Pressure-Volume Isotherms

BOYLE S LAW. This law, attributed to Robert Boyle (1662) but also known as Mariottc s law, expresses the isothermal pressure-volume relation for abody of ideal gas. That is, if the gas is kept at constant temperature, the pressure and volume are in inverse proportion, or have a constant product. The law is only approximately true, even for such gases as hydrogen and helium nevertheless it is very useful. Graphically, it is represented by an equilateral hyperbola (see Fig. I). If the temperature is not constant, the behavior of die ideal gas must be expressed by die Boyle-Charles law. 

Since, for reversible cycles in the isothermal pressure-volume change of pure substances we know that dQrev = TdS and dWrev = -PdV, from the First and Second Laws of Thermodynamics, we may then obtain by rearrangement... 

Adsorption isotherms conventionally have been determined by means of a vacuum line system whereby pressure-volume measurements are made before and after admitting the adsorbate gas to the sample. For some recent experimental papers, see Refs. 24 and 25. 

Fig. XVII-29. Nitrogen isotherms the volume adsorbed is plotted on an arbitrary scale. The upper scale shows pore radii corresponding to various relative pressures. Samples A, Oulton catalyst B, bone char number 452 C, activated charcoal F, Alumina catalyst F12 G, porous glass S, silica aerogel. (From Ref. 196).

Figure 4.2. Pressure-volume compression curves. For isentrope and isotherm, the thermodynamic path coincides with the locus of states, whereas for shock, the thermodynamic path is a straight line to point Pj, V, on the Hugoniot curve, which is the locus of shock states.

Another useful teehnique in kinetie studies is the measurement of the total pressure in an isothermal eonstant volume system. This method is employed to follow the eourse of homogeneous gas phase reaetions that involve a ehange in tlie total number of gaseous moleeules present in the reaetion system. An example is the hydrogenation of an alkene over a eatalyst (e.g., platinum, palladium, or niekel eatalyst) to yield an alkane ... 

A reversible isothermal expansion of the ideal gas is made from an initial volume V to a volume Vz at an absolute (ideal gas) temperature 73. The amount of pressure-volume work in done by the system is obtained by substituting into Equation (2.16). The result is... 

Thus, in an isothermal reversible process, dA equals the reversible work. Note that <5vr in equation (3.92) is the total work. It includes pressure-volume work and any other forms, if present."1... 

Thus, in a reversible process that is both isothermal and isobaric, dG equals the work other than pressure-volume work that occurs in the process." Equation (3.96) is important in chemistry, since chemical processes such as chemical reactions or phase changes, occur at constant temperature and constant pressure. Equation (3.96) enables one to calculate work, other than pressure-volume work, for these processes. Conversely, it provides a method for incorporating the variables used to calculate these forms of work into the thermodynamic equations. 

Physical Methods that have been Used to Monitor Reaction Kinetics. In this section some physical property measurements of general utility are discussed. One of the oldest and most useful techniques used in kinetics studies involves the measurement of the total pressure in an isothermal constant volume reactor. This technique is primarily used to follow the course of homogeneous gas phase reactions that involve a change in the total number of gaseous molecules present in the reaction vessel (e.g., the hydrogenation of propylene). 

Water intrusion-extrusion isotherms performed at room temperature on hydrophobic pure silica chabazite show that the water-Si-CHA system displays a real spring behavior. However, Pressure/Volume differences are observed between the first and the second cycle indicating that some water molecules interact with the inorganic framework after the first intrusion. 29Si and especially H solid state NMR and powder X-ray diffraction demonstrated the creation of new defect sites upon the intrusion-extrusion of water and the existence of two kinds of water molecules trapped in the super-cage of the Si-CHA a first layer of water strongly hydrogen bonded with the silanols of the framework and a subsequent layer of liquid-like physisorbed water molecules in interaction with the first water layer. 

Figure 2.15 Pressure-volume data for diamond, SiC>2-stishovite, MgSiC>3 and 8102-quartz based on third order Birch-Murnaghan equation of state descriptions. The isothermal bulk modulus at 1 bar and 298 K are given in the figure.

Such a cycle is represented as a pressure-volume diagram in Figure 6.2. The representation of a temperature-volume diagram in Figure 6.3 emphasizes the isothermal nature of Steps I and III. 

Such a diagram is presented in Fig 4.1-1, p 238 of Ref 3. The adjective "critical is also applied to temperature, pressure, volume and density existing at that point (Ref 1, p 269). Methods for determining critical point on the "critical isotherm are given in Ref 3, pp 357-63 (See also under "Corresponding States and under "Critical Phenomena )... 

Typical magnetization-volume and pressure-volume isotherms obtained by this method are shown in Fig. 12. The magnetization-volume isotherm is linear i.e., each hydrogen molecule adsorbed causes the same decrease in magnetization. In the interpretation of these results, and those for other gases to which this method has been applied, it has been assumed on the basis of reasonable, but not conclusive, evidence that the slope of this isotherm corresponds to the entry of one electron into the nickel d-band for each chemisorhed H atom. From Fig. 12 it is evident that chemisorption of hydrogen is not complete at one atmosphere, and recent measurements (Vaska and Selwood, 102) indicate that it continues to much higher pressures. 

Fiq. 12. Magnetization-volume and pressure-volume adsorption isotherms for hydrogen on niokel-kieselguhr at 27 . [Selwood, P.W.,y. Am. Chem. Soc.79,3346 (1957).]... 

EXAMPLE 9.1 Construction of Adsorption Isotherms The following pressure-volume (p-V) data were collected at a temperature of 22°C. The l/ s are volumes in the gas burette, and the numerical subscripts refer to the steps itemized above. 

The results of the process described in Figure 2-2 may be presented in the form of a pressure-volume diagram. Figure 2-9 shows two isotherms of a typical pressure-volume diagram for a pure substance. Processes 1-3 and 4-5 correspond to the processes indicated in Figure 2-3. 

Fig. 2-9. Typical pressure-volume diagram of a pure substance showing two isotherms 13 below critical temperature, 45 above critical temperature.

Figure 2-10 shows a more nearly complete pressure-volume diagram.2 The dashed line shows the locus of all bubble points and dew points. The area within the dashed line indicates conditions for which liquid and gas coexist. Often this area is called the saturation envelope. The bubble-point line and dew-point line coincide at the critical point. Notice that the isotherm at the critical temperature shows a point of horizontal inflection as it passes through the critical pressure. 

Already we have seen that the critical temperature isotherm on a pressure-volume diagram for a pure substance has a horizontal point of inflection as it passes through the critical pressure. The data of Figure 2-10 clearly show this. Thus, for a pure substance at the critical point... 

Pressure is plotted against total volume, as in Figure 10-2. The plot reproduces part of an isotherm of a pressure-volume diagram. The shape is similar to that shown in Figure 2-20. 

The Gibbs free energy change during a reaction is a measure of the reversible work (other than pressure-volume work) that can be obtained from the process at constant T and p. Since cellular processes are isothermal and isobaric, free energies are the quantities of choice in studying metabolic processes with respect to their ability to carry out the work of cells. 

We now consider the isothermal, quasistatic expansion and compression of the gas for several processes. The curve AE in Figure 3.1 represents a pressure-volume isotherm of an ideal gas at a given temperature. The... 

From Fig. XII-3, it is plain that the critical point is the point C of Fig. XII-2, at which the maximum and the minimum of the isothermal coincide. We can easily find the pressure, volume, and temperature of the critical point in terms of the constants a and 5, from this condition. The most convenient way to state the condition analytically is to demand that the first and second derivatives of P with respect to F for an isothermal vanish simultaneously at the critical point. Thus, denoting the... 

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