Irreversible expansion of a gas

In this discussion, we will limit our writing of the Pfaffian differential expression bq, for the differential element of heat flow in thermodynamic systems, to reversible processes. It is not possible, generally, to write an expression for bq for an irreversible process in terms of state variables. The irreversible process may involve passage through conditions that are not true states" of the system. For example, in an irreversible expansion of a gas, the values of p. V, and T may not correspond to those dictated by the equation of state of the gas. 

Derive an explicit expression for the work performed in the irreversible expansion of a gas from volume Vi to volume V2 against a constant external pressure P that is less than the pressure of the gas throughout the expansion. 

Figure 3.10 Schematic of Joule s experiment for irreversible expansion of a gas into an evacuated chamber in a water bath.

FIGURE 12.2 Stages in an irreversible expansion of a gas from an initial state (a) of volume V to a final state (c) of volume Vj. In the intermediate stage shown (b), the gas is not in equilibrium because of turbulence, pressure and temperature cannot be defined. 

Fig. 4.6. Irreversible expansion of a gas from external pressure Pe t, to Pext-i- During expansion, external pressure is fixed at P by the weight of the piston.

All non-reversible processes are called irreversible. An example of an irreversible process is expansion of a gas into a vacuum during the expansion process the system is in a state of non-equiUbrimn and cannot be described by the usual macroscopic state variables such as temperature T and pressure p. The irreversible expansion of a gas into a vacuum can therefore not be shown as a process curve in a pV diagram. 

As an example of conditional irreversibility may be taken the expansion of a gas. Work is done by the change of bulk in opposition to the external forces, and heat is absorbed from the environment. The conditions which must hold in order that the process actually occurs are ... 

Expansion of a Gas into a Vacuum.—If a gas is allowed to rush into a vacuous space, or into a space containing a gas under a less pressure, we have an example of a process attended by conditional irreversibility. 

For a scientist, the primary interest in thermodynamics is in predicting the spontaneous direction of natural processes, chemical or physical, in which by spontaneous we mean those changes that occur irreversibly in the absence of restraining forces—for example, the free expansion of a gas or the vaporization of a hquid above its boiling point. The first law of thermodynamics, which is useful in keeping account of heat and energy balances, makes no distinction between reversible and irreversible processes and makes no statement about the natural direction of a chemical or physical transformation. 

We conclude by summarizing in Table 3.1 some key distinctions between reversible and irreversible processes, taking as an example the expansion of a gas against a piston, with external pressure Fext ... 

We note here, as was found in the example of the calculation of the work done during the irreversible adiabatic of expansion of a gas (equation 9.4 Frame 9), that under specific conditions a path dependent function can become identically equal to the change in a state function. 

For an irreversible expansion of a real gas at constant temperature due to a heat reservoir, the change of entropy flow is d,.S = 8q/T, where 8q is the heat flow between the gas and the reservoir to maintain the constant temperature. The increase of entropy during the expansion is... 

Here, <55 s 0 is the entropy change arising from irreversible processes occurring within a completely closed system. As Eq. (1.12.9a) shows, S can then only increase. As soon as these processes have ceased, 50 = 55 = 0, so that 5 has assumed an extremal value which is a maximum under the present constraints. For example, the entropy change in the free expansion of a gas can be determined by finding AS under quasistatic conditions, as specified later in Section 2.3. Since 5 is a function of state the same entropy change takes place in a free expansion under the same conditions. All this, of course, merely repeats what has been well established in earlier sections. 

In order to avoid the possibility of misunderstanding, it may be pointed out here that many changes which occur spontaneously in nature, e g., expansion of a gas, evaporation of a liquid and even chemical reactions, can be carried out reversibly, at l t in principle, as described in 8a However, when they do occur spontaneously, without external intervention, they are thermodynamically irreversible. 

Many changes which are naturally spontaneous, e.g., expansion of a gas, solution of zinc in copper sulfate, etc., can be carried out, actually or in principle, in a reversible manner. It should be clearly understood that in the latter event the total entropy of the system and its surroundings remains unchanged. There is an increase of entropy only when the change occurs spontaneously and hence irreversibly. 

In the isothermal expansion of a gas, the final volume V2 is greater than the initial volume Vi and, consequently, by equation (19.26), AS is positive that is to say, the expansion is accompanied by an increase of entropy of the system. Incidentally, when an ideal gas expands (irreversibly) into a vacuum, no heat is taken up from the surroundings ( 9d), and so the entropy of the latter remains unchanged. In this case the net entropy increase is equal to the increase in entropy of the system, i.e., of the gas, alone. 

However, if we consider the irreversible expansion of a perfect gas into a vacuum (doing no work) then ASovwaH will no longer be zero. For the gas itself AS = nRln VB/VA however, the changes in the surroundings will be different. The gas does no work and as q - — w = 0, the system absorbs no heat... 

We have already noted that work done is not a state function this is also true of the mechanical work of expansion. The derivation above has shown that the work is related to the process carried out rather than to the initial and final states. We can consider the reversible expansion of a gas from volume Vy to volume V2, and can also consider an irreversible process, in which case less work will be done by the system. 

In discussing Figure 19.2, we talked about the expansion of a gas into a vacuum as a spontaneous process. We now understand that it is an irreversible process and that the entropy of the universe increases during the expansion. How can we explain the spontaneity of this process at the molecular level We can get a sense of what makes this expansion spontaneous by envisioning the gas as a collection of particles in constant motion, as we did in discussing the kinetic-molecular theory of gases. (Section 10.7) When the stopcock in... 

In discussing Figure 19.2, we talked about the expansion of a gas into a vacuum as a spontaneous process. We now understand that it is an irreversible process and that the entropy of the universe increases during the expansion. How can we explain... 

Consider the free expansion of a gas shown in Fig. 3.8 on page 79. The system is the gas. Assume that the vessel walls are rigid and adiabatic, so that the system is isolated. When the stopcock between the two vessels is opened, the gas expands irreversibly into the vacuum without heat or work and at constant internal energy. To carry out the same change of state reversibly, we confine the gas at its initial volume and temperature in a cylinder-and-piston device and use the piston to expand the gas adiabatically with negative work. Positive heat is then needed to return the internal energy reversibly to its initial value. Because the reversible path has positive heat, the entropy change is positive. 

Figure 3.6 Reversible and irreversible processes. (A) The system reaches the state X from the standard state O through a path I involving irreversible processses. It is assumed that the same transformation can be achieved through a reversible transformation R. (B) An example of an irreversible process is the spontaneous expansion of a gas into vacuum. The same change can be achieved reversibly through an isothermal expansion of a gas that occurs infinitely slowly so that the heat absorbed from the reservoir equals the work done on the piston. In a reversible isothermal expansion the change in entropy can be calculated using dS = dQ/T.

---