Solved Problems Ideal Gases

In this section, we have intentionally omitted mass balance problems. As you will see in Chaps. 7 and 8, this topic is treated in detail and with an emphasis on being quantitative. 

This problem involves several concepts beyond the ideal gas law. First, the mixmre of gases will be treated as an ideal gas. Second, remember that a % / composition is equal to a % mol/mol composition. Finally, to answer questions (a) and (c) we will assume that Dalton s and Amagat s laws are valid. 

Fn2 partial pressure of N2 Vn2 partial volume of N2 / m density of mixture 

We have three unknowns, and so we need three equations. The equations are Dalton s equation to estimate the partial pressure of N2, Amagat s law to estimate the partial volume of N2, and, finally, the ideal gas law to estimate the density of the mixture. 

Dalton s law indicates (6.11) that the partial pressure of component i is equal to the product of the molar fraction of component i and the total pressure. Therefore, 

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However, suppose that you have forgotten the convenient formula for the density of an ideal gas You can still solve a problem such as this one. The density of 1.00 g/L indicates a mass of 1.00 g of gas in a 1.00-L volume. Of course, the mass of a gas, given its identity (oxygen in this case) enables us to determine the amount in moles of the gas (n in the ideal gas equation). Then, we can solve the ideal gas equation for the desired property, such as temperature, as follows ... 

Now the problem has become, What is the volume of 0.653 mol H2, measured at 739 torr and 21°C This is just like Example 14.3. Solve the ideal gas equation for... 

Your ability to solve the gas problems in this chapter depends largely on your algebra skills. Most students find it easiest to determine what is wanted and then solve the ideal gas equation for that variable. If the wanted quantity is density or molar volume, solve the equation for the combination of variables that represents the desired property. Then substitute the known variables, including units, and calculate the answer. Units are important If they don t come out right, you know there is an error in the algebra. 

Be sure to match the units of the known quantities and the units of R. In this book, you will be using R = 0.0821 L atm/(mol K). Your first step in solving any ideal gas law problem should be to check the known values to be sure you are working with the correct units. If necessary, you must convert volumes to liters, pressures to atmospheres, temperatures to kelvins, and masses to numbers of moles before using the ideal gas law. 

SOLVE To solve the problem, first solve the ideal gas law for V. 

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

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

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... 

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 ... 

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

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

You will need to use the ideal gas law to solve the problem PV = nRT. Because moles can be calculated by dividing the mass of the sample by its molecular weight, the ideal gas law becomes... 

D) Let 1 represent the initial state and 2 represent the final state. The nnmber of moles of gas remains constant in this problem. Use the combined form of the ideal gas law to solve for the pressnre change. 

When solving gas law problems using the combined gas law, the pressure and volume units do not have to be as indicated by the authors of the laws and by the ideal gas law they don t even have to be in the metric system. However, temperature must be in the Kelvin scale. Explain. 

C) The problem states that 135 ml of an ideal gas are supplied to the stove each second. The pressure and temperature are also known. The ideal gas equation, PV = nRT, may be rearranged to solve for the number of moles of gas flowing in a second ... 

Gas Theory, Problem Solving, Applications, Absolute Zero, and the Ideal Gas Law... 

Solve the following problems dealing with the ideal gas law. 

Answer Using the ideal gas equation, we know that PV = nRT. We also know that when we solve for P, our answer should come out to be in units of atmospheres (atm). When we set up and solve the problem, all units should cancel to give us atm. 

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). 

A) This is a laboratory procedure that you are expected to have performed and know for the AP test. While setting up this calculation, we will actually be answering some of the following questions. You will use the ideal gas equation to solve the problem. Therefore, you will need to know the pressure, volume, and temperature of the gas. The pressure of the gas will equal the total pressure of the mixture minus the vapor pressure of water (which is present in the gas collection tube, along with the hydrogen). The total pressure of the gas is equal to the atmospheric pressure in the room. Therefore, the pressure of the hydrogen gas is equal to 745 mm Hg 19.4 mm Hg = 725.6 mm Hg (726 mm Hg). Remember, the ideal gas equation requires atmospheres, so this will need to be converted to atmospheres ... 

The ideal gas law gives great flexibility for solving many different types of problems. After the investigation, you will find another Sample Problem. It illustrates how you can use the ideal gas law with methods you have previously learned, to identify an unknown gas. Practice problems are located at the end of this Sample Problem. 

You have already learned that the ideal gas law can be used to solve for different variables in several different types of situations. As you may recall, the term stoichiometry" refers to the relationship between the number of moles of the reactants and the number of moles of the products in a chemical reaction. In this section, you will learn how to use Gay-Lussac s law of combining volumes and the ideal gas law to solve stoichiometric problems that involve gases. 

Earlier in this course, you learned how to do stoichiometry calculations. To solve gas stoichiometry problems, you will incorporate the ideal gas law into what you learned previously. The following steps will help you do this. 

The ideal gas equation, PV = nRT is needed to solve this problem. Solving for moles, n = PV / RT. Remember that the value for T must be in Kelvin. 

This problem requires the use of the ideal gas equation, PV = nRT. Rearranging to solve for V gives ... 

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