Gas Density and Molar Volume

This time pressure is given in torr, so you use 62.4 L torr/mol K for R. Molar mass is the Wanted quantity, and it is the only unknown in the PV = mRT/MM form of the ideal gas equation (Equation 14.8). Complete the example. 

At a time at which the temperature is 22°C and the pressure is 0.988 atm, the air in a 0.50-liter container is found to have a mass of 0.59 gram. Find the effective molar mass of air—the mass per mole of mixed molecules in the container. 

With oxygen at 32 g/mol and nitrogen at 28 g/mol, we would expect a mixture to have a value between them. Nitrogen is present in the larger amount, so we would expect the in-between value to be closer to 28 than 32. The result, 29 g/mol, is reasonable. 

3 Calculate the density of a known gas at any specified temperature and pressure. 

4 Given the density of a pure gas at speeified temperature and pressure, or information from which it may be found, calculate the molar mass of that gas. 

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Table 11.12 shows the densities and molar volumes for several common gases. Since the volume and, therefore, the density of a gas changes significantly with changes in temperature and pressure, all these values were measured for gases at 25 °C and 100 kPa atmosphere pressure. 

The mathematical relationship between pressure, volume, temperature, and number of moles of a gas at equilibrium is given by its equation of state. The most well-known equation of state is the ideal gas law, PV=RT, where P = the pressure of the gas, V = its molar volume (V/n), n = the number of moles of gas, R = the ideal gas constant, and T = the temperature of the gas. Many modifications of the ideal gas equation of state have been proposed so that the equation can fit P-V-T data of real gases. One of these equations is called the virial equation of state which accounts for nonideality by utilizing a power series in p, the density. 

Still other applications of the ideal gas law make it possible to calculate such properties as density and molar mass. Densities are calculated by weighing a known volume of a gas at a known temperature and pressure, as shown in Figure 9.10. Using the ideal gas law to find the volume at STP and then dividing the measured mass by the volume gives the density at STP. Worked Example 9.7 gives a sample calculation. 

In this section, you will learn more about two properties of a gas that are closely related to molar volume density and molar mass. You have already encountered these properties, but now you will use them to help you with your gas calculations. 

The molar mass of a gas refers to the mass (in g) of one mole of the gas. You can calculate molar mass by adding the masses of atoms in the periodic table. You can also calculate molar mass by dividing the mass of a sample by the number of moles that are present. Molar mass is always expressed in the units g/mol. Table 12.2 summarizes molar volume, density, and molar mass. 

Gases Gases/vapors are compressible and their densities are strong functions of both temperature and pressure. Equations of state (EoS) are commonly used to correlate molar densities or molar volumes. The most accurate EoS are those developed for specific fluids with parameters regressed from all available data for that fluid. Super EoS are available (or some of the most industrially important gases and may contain 50 or more constants specific to that chemical. Different predictive methods may be used for gas densities depending upon the conditions ... 

Plan We need to use the mass information given to calculate the volume of the container and the mass of the gas in it From this we calculate the gas density and then apply Equation 10.11 to calculate the molar mass of the gas. 

Properties near the critical point are quite different compared to states at lower temperatures and pressures. As the difference between vapor and liquid becomes less clear near the critical point, the liquid becomes substantially more compressible than typical liquids. This is indicated on the PVgraph by the gentle slope of the isotherm as it approaches the critical point. Isotherms below but near the critical temperature (not shown in Figure 2-2) show similar behavior. The usual approximation that treats liquids as incompressible is acceptable only at temperatures well below the critical. In the supercritical region, the behavior of a fluid is somewhere between that of a liquid and a gas. The gentle slope of the isotherms indicates that the fluid is quite compressible, even at high, liquid-like densities Qow molar volumes). 

Another frequent mistake among students is to try to apply the ideal gas law to calculate the concentrations of species in condensed-matter phases (e.g., liquid or solid phases). Do not make this mistake] The ideal gas law only applies to gases. To calculate concentrations for liquid or solid species, information about the density (pj) of the liquid or solid phase is required. Both mass densities and molar densities (concentrations) as well as molar and atomic volumes may be of interest. The complexity of calculating these quantities tends to increase with the complexity of the material under consideration. In this section, we will consider three levels of increasing complexity pure materials, simple compounds or dilute solutions, and more complex materials involving mixtures of multiple phases/compounds. 

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. 

Applications of the Ideal Gas Law Molar Volume, Density, and Molar Mass of a Gas... 

We just examined how we can use the ideal gas law to calculate one of the variables (P, V,T, or n) given the other three. We now turn to three other applications of the ideal gas law molar volume, density, and molar mass. 

Figure 8.4 Graph of temperature against molar volume (a), and density (b). for CO (gas) and C02 (liquid) in the temperature range from the triple point to the critical point. The dashed line in (b) is the average density. The area enclosed within the curves is a two-phase region, with the molar volume or the density of the gas and liquid at a particular temperature given by the horizontal (dotted) tie-lines connecting the gas and liquid sides of the curve.

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