Tire Ideal Gas

The question often arises as to when tire ideal-gas eqnatioir irray be nsed as a reasoirable approxiirrationto reality. Figure 3.16 can serve as a gnide. 

Finally, a new class of solution properties is introduced. Known as excesspropevties, they are based on an idealization of solution behavior called tire ideal solution. Its role is like that of tire ideal gas in that it serves as a reference for real-solution belravior. Of particular interest is tire excess Gibbs energy, a property wlriclr underlies tire activity coefficient, introducedfrom a practical point of view hr tire preceding clrapter. 

The definition of tlie fugacity of a species in solution is parallel to the definition of tire pure-speciesfugacity. For species i m a mixture of real gases or in a solution of liquids, the equation analogous to Eq. (11.28), tire ideal-gas expression, is ... 

Practically, in environmental applications, the temperature ranges from 20 to 400 °C and pressme from 1 to 40 atm. According to available data (Peiiy and Green, 1999) for air, nitrogen, oxygen, and hydrogen, the compressibility factor is practically unity and only in severe conditions of pressure and temperatoe varies from 0.98 to 1.02. Tlius, tire ideal gas law can be safely used in most envuonmental applications. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

One national news story in the year 2000 involved the defect in certain brands of tires and how tread separation caused vehicle accidents leading to personal injury and in some cases deaths. One prominent aspect of the tire story involved whether the tires were properly inflated to the correct pressure. The air inside a tire can be considered an ideal gas, and we can apply the gas laws to the air inside. The inflation pressure recommendations stamped on the sides of tires call for tires to be inflated under cold conditions, or before the vehicle is driven. Because the most important aspect of tire performance is correct inflation pressure, it is important to adhere to the recommended tire pressure. Many individuals disregard this recommendation. It is not uncommon for someone to pull into a gas station and fill... 

Since C and A are both constants which are not properties of a particular ideal gas, R is a constant independent of tire particular ideal gas. This is the equation of state for an ideal gas which was previously derived from the equations of Boyle, Charles, and Avogadro. 

We recognize that the equation of state for an ideal gas does not describe adequately tire behavior of gases at temperatures and pressures normally encountered in petroleum reservoirs. However, Equation 6-6 does illustrate that we can expect the coefficient of isothermal compressibility of a gas to be inversely proportional to pressure. Equation 6-6 can be used to determine the expected order of magnitude of gas compressibility. 

The ideal gas law has many uses in chemistry, some of which we shall meet later in this text. To begin to see how useful the law can be, recall that we have seen how to use the individual laws to make predictions when only one variable is changed, such as heating a fixed amount of gas at constant volume. The ideal gas law enables us to make predictions when more than one variable is changed. For example, when we pump up an actual bicycle tire, the temperature of the gas in the pump increases as we press in the piston, so the compression is not strictly isothermal as we assumed in Example 4.3. 

We are given V, P, and T, and we need to use the ideal gas law to calculate n, the number of moles of helium in the tire. With n known, we then do a mole-to-mass conversion. 

Solution for V may be by trial, iteration, or with the solve routine of a software package. An initial estimate for V is the ideal-gas value 7 // . For iteration, tins value is substituted on tire right side of Eq. ( 5.4(5). The resulting value of V on the left is then returned to the right side, and the process continues until the clrange in V is suitably small. 

An ideal gas initially at 600 K and 10 bar rmdergoes a fonr-step mechanically reversible cycle in aclosedsystem.hr step 12, pressure decreases isodrennallyto 3 bar in step 23, pressure decreases at constant volnine to 2 bar in step 34, volnine decreases at constant pressure and in step 41, tire gas rehinrs adiabatically to its iiritial state. 

Tire generalized correlations for and S, together with ideal-gas heat capacities, allow calculation of enthalpy and entropy values of gases at ary temperature and pressure by Eqs. (6.49)and (6.50). For a change from state 1 to state 2, write Eq. (6.49) for both states ... 

Propane gas at 1 bar and 308.15 K (35°C) is coirrpressed to a fiiral state of 135 bar and 468.15K(195°C). Estimate the irrolar voluirre of the propaire hr tire final state and the enthalpy and entropy changes for tire process. In its iiritial state, proparre may be assumed air ideal gas. 

Since tire enthalpy of an ideal gas depends on temperature only, a tlirottling process does not clrange the temperature of an ideal gas. For most real gases at moderate conditions of temperature and pressure, a reduction in pressure at constant enthalpy results in a decrease in temperature. For example, if steam at 1000 kPa and 573.15 K (300°C) is throttled to 101.325 kPa (atmospheric pressure),... 

For tire special case of an ideal gas witli constant heat capacities,... 

Air expands adiabatically tlnough a nozzle from a negligible initial velocity to a final velocity of 325 m s . Wliat is tire temperature drop of the air, if air is assuirred to be an ideal gas for wlriclr Cp = (7/2)/ ... 

An ideal gas with constant heat capacities enters a convergingldiverging nozzle witli negligible velocity. If it expands isentropically witlhn tlie nozzle, show tliat tire tliroat velocity is given by ... 

The application of Eq. (11.6) in later chapters to specific phase-equilibrium problems requiresuse of models of solutionbeliavior, wliichprovideexpressionsfor G and (ii as functions of temperature, pressure, and composition. The simplest of these, the ideal-gas mixture and tire ideal solution, are treated in Secs. 11.4 and 11.8. 

If n moles of an ideal-gas mixture occupy a total volume V at temperature T, tire pressure is ... 

Since tire entlialpy of an ideal gas is independent of pressure,... 

As evident from Eq. (11.6), the chemical potential jiif provides the fundamental criterionfor plrase equilibria. This is true as well for chemical-reaction equilibria. However, it exhibits characteristics which discourage its use. The Gibbs energy, and hence /i. , is defined in relation to tire internal energy and entropy, both primitive quantities for wliich absolnte valnes are unknown. TVs a resnlt, absolute values for fn do not exist. Moreover, Eq. (11.28) shows that for an ideal-gas mixhire m approaches negative infinity when either P or y,- approaches zero. Tliis is in fact tme for any gas. While these characteristics do not preclude the use of chemical potentials, the application of eqnilibrinm criteria is facilitated by introdnctionof the fugacity, a quantity that takes the place of bnt which does not exliibit its less desirable characteristics. 

The ideal gas is a useful model of tlie behavior of gases, and serves as a standard to wliich real-gas belravior can be compared. Tliis is formalized by the introdnction of residual properties. Another useful model is tire ideal solution, wliich serves as a standard to wliich real-solution behavior can be compared. We will see in the following section how tliis is formalized by introdnction of excess properties. 

Equation (11.26) establishes tire belravior of species i m an ideal-gas mixture ... 

Tliis equation takes on a new dimension when G f, the Gibbs energy of pure species i m the ideal-gas state, is replaced by Gj, tire Gibbs energy of pure species i as it actually exists at tire mixhire T and P and hr tire same physical state (real gas, liquid, or solid) as the mixture. It then applies to species in real solutions. We therefore define an ideal solution as one for which ... 

The standard state for a gas is the ideal-gas state of the pure gas at tire standard-state pressure P° of 1 bar. Since tire fugacity of an ideal gas is equal to its pressure, /,-° = P° for each species/. Thus for gas-pliase reactions= fi/P°, etnd Eg. (13.10) becomes ... 

The seeond tenn arises from Eq. (13.35) applied to species C, Eq. (13.29) applied to B with /b //b = 15 the fact that / = P ° for species A in tire gas phase. Since K depends on tlie standard states, tliis value of K is not tlie same as that obtained when, for example, the standard state for each species is chosen as the ideal-gas state at 1 bar. However, all methods tlieoretically lead to the same equilibrium composition, provided Henry s law as applied to species C in solution is valid. In practice, a particular choice of standard states may simplify calculations or yield more accurate results, because it makes better use of the available data. The nature of the calculations required for heterogeneous reactions is illustrated in the following example. 

---