The Ideal Gas Equation

F ure 5.10 Volume relationship of gases in a chemical reaction. The ratio of the volumes of molecular hydrogen to molecular nitrogen is 3 1, and that of ammonia (the product) to molecular hydrogen and molecular nitrogen combined (the reactants) is 2 4, or 1 2. 

Because, at the same temperature and pressure, the volumes of gases are directly proportional to the number of moles of the gases present, we can now write 

Worked examples illustrating the gas laws are presented in Section 5.4. 

We can combine all three expressions to form a single master equation for the behavior of gases  

Keep in mind that the ideal gas equation, unlike the gas laws discussed in Section 5.3, applies to systems that do not undergo changes in pressure, volume, temperature, and amount of a gas. 

A comparison of the molar volume at STP (which is approximately 22.4 L) with a basketball. 

Sulfur hexafluoride (SFg) is a colorless, odorless, very unreactive gas. Calculate the pressure (in atm) exerted by 1.82 moles of the gas in a steel vessel of volume 5.43 L at 45°C. 

Strategy The problem gives the amount of the gas and its volume and temperature. Is the gas undergoing a change in any of its properties What equation should we use to solve for the pressure What temperature unit should we use  

RecaU that the state of a sample of gas is described completely using the four variables T, P, V, and n. Each of the gas laws introduced in Section 11.2 relates one variable of a sample of gas to another while the other two variables are held constant. In experiments with gases, however, there are usually changes in more than just two of the variables. Therefore, it is useful for us to combine the equations representing the gas laws into a single equation that will enable us to account for changes in any or all of the four variables. 

Deriving the Ideai Gas Equation from the Empiricai Gas Laws 

We can combine these equations into the following general equation that describes the physical behavior of all gases  

The proportionality constant, R, in Equation 11.6 is called thegas constant. Its value and units depend on the units in which P and V are expressed. (The variables n and T are always expressed in mol and K, respectively.) Recall from Section 11.1 that pressure is commonly expressed in atmospheres, mmHg (ttjrr), pascals, or bar. Volume is typically expressed in liters or milliliters, but can also be expressed in other units, such as m. Table 11.4 lists several different expressions of the gas constant. R. 

V fe wl ctsoiss the condtions that result in deviation from ideal behavior in Section 11.7. 

If you take Bojde s Law, Charles s Law, Gay-Lussac s Law, and Avogadro s Law and throw them into a blender, turn the blender on high for a minute, and then pull them out, you get the ideal gas equation — a way of working in volume, temperature, pressure, and amount. The ideal gas equation has the following form  

The P represents pressure in atmospheres (atm), the V represents volume in liters (L), the n represents moles of gas, the T represents the temperature in Kelvin (IQ, and the R represents the ideal gas constant, which is 0.0821 liters atm/K mol. 

Using the value of the ideal gas constant, the pressure must be expressed in atm, and the volume must be expressed in liters. You can calculate other ideal gas constants if you really want to use torr and milliliters, for example, but why bother It s easier to memorize one value for R and then remember to express the pressure and volume in the appropriate units. Naturally, you ll always express the temperature in Kelvin when working any kind of gas law problem. 

That said, now 1 want to show you an easy way to convert a gas from a mass to a volume if the gas is not at STP. What s the volume of 50.0 grams of oxygen at 2.00 atm and 27.0 degrees Celsius  

The first thing you have to do is convert the 50.0 grams of oxygen to moles using the molecular weight of O2  

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The importance of the van der Waals equation is that, unlike the ideal gas equation, it predicts a gas-liquid transition and a critical point for a pure substance. Even though this simple equation has been superseded, its... 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

But since the ideal gas equation requires that = pRT/M, this reduces finally to... 

The behavior of all gases that obey the laws of Boyle and Charles, and Avogadro s hypothesis, can be expressed by the ideal gas equation ... 

The Ideal Gas The simplest equation of state is the ideal gas equation ... 

As long as the volume flow is kept near design point, both the deflection angle and pressure drop can be corrected. Temperature differential increase is limited by metallurgy, so it is neglected in analytical calculations. This evaluation is based on inlet pressure changes. The new volume at a different pressure is calculated by the ideal gas equation ... 

Compressibility is experimentally derived from data about the actual behavior of a particular gas under pVT changes. The compressibility factor, Z, is a multiplier in the basic formula. It is the ratio of the actual volume at a given pT condition to ideal volume at the same pT condition. The ideal gas equation is therefore modified to ... 

If the gas is not ideal, so that the ideal gas equation cannot be used, we replace the pressurep in equations 20.198 and 20.199 by the fugacity,/, such that the ideal gas equation still holds if the pressure p is replaced by the fugacity, an effective pressure, when the real pressure is p. This form is most convenient because of the numerous ways in which non-ideality can be expressed, and we note that the fugacity is related to, but not necessarily proportional to the pressure. We can express the fugacity as a function of the pressure by introducing the fugacity coefficient, 7p, as / = y p, which then replacesp in equation 20.199 for the non-ideal case. The value of 7p tends to unity as the gas behaves more ideally, which means as the pressure decreases. 

In the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

Thus, p can be estimated from the observed pressure and the ideal pressure calculated from the molar volume and the ideal gas equation. Klotz and Rosenburg3 report that the error in using equation (6.23) to calculate p is less than 1% for O2 up to a pressure of 10 MPa. For CO2 (g) the error is 1% at 2.5 MPa and 4% at 5 MPa. 

Equation (7.123)w is often referred to as the law of Van t Hoff, since it was originally proposed by J. H. Van t Hoff. It is interesting to note that equation (7.123) is of the same form as the ideal gas equation, if one takes c as njVx Table 7.3 compares experimental values of II for aqueous sucrose solutions with those calculated using equations (7.120) and (7.123). We see that neither of the equations predicts n with high accuracy. However, the superiority of equation (7.120), especially at higher concentrations, is apparent. 

Example 10.4 Use Zm to derive the ideal gas equation Solution From Table 3.1... 

We call Equation the ideal gas equation, because it refers to the behavior of a so-called ideal gas. As we describe in the following section, no gas is truly ideal, but under conditions close to 1 atm and room temperature, the ideal gas equation is adequate to describe most real gases. 

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

To use the ideal gas equation for quantitative calculations, we must express each quantity in appropriate units. The ideal gas equation holds only when temperature is expressed using an absolute scale. We will always use the Kelvin scale, applying the conversion introduced in Chapter E 7 (K) = T(° C) + 273.15. Typical laboratory pressures are expressed in atmospheres, and typical laboratory volumes are expressed in liters. For this choice of... 

We can calculate the pressure of the gas using the ideal gas equation, but we need to make sure all the variables are expressed in consistent units. Temperature must be in kelvins, amount of methane in moles. For R = 0.08206 L atm mol K, we need the volume in liters, and the units of the calculated pressure will be atmospheres. 

We rearrange the ideal gas equation so that pressure is isolated on the left ... 

The question asks for the mass of oxygen. We can use the ideal gas equation to calculate the number of moles of oxygen, and then molar mass leads us from moles to grams. 

During chemical and physical transformations, any of the four variables in the ideal gas equation P, V, n, T) may change, and any of them may remain constant. The experiments carried out by Robert Boyle are a good example. Boyle worked with a fixed amount of air trapped in a glass tube, so the number of moles of gas remained the same during his experiments. In other words, n was held constant. Boyle also worked at only one temperature, so T remained constant. Example applies the ideal gas equation to this situation. 

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. 

To understand why all gases can be described by a single equation, we need to explore how gases behave at the molecular level, hi this section we examine the molecular properties of gases and how they result in the ideal gas equation. 

First, consider increasing the amount of the gas while keeping the temperature and volume fixed. Figure 5-10 shows that doubling the amount of gas in a fixed volume doubles the number of collisions with the walls. Thus, pressure is directly proportional to the amount of gas. This agrees with the ideal gas equation. 

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