Ideal gases reversible processes

TiVi = T2V2 (ideal gas, reversible adiabatic process) (Section 12.6)... 

The total energy of condensation from the ideal gas to the liquid state (the reverse process of vaporization) as a consequence of 1-1 contacts (i.e., intermolecular interactions of component 1 with like molecules) is the product of the energy of condensation per unit volume, the volume of liquid, and the volume fraction of component 1 in the liquid, or... 

We shall suppose the solute to be a mol of an ideal gas, occupying a volume v at the pressure o and the solvent a volume Y of Iig. 56. liquid just sufficient to dissolve all the gas under the pressure j)o- If the gas is brought directly into contact with the liquid, an irreversible process of solution occurs, but if it is first of all expanded to a very large volume, the dissolution may be made reversible, except for the first trace of gas entering the... 

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

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... 

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

The above relations apply for an ideal gas to a reversible adiabatic process which, as already shown, is isentropic. 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

To evaluate the integral in Equation B.l requires the pressure to be known at each point along the compression path. In principle, compression could be carried out either at constant temperature or adiabatically. Most compression processes are carried out close to adiabatic conditions. Adiabatic compression of an ideal gas along a thermodynamically reversible (isentropic) path can be expressed as ... 

If there is no heat transfer or energy dissipated in the gas when going from state 1 to state 2, the process is adiabatic and reversible, i.e., isentropic. For an ideal gas under these conditions,... 

Temperature and enthalpy are not the only conditions that determine whether a change is favourable. Consider the process shown in Figure 7.5. A closed valve links two flasks together. The left flask contains an ideal gas. The right flask is evacuated. When the valve is opened, you expect the gas to diffuse into the evacuated flask until the pressure in both flasks is equal. You do not expect to see the reverse process—with all the gas molecules ending up in one of the flasks—unless work is done on the system. 

Because the reversible process is a succession of equilibrium states, P is given by the equation of state, which is Equation (5.1) for an ideal gas. Substituting from Equation (5.1) into Equation (5.19), we obtain... 

So far we have not specified whether the adiabatic expansion under consideration is reversible. Equations (5.40), (5.42), and (5.44) for the calculation of the thermodynamic changes in this process apply to the reversible expansion, the free expansion, or the intermediate expansion, so long as we are dealing with an ideal gas. However, the niunerical values of W, AU, and AH will not be the same for each of the three types of adiabatic expansion because T2, the final temperature of the gas, will depend on the type of expansion, even though the initial temperature is identical in aU cases. 

As we noticed in Table 5.1, AC/ = 0 both for the free expansion and for the reversible expansion of an ideal gas. We used an ideal gas as a convenient example because we could calculate easily the heat and work exchanged. Actually, for any gas, AC/has the same value for a free and a reversible expansion between the corresponding initial and final states. Furthermore, AC/ for a compression is equal in magnitude and opposite in sign to AC/ for an expansion no indication occurs from the first law of which process is the spontaneous one. 

ENTROPY CHANGES IN REVERSIBLE PROCESSES 131 For the specific case of the expansion of an ideal gas, because AU=0,... 

To calculate the change in entropy in this irreversible flow, it is necessary to consider a corresponding reversible process. One process would be to allow an ideal gas to absorb reversibly the quantity of heat Q at the temperature T2. The gas then can be expanded adiabatically and reversibly (therefore with no change in entropy) until it reaches the temperature Ti. At Ti the gas is compressed reversibly and evolves the quantity of heat Q. During this reversible process, the reservoir at T2 loses heat and undergoes the entropy change... 

A mole of steam is condensed reversibly to liquid water at 100°C and 101.325-kPa (constant) pressure. The heat of vaporization of water is 2256.8 Jg. Assuming that steam behaves as an ideal gas, calculate W, Q, AUra, AH, ASra, AGm. and AAm for the condensation process. 

The Kelvin scale is thus defined in terms of an ideal reversible heat engine. At first such a scale does not appear to be practical, because all natural processes are irreversible. In a few cases, particularly at very low temperatures, a reversible process can be approximated and a temperature actually measured. However, in most cases this method of measuring temperatures is extremely inconvenient. Fortunately, as is proved in Section 3.7, the Kelvin scale is identical to the ideal gas temperature scale. In actual practice we use the International Practical Temperature Scale, which is defined to be as identical as possible to the ideal gas scale. Thus, the thermodynamic scale, the ideal gas scale, and the International Practical Temperature Scale are all consistent scales. Henceforth, we use the symbol T for each of these three scales and reserve the symbol 9 for any other thermodynamic scale. 

Consequently, the energy of the gas is constant for the isothermal reversible expansion or compression and, according to the first law of thermodynamics, the work done on the gas must therefore be equal but opposite in sign to the heat absorbed by the gas from the surroundings. For a reversible process the pressure must be the pressure of the gas itself. Therefore, we have for the isothermal reversible expansion of n moles of an ideal gas between the volumes F and V... 

In a reversible, adiabatic process of a system comprising of one mole of an ideal gas, prove the following relationships ... 

Of particular relevance to understanding the influence of solvation on chemical processes is the concept of the free energy of solvation, AGsol, which can be defined as the reversible work required to transfer the solute from the ideal gas phase to solution at a given temperature, pressure and chemical composition [10]. This definition is well suited for molecular formulations of the solvation problem, because it permits AGsol to be related to the difference in the reversible works necessary to build up the solute both in solution and in the gas phase. 

One specific reversible expansion of an ideal gas that will be of particular interest to us is the one in which the system remains at constant temperature (by being immersed in a thermostat). Such a process is called isothermal. For this case, we can use Eq. (6), with P = nRT/V ... 

Example 3. The ideal gas discussed in Example 2 could also undergo a single-stage expansion to the final state f, by suddenly reducing the pressure on the confining piston to Pj. How does this process differ from the reversible path i-3-f discussed in Example 2 ... 

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