Ideal gas problem

Volume does not need such a lengthy discussion. The volume of a substance is simply the amount of space that it occupies. There are numerous units of measure that can describe volume, including cubic centimeters (cm3), cubic meters (m3), milliliters (mL), and liters (L). One cm3 is equal to 1 mL. Ideal gas problems need to be solved in liters, since the ideal gas constant R uses liters as a unit (more on this later). 

You might have noticed that torr and mm Hg have the same value. On the AP test, units of pressure are usually expressed as atm in the ideal gas problems (R uses atm as the pressure unit) and mm Hg in partial pressure problems. 

Madden SP, Jones LL (2002) Multiple representations and their role in solving ideal gas problems, presented at the Annual Meeting of the National Association for Research in Science Teaching. April New Orleans LA. Forwarded electronically by one of the authors (LL Jones) November 2005. 

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. 

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 (... 

So far, so good. The situation is really no different, say, than the ideal gas law, in which the gas constant is numerically different and has different units depending on the units chosen for p and V, The unit change in Example 10.1 is analogous to changing the gas constant from liter-atmospheres to calories it is apparent that one system is physically more meaningful than another in specific problems. Several considerations interfere with this straightforward parallel, however, and cause confusion ... 

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 ... 

Click Coached Problems for a self-study module and a simulation on the ideal gas law. 

The ideal gas law can be used to solve a variety of problems. We will show how you can use it to find—... 

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. 

Similar two-point equations can be derived from the ideal gas law to solve any problem of this type. 

Only one way of working gas-law problems, using the ideal gas law in all cases (Chapter 5). 

Data summarized in Tables 10.1 to 10.3 can be used to solve the exercises and problems given in this chapter. Unless specifically stated otherwise, the rigid rotator and harmonic oscillator approximations (and hence. Table 10.4) and the assumption of ideal gas can be used. 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler s method, V is determined at each time step... 

For gas-phase reactions, the molar density is more useful than the mass density. Determining the equation of state for a nonideal gas mixture can be a difficult problem in thermod5mamics. For illustrative purposes and for a great many industrial problems, the ideal gas law is sufficient. Here it is given in a form suitable for flow reactors ... 

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

Solution The obvious way to solve this problem is to choose a pressure, calculate Oq using the ideal gas law, and then conduct a batch reaction at constant T and P. Equation (7.38) gives the reaction rate. Any reasonable values for n and kfCm. be used. Since there is a change in the number of moles upon reaction, a variable-volume reactor is needed. A straightforward but messy approach uses the methodology of Section 2.6 and solves component balances in terms of the number of moles, Na, Nb, and Nc-... 

Solution Ideal gas behavior is a reasonable approximation for the feed stream. The inlet concentrations are 287mol/m of methane and 15mol/m of carbon dioxide. The column pressure drop is mainly due to the liquid head on the trays and will be negligible compared with 8 atm unless there are an enormous number of trays. Thus, the gas flow rate F will be approximately constant for the column as a whole. With fast reaction and a controlling gas-side resistance, c = 0. The gas-phase balance gives everything that is necessary to solve the problem ... 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

Solving quantitative problems about gases requires only one equation, the ideal gas equation. 

The problem asks us to determine how the gas Is distributed between the two tanks. This Is a gas problem, so we use the Ideal gas equation. 

We can use the ideal gas equation to calculate the molar mass. Then we can use the molar mass to identify the correct molecular formula among a group of possible candidates, knowing that the products must contain the same elements as the reactants. The problem involves a chemical reaction, so we must make a connection between the gas measurements and the chemistry that takes place. Because the reactants and one product are known, we can write a partial equation that describes the chemical reaction CaC2(. ) +H2 0(/) Gas -I- OH" ((2 q) In any chemical reaction, atoms must be conserved, so the gas molecules can contain only H, O, C, and/or Ca atoms. To determine the chemical formula of the gas, we must find the combination of these elements that gives the observed molar mass. 

We have a mixture of two gases in a container whose volume and temperature are known. The problem asks for pressures and mole tractions. Because molecular interactions are negligible, each gas can be described independently by the ideal gas equation. As usual, we need molar amounts for the calculations. 

In any stoichiometry problem, work with moles. This problem involves gases, so use the ideal gas equation to convert P-V-T information into moles. 

Any of the types of problems discussed in Chapters 3 and 4 can involve gases. The strategy for doing stoichiometric calculations is the same whether the species involved are solids, liquids, or gases. In this chapter, we add the ideal gas equation to our equations for converting measured quantities into moles. Example is a limiting reactant problem that involves a gas. 

One problem remains. The reference enthalpy must be defined at temperature T and pressure P0. The reference state for enthalpy can be taken as an ideal gas. At zero pressure, fluids are in their ideal gaseous state and the enthalpy is independent of pressure. The ideal gas enthalpy can be calculated from ideal gas heat capacity data3 ... 

Solution The ideal gas equilibrium constants can be corrected for real gas behavior by multiplying the ideal gas equilibrium constant by K,f as defined by Equation 6.23, which for this problem is ... 

In the simplest ideal gas law problems, values for three of the four variables are given, and you are asked to calculate the value of the fourth. As usual with the gas laws, the temperature must be given as an absolute temperature, in kelvins. The units of P and V are most conveniently given in atmospheres and liters because the units of R with the value given above are in terms of these units. If other units are given for pressure or volume, convert them to atmospheres and liters, respectively. 

Ans. Ideal gas law problems involve moles. If the number of moles of gas is given or asked for, or if a quantity that involves moles is given or asked for, the problem is most likely an ideal gas law problem. Thus, any problem involving masses of gas (which can be converted to moles of gas) or molecular weights (grams per mole) or numbers of individual molecules (which can be converted to moles) and so forth is an ideal gas law problem. Problems that involve an unchanging mass of gas are most likely not ideal gas law problems. 

Which temperature scale must be used in (a) Charles law problems (h) ideal gas law problems (e) combined gas law problems (<7) Boyle s law problems ... 

Consider a problem, in which we are interested in the mass-dependent partitioning of a solute between an ideal gas phase and a condensed phase. The ratio of the densities in the two phases or the partition coefficient for species a is then... 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

In 1873, J. D. van der Waals recognized deficiencies in the ideal gas equation and developed an equation to eliminate two problems. First, the volume of the container is not the actual volume available to the molecules of the gas because the molecules themselves occupy some volume. The first correction to the ideal gas equation was to subtract the volume of the molecules from V, the volume of the container, to give the net volume accessible to the molecules. When modified to include the number of moles, n, the corrected volume is (V - nb) where b is a constant that depends on the type of molecule. 

Note However, it is standard procedure to solve this problem using the variation of the ideal gas law. See Example 12-12. 

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

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