Ideal gases model

The number of particles in dV is ndV, where n is the number density. Let v be the collision frequency so that the number of colliding particles emerging from all directions from dV in time dt will be nvdVdt. (Actually the number of collisions per unit time will be 1/2 nvdVdt, but each collision produces two particle paths.) If the collisions are random. 

Note that the mean free path A is just the average velocity v times the average time between collisions, which is just the reciprocal of the collision frequency v. Thus the above result is identical to the well-known equation for the number of particles incident on a surface per unit time, which is given by 1 /4 i . 

However, the reason for going through all this trouble is that we need the average height above the surface dA at which the molecules make their last collision. To get this, we put h = rcos B into the above integral and normalize by dividing by the integral without the rcos B. 

Now assume there is a uniform vertical temperature gradient, dT/dy in the system. Molecules crossing the surface from above will have a temperature given by T - - h) dT/dy, while molecules crossing the surface from below will have temperature given by T — h)dT/dy. Therefore, the net heat transferred to surface dA in time df will be given by 

This important result should be committed to memory for it applies not only to the thermal conductivity of gases, but also to thermal transport from phonons and, with some modification, to thermal transport from electrons. 

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Many gases are mixtures of two or more species. The atmosphere, with its mixture of nitrogen, oxygen, and various trace gases, is an obvious example. Another ex-ample is the gas used by deep-sea divers, which contains a mixture of helium and oxygen. The ideal gas model provides guidance as to how we describe mixtures of gases. 

Suppose we pump 4.0 mol of helium into a deep-sea diver s tank. If we pump in another 4.0 mol of He, the container now contains 8.0 mol of gas. The pressure can be calculated using the ideal gas equation, with n = 4.0-1-4.0 = 8.0 mol. Now suppose that we pump in 4.0 mol of molecular oxygen. Now the container holds a total of 12.0 mol of gas. According to the ideal gas model, it does not matter whether we add the same gas or a different gas. Because all molecules in a sample of an ideal gas behave independently, the pressure increases in proportion to the increase in the total number of moles of gas. Thus, we can calculate the total pressure from the ideal gas equation, using n — 8.0 + 4.0 = 12.0 mol. 

Given that every gas deviates from ideai behavior, can we use the ideal gas model to discuss the properties of real gases The answer is yes, as iong as conditions do not become too extreme. The gases with which chemists usuaiiy work, such as chiorine, heiium, and nitrogen, are nearly ideal at room temperature at pressures below about 10 atm. 

Should we regard 407.1 2.0 kJ mol-1 as the final value for the enthalpy of reaction 2.13 under the experimental conditions Recall that the starting assumption was p = 1 bar and the standard state conditions refer to the ideal gases at that pressure or to the real gases at zero pressure. The ideal gas model (or the ideal gas equation) describes very well the behavior of most gases at 1 bar, so it is... 

The vapor pressure against temperature data obtained with a Knudsen cell set-up are handled as already described for a low boiling temperature liquid. The main difference stems from the very low pressure of the vapor in equilibrium with the solid, which justifies the adoption of the ideal gas model in this case. ASub ° at the mean temperature can then be derived from equation 2.40 (with Z = 1) and the correction to 298.15 K can be made with an equation similar to 2.41. 

In other words, Aiatf/°(LiOCH3) is the energy required to destroy the crystalline network, yielding the gas-phase ions. As the cation and the anion are infinitely separated, the gas phase can be described by the ideal gas model and the enthalpy corresponding to the same process is given by... 

The equilibrium pressure of hydrogen in the experiments by Bercaw and co-workers was rather small, allowing the use of the ideal gas model in equation 14.14. Although that pressure was not reported by the authors, the conclusion is obvious from the very low values of Kc shown in figure 14.3. Higher hydrogen pressures, even as low as 1 bar, would lead to the formation of Sc(Cp )2H with nearly 100% yield, and thus equilibrium 14.12 could not be examined. 

The ideal gas model cannot be used at high pressures. Under these conditions, as pointed out in section 2.9, we have to deal with fugacities. Neglecting this and other correction parameters may lead to large errors. The point can be illustrated with results obtained by Oldani and Bor, who studied equilibrium 14.21 in hexane over the temperature range of -21.5 to 19.6°C and under a CO pressure of 198 bar [317],... 

To determine the amount of substance and the concentration of carbon monoxide in solution, we have to relate these quantities to the CO pressure. That can be done as described by using Henry s law. The only (important) difference is that now the CO pressure is too high to justify use of the ideal gas model. Hence, for the present case, equation 14.15 becomes [316]... 

It is instructive to compare the thermochemical values with those obtained by the same authors when they assumed the ideal gas model (i.c.,/co = pco) and neglected the CO solubility change with temperature ArH%12 = 45.5 kJ mol-1 and ATS 72 = 267 J K-1 mol-1. 

Equation (49) must be substituted for V in eqn. (46) and dV/dt is to be expressed through reaction rates using equations of states under given conditions. For isobaric isothermal conditions and in the case of the applicability of the ideal gas model... 

For an ideal gas modeled as rigid spheres, the mean free path of the molecules, X, can be related to the temperature, T, and pressure, P, via the following equation [2,3]... 

The Ideal Gas Model An ideal gas is a model gas comprising imaginary molecules of zero volume that do not interact. Its PVT behavior is represented by the simplest of equations of state PV = PiT, where i is a universal constant, values of which are given in Table 1-9, The following partial derivatives, all taken at constant composition, are obtained from this equation ... 

The ideal gas model may serve as a reasonable approximation to reality under conditions indicated by Fig. 4-1. 

Ideal Solution Model The ideal gas model is useful as a standard of comparison for real gas behavior. This is formalized through residual properties. The ideal solution is similarly useful as a standard to which real solution behavior may be compared. 

In the fabrication of practical E-O devices, all of the three critical materials issues (large E-O coefficients, high stability, and low optical loss) need to be simultaneously optimized. One of the major problems encountered in optimizing polymeric E-O materials is to efficiently translate the large P values of organic chromophores into large macroscopic electro-optic activity (r33). According to an ideal-gas model, macroscopic optical nonlinearity should scale as (M is the chromo-... 

For low pressures (a few atmospheres and lower) we can apply the ideal gas model for gases and ideal mixture models for liquids. This formulation is very common in reactor technology. In some cases at higher pressures, the pressure effect on the gas phase is important. A suitable model for these systems is to use an EOS for the gas phase, and an ideal mixture model for liquids. However, in most situations at low pressures the liquid phase is more non-ideal than the gas phases. Then we will rather apply the ideal gas law for the gas phase, and excess properties for liquid mixtures. For polar mixtures at low to moderate pressures we may apply a suitable EOS for gas phases, and excess properties for liquid mixtures. All common models for excess properties are independent of pressure, and cannot be used at higher pressures. The pressure effect on the ideal (model part of the) mixture can be taken into account by the well known Poynting factor. At very high pressures we may apply proper EOS formulations for both gas and liquid mixtures, as the EOS formulations in principle are valid for all pressures. For non-volatile electrol3d es, we have to apply a suitable EOS for gas phases and excess properties for liquid mixtures. For such liquid systems a separate term is often added in the basic model to account for the effects of ions. For very dilute solutions the Debye-Htickel law may hold. For many electrolyte systems we can apply the ideal gas law for the gas phase, as the accuracy reflected by the liquid phase models is low. 

Equation [2] and subsequent equations are for ideal-gas models. The usual methods of correcting for real-gas behavior may be employed to evaluate the available energy as given by equation [1]. 

When we add these assumptions to our model for gases, we call it the ideal gas model. As the name implies, the ideal gas model describes an ideal of gas behavior that is only approximated by reality. Nevertheless, the model succeeds in explaining and predicting the behavior of typical gases under typical conditions. In fact, some actual gases do behave very much in accordance with the model, and scientists may call them ideal gases. The ideal gas assumptions make it easier for chemists to describe the relationships between the properties of gases and allow us to calculate values for these properties. 

Because the ideal gas equation applies to all ideal gases, the molar volume at STP applies to all gases that exhibit the characteristics of the ideal gas model. In equation stoichiometry, the molar volume at STP is used in much the way we use molar mass. Molar mass converts between moles and the measurable property of mass molar volume at STP converts between moles and the measurable property of volume of gas. Note that while every substance has a different molar mass, all ideal gases have the same molar volume at STP. Example 13.5 provides a demonstration. 

Ideal gas model The model for gases that assumes (1) the particles are point-masses (they have mass but no volume) and (2) there are no attractive or repulsive forces between the particles. 

Ideal gas A gas for which the ideal gas model is a good description. 

The ideal gas model is used to predict changes in four related gas properties ... 

Gases exhibit more ideal behavior at low pressures. At low pressures, gas particles are more widely separated and therefore the attractive forces between particles are less. The ideal gas model assumes negligible attractive forces between gas particles. 

The price paid for this simplification is that certain physical features of the porous medium are lost. One of the more important issues is that the model is no longer dealing with channels of finite size corresponding to the pores in the real medium. Thus, there is no introduction of the hydrodynamic fluxes in formal development of the equations. These have to be added empirically. The working form of the dusty ideal gas model equations is (Higler et al., 2000)... 

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