a mild effect on the induction time. For hydrocarbon-air mixtures, the overall order is approximately 2, so Eq. (7.46) becomes... 

Liquid densities can be assumed constant in many systems unless large changes in composition and temperature occur. Vapor densities usually caimot be considered invariant and some sort of FVr relationship is almost always required. The simplest and most often used is the perfect-gas law ... 

An equation of state for the vapor is needed to be able to calculate density from the pressure or temperature, lowing any one property (T, P, or pj pins down all the other properties since there is only one component, and two pMses are present in the tank. The perfect-gas law is used. 

The equations of state for steam (or the steam tables) can be used to calculate temperature 7 and pressure Pj from density pj. For example, if the perfect-gas law and a simple vapor-pressure equation ean be used,... 

Assume the perfect gas law can be used 8.314 kPa m /kg mol K). Assume the pressure transmitter range is 1800-2000 kPa and that the valve has linear installed characteristics with a maximum flow rate of 2000 kg/h. 

The perfect gas law is also used to relate the assumed spatially constant density to p and Tg ... 

The values of some of these parameters at room temperature and pressure are given in table 4.26. These values are obtained from measurements of viscosity, thermal conductivity, diffusion, and from deviations from the perfect gas law. 

The perfect gas law applies when the gaseous molecules are far apart and have very little interaction with each other. At 0 °C and 1 atm, the air molecules have a an average diameter of 3.6 A and are at an average distance of 33 A from their nearest neighbor. When the density is increased, the molecules draw closer to each other and the ideal... 

This parameter is equal to one when the perfect gas law applies. When the pressure is increased to 10 atm, the predicted perfect gas volume is reduced to 2.24 L and the average distance between the molecules shrinks to 15.5 A when the pressure is increased to 100 atm, the predicted volume is reduced to 0.224 L and the average distance shrinks further to 7.2 A when the pressure is further increased to 1000 atm, the volume predicted by the perfect gas law is 0.0224 L and the average distance shrinks further to 3.3 A. By then, the intermolecular distance is roughly equal to the molecular diameters. As the molecules come together, the attractive forces are first felt, which manifests itself by a drop from the ideal gas volume, which is represented by Z < 1. With further compression, the molecules begin to touch, and the repulsive forces become dominant, which manifests itself as a resistance to further volume reduction and Z > 1. 

The van der Waals equation is not a particularly accurate tool for prediction of compressibility Z, but it is the first theory to illuminate the nature of the attractive and repulsive forces that lead to departure from the perfect gas law. There are many more accurate equations of state that use more parameters, including the Benedict-Webb-Rubin equation, the Redlich-Kwong equation, and the Peng-Robinson equation. The compressibility factor can also be expanded into the virial form... 

One has to design the experiment to take a set of data designed to facilitate the task of parameter extraction. If a set of data is taken under constant volume conditions, and the pressure is plotted against the temperature, then there will be an intercept of —alV and a slope of R/ V — b). The van der Waals equation of state is the simplest of the equations of state beyond the perfect gas law, and the task of extracting parameter values from experimental data for the more complicated equations of state would require more ingenuity. The Redlich-Kwong equation has two parameters, A and B ... 

---