Free expansion of a gas

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. 

This is an example of an irreversible process in an isolated system for which a reversible path between the initial and final states has both heat and work. 

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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. 

Another instructive exercise consists in examining the free expansion of a gas, depicted in Fig. 1.8.2. The gas is initially confined to a space of volume VA, volume VB being totally evacuated. A hole of macroscopic dimensions is now opened in the diaphragm separating the two volumes the gas ultimately occupies the total volume VA + VB. What is the work involved in this process The answer is not absolutely straightforward If the system is taken with boundaries encompassing only the volume VA, complications arise because... 

FIGURE 1.8.2 Schematic diagram pertaining to the free expansion of a gas. The gas is initially confined in compartment A of the system and is allowed to expand into compartment B through a hole opened in the diaphragm separating the two regions. 

Fig. 1.7.2. Schematic diagram illustrating the free expansion of a gas initially confined to compartment A. On opening the partition the gas then expands into compartment B.

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. 

FIGURE 13.2 The free expansion of a gas into a vacuum, (a) The stopcock is ciosed with aii gas in the ieft buib. (b) With the stopcock open, haif the gas is in each buib. 

COMMENT. An isothermal free expansion of a perfect gas is also adiabatic,... 

The interpretation of this experiment is as follows. To begin with, no work is produced in the surroundings. The boundary, which is initially along the interior walls of vessel A, moves in such a way that it always encloses the entire mass of gas the boundary therefore expands against zero opposing pressure so no work is produced. This is called a free expansion of the gas. Setting 4W = 0, we see that the first law becomes dU = 4Q>... 

Inversion temperature (1921) n. A temperature at which free expansion of a real gas produces neither heating nor cooling of the gas. 

The I2 (C02) cluster ions can be formed by supersonic expansion of a gas mixture produced by flowing pure CO2 over solid iodine crystal. The I2 (C02) clusters are formed in the pulsed free-jet expansion by an electron beam. As shown... 

In an isothermal expansion of a gas from volume Vi to Vf, what is the change in the Helmholtz free energy F ... 

Figure 2.3 illustrates a condition that occasionally occurs with gases the expansion of a gas into a larger volume, which is initially a vacuum. In such a case, because the gas is expanding against a of 0, by the definition of work in equation 2.4 the work done by the gas equals zero. Such a process is called a free expansion ... 

Self-Test 7.16A Determine AS, ASsllrr, and AStot for (a) the reversible, isothermal expansion and (b) the isothermal free expansion of 1.00 mol of ideal gas molecules... 

As 5 is a thermodynamic property, ASsys is the same in an irreversible isothermal process from the same initial volume Vi to the same final volume V2. However, the change in entropy of the surroundings differs in the two types of processes. First let us consider an extreme case, a free expansion into a vacuum with no work being performed. As the process is isothermal, AU for the perfect gas must be zero consequently, the heat absorbed by the gas Q also is zero ... 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. Note that even though entropy is a state function, we cannot just disregard the rev subscript in Eq. 1 and calculate AS directly from q/T for any path. If we want to calculate the entropy difference between a pair of states joined by an irreversible path—such as between the initial and final states of an ideal gas that undergoes free expansion—then we can calculate that entropy difference from a reversible path between the same two states. For example, a slow, careful, reversible isothermal expansion of the gas would be a reversible path for which we can calculate AS from qrev/T. The values of q and w will differ for the two paths, but AS for the reversible path will be the same as for the irreversible path. 

Find a formula for the change of Gibbs free energy of 1 mol of a gas that expands from a volume V to a volume V2 at constant temperature T. Use terms in the virial expansion up to and including the second virial coefficient to describe the equation of state of the gas. 

Joule s experiments on the free expansion of an ideal gas showed that the internal energy of such a system is a function of temperature alone. For a real gas, this is only approximately true. For condensed phases, which are effectively incompressible, the volume dependence on the change in internal energy is negligible. As a result, the internal energies of liquids and solids are also considered a function of temperature alone. For this reason, the internal energy of a system may loosely be referred to as the thermal energy . 

Gas saturated solutions possess lower viscosity (e.g. in the region of viscosity of water). The surface tension between gas and liquid phase is also lowered. By the expansion of such gas saturated solution over a nozzle or other expansion device the compressed medium is set free, due to its high vapour pressure. Therefore the solution is cooled. At the same time high... 

Fig. 2 Schematic diagram of the pulsed supersonic nozzle used to generate carbon cluster beams. The integrating cup can be removed at the indicated line. The vaporization laser beam (30-40 mJ at 532 nm in a 5-ns pulse) is focused through the nozzle, striking a graphite disk which is rotated slowly to produce a smooth vaporization surface. The pulsed nozzle passes high-density helium over this vaporization zone. This helium carrier gas provides the thermalizing collisions necessary to cool, react and cluster the species in the vaporized graphite plasma, and the wind necessary to carry the cluster products through the remainder of the nozzle. Free expansion of this cluster-laden gas at the end of the nozzle forms a supersonic beam which is probed 1.3 m downstream with a time-of-flight mass spectrometer.

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