The Ideal Gas Law and Its Applications

Gas Stoichiometry Molar Volume Method Gas Stoichiometry Ideal Gas Equation Method Volume-Volume Gas Stoichiometry 

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In Chapter 4 you learned about several proportionalities among the volume, pressure, and temperature of a fixed amount of gas. In this chapter the amount becomes a fourth variable. The result is a single mathematical relationship, the ideal gas law, that summarizes all the measurable properties of gases. 

These properties of gases were introduced in Section 4.1  

Gases expand to fill their containers uniformly. 

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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. 

Obtain m> expression for the concentration of gas A in that half of she pipe in which it is increasing, as a (unction of distance y from the valve and time t after opening. The whole system is at a constant pressure and the ideal gas law is applicable to both gases. It may be assumed that the rate of mixing in the vessels is high so that the gas concentration at the two ends of the pipe do not change. 

It should be quite obvious that, although the model provided in the form of the ideal gas law does a reasonable job at lower pressures, it rapidly deviates as the pressure increases and the volume decreases. We can see this more clearly in Figure 2.5, where we compare the real data with that derived from the ideal gas law in a scatter plot of p versus 1/v. We can see from our plot that the experimental data, shown as solid circles, are modelled reasonably well by a linear (straight line) function, but only for pressures less than 50 atm. The Boyle model is clearly of limited applicability in this case. 

The ideal-gas law is a simplistic model that is applicable to simple molecules at low pressure and high temperature. As for Kay s method, which in general is superior to the others, it is basically suitable for nonpolar/nonpolar mixtures and some polar/polar mixtures, but not for nonpolar/polar ones. Its average error ranges from about 1 percent at low pressures to 5 percent at high pressures and to as much as 10 percent when near the critical pressure. 

The ideal gas law was derived from experiments in which tlie effects of pressure and temperature on gaseous volumes were measured over a moderate range of temperatures and pressures. As a general rule, tliis law works best when the molecules of the gas are far apart, that is, when tlie pressure is low and tlie temperature is liigli. Under these conditions, the gas is said to behave ideally. For engineering calculations the ideal gas hnv is almost always assumed to be valid, since it generally works well for the temperature and pressure ranges used in most applications. 

One of the most important applications of the gas laws in chemistry is to calculate the volumes of gases consumed or produced in chemical reactions. If the conditions of pressure and temperature are known, the ideal gas law can be used to convert between the number of moles and gas volume. Instead of working with the mass of each gas taking part in the reaction, we can then use its volume, which is easier to measure. This is illustrated by the following example. 

Further, in regions well below the critical point, the vapor pressure is relatively small, so that the ideal gas law may be assumed to be applicable, i.e., pVv = RT, where Vv is the molar volume of the vapor and p is its pressure at the temperature T. Substituting RT/p for Vv in equation (27.11), this becomes... 

This is known as the ideal gas law, called ideal because, under conditions of low pressure and high temperature, all gases obey it regardless of types or mass of molecule. It is universally applicable. You will find it necessary to use this taw in gas problems which involve moles or masses of gas. To use this equation, it is necessary to evaluate the ideal gas constant R. Experimentally it has been found that at 1.0 atm, and 273 K, (standard temperature and pressure or STP), 1.0 mol of a gas occupies 22.4 L. Solving the equation for R, and inserting those values ... 

In Example 1 the ideal gas law was used. To test if this law can be used, another, more accurate law, must be available. The REDLICH and KWONG equation appears to be well adapted for this purpose, both by its simplicity and its ease of application. This analytical state equation relates the pressure p, to the temperature T and the molar volume V ... 

For engineering calculations it is important to have equations of state that are accurate over a wide range of pressures and temperatures. The ideal gas law is very simple to use, but its validity is restricted to gases at low pressures. The truncated virial equation is applicable over a somewhat wider range of pressures, but only for gases. If the pressure is high or the phase liquid, neither of these equations can be used. 

A schematic, three-dimensional, one-component, p-V-T diagram is reproduced in Fig. 4.20. Its surface represents all possible equilibrium states of the system. The gas area, especially at high temperature and volume, is well described by the ideal gas law, at lower temperatures, the van der Waals equation is applicable as seen in Fig. 2.99. 

Equation (16) has also been applied to lAST [41]. When it is assumed that the gas mixture obeys the perfect gas law and that the adsorbed phase on each patch is ideal, the resulting model is termed HI AST (heterogeneous lAST). Since the HIAST requires the evaluation of the local equilibria on each energy patch with lAST, much more computational effort is required, and yet the increase in performance over the original lAST is only modest for most cases. Therefore, this theory has not been widely used in practical applications. 

Throughout this textbook, the application of the ideal gas law will come up again and again in one form or another. While you have almost certainly encountered the ideal gas law many times already, it is worth briefly reviewing it here. The ideal gas law is most commonly expressed as... 

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