The equilibrium pressure

We have already defined Dalton s pressures and the partial pressures. There is still one more pressure of a gas in a gas mixture that must be considered. This is the equilibrium pressure, and was first discussed by Gibbs [15]. The equilibrium pressure of a component is the pressure exerted by the pure component when it is in equilibrium with a gas mixture through a rigid, diathermic membrane that is permeable only to that component. The temperature of the pure phase and that of the gas mixture must be the same, but the pressure of the two phases will be different. We may conceive of two phases at the same temperature separated by such a membrane, one phase being the pure fcth component and the other the gas mixture. The condition of equilibrium is that the chemical potential of the fcth component must be identical in each phase. If P is the equilibrium pressure, then by Equation (7.69) 

This equation is transcendental and the equilibrium pressure must be calculated by approximation methods. The first approximation could be obtained by letting the equilibrium pressure be the partial pressure and evaluating the terms corresponding to the nonideality of the gas by this approximation. The solution of the resultant equation for the equilibrium pressure in the logarithmic term would give a second approximation. By such successive approximations the actual value of the equilibrium pressure can be obtained easily by using computer calculations. 

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If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

Two other examples will sufhce. Methane physisorbs on NaCl(lOO) and an early study showed that the symmetrical, IR-inactive v mode could now be observed [97]. In more recent work, polarized FTIR rehection spectroscopy was used to determine that on being adsorbed, the three-fold degeneracies of the vs and v modes were partially removed [98]. This hnding allowed consideration of possible adsorbate-adsorbent geometries one was that of a tripod with three of the methane hydrogens on the surface. The systems were at between 4 and 40 K so that the equilibrium pressure was very low, about 10 atm. 

Now, considering process XVII-93, where P is the equilibrium pressure of the adsorbate, we have... 

For an absorption system in which the solute is highly soluble in the solvent, such that the equilibrium pressure of the solute over the absorbing liquid is very low, we may write ... 

What is the equilibrium pressure of hydrogen sulfide if those of hydrogen and sulfur gases are 0.103 atm and 0.417 atm, respectively ... 

If we start with hydrogen and oxygen, equilibrium is attained after most of the hydrogen and oxygen have united to form water. More important, though, the partial pressures at equilibrium are the same as those obtained beginning with pure H20. The equilibrium pressures are fixed by the temperature, the composition, and the total pressure they do not depend upon the direction from which equilibrium is approached. The balanced equation does not indicate the concentrations (or partial pressures) at equilibrium. 

Although we have no way of determining the moisture content of the individual components, it is not difficult to measure a property which is directly related to it and is common to all the components—the equilibrium pressure of water vapor. It seems logical, therefore, to use it as an index of moisture, as suggested by Makower and Myers (20). 

An example is the complex with argon which can be kept indefinitely in an ordinary bottle, although the equilibrium pressure of argon over the crystal amounts to several atmospheres at room temperature. Powell31 named these complexes "clathrate compounds, which according to him are those compounds "in which two or more components are associated without ordinary chemical union but through complete enclosure of one set of molecules in a suitable structure formed by another. ... 

Since the molecules of SFt and propane only fit into the larger cavities, their equilibrium hydrates must have identical compositions. From the experimental value of the equilibrium pressure, pp = 1.48 atm for propane... 

It is essentially a phase diagram which consists of a family of isotherms that relate the equilibrium pressure of hydrogen to the H content of the metal. Initially the isotherm ascends steeply as hydrogen dissolves in the metal to form a solid solution, which by convention is designated as the a phase. At low concentrations the behaviour is ideal and the isotherm obeys Sievert s Law, i.e.,... 

PCT diagrams of AB2 (electrode alloys) /H systems reflect multiphase or nonideal behaviour [54], This is illustrated in Fig. 19, in which both the equilibrium pressure and the open—circuit equilibrium voltage, Er are plotted for Zr()5Ti05 Vo.sNij Fe02Mn02. 

The isotherms represented in Fig. 1 give a general idea of the equilibria in the Pd-H system under different p-T conditions. Most experimental evidence shows, however, that the equilibrium pressure over a + /3 coexisting phases depends on the direction of the phase transformation process p a-p > pp-a (T, H/Pd constant). This hysteresis effect at 100°... 

Kinetic-molecular theory provides an explanation on a molecular level for this equilibrium. Evaporation from the liquid occurs as fast moving molecules on the surface escape from the liquid. In turn, molecules in the gas phase strike the liquid and condense, As the concentration (pressure) of gas molecules builds up in the gas phase, the rate of condensation increases. Eventually, a pressure is reached where the rate of condensation and rate of evaporation just balance, and equilibrium is achieved. The equilibrium pressure is denoted by p and is known as the vapor pressure. The magnitude ofp depends upon the substance, composition of the liquid, and any two of our thermodynamic variables such as temperature and total pressure. The criteria for equilibrium that we will now derive provide the thermodynamic relationships that will help... 

The equilibrium pressure when (solid + vapor) equilibrium occurs is known as the sublimation pressure, (The sublimation temperature is the temperature at which the vapor pressure of the solid equals the pressure of the atmosphere.) A norma) sublimation temperature is the temperature at which the sublimation pressure equals one atmosphere (0.101325 MPa). Two solid phases can be in equilibrium at a transition temperature (solid + solid) equilibrium, and (liquid + liquid) equilibrium occurs when two liquids are mixed that are not miscible and separate into two phases. Again, "normal" refers to the condition of one atmosphere (0.101325 MPa) pressure. Thus, the normal transition temperature is the transition temperature when the pressure is one atmosphere (0.101325 MPa) and at the normal (liquid + liquid) solubility condition, the composition of the liquid phases are those that are in equilibrium at an external pressure of one atmosphere (0.101325 MPa). 

Line db in Figure 8.1 represents the equilibrium melting line for C02. Note that the equilibrium pressure is very nearly a linear function of T in the (p, T) range shown in this portion of the graph, and that the slope of the line, (d/ /d7 )s ], is positive and very steep, with a magnitude of approximately 5 MPa-K-1. These observations can be explained using the Clapeyron equation. For the process... 

Hydrogen may also be determined by both electrochemical and diffusion meters. The electrochemical meter is a hydride-activated concentration cell that employs an electrolyte consisting of a CaH2-CaCl2 mixture. The diffusion meter is based on the equilibrium pressures attained on either side of a thin membrane, usually nickel. 

Here, P is the equilibrium pressure. Therefore, Dj of the anode electrode can be also calculated through the slope of a log-log plot of Eq. (4) by the data from the mercury porosimeter. 

C12-0061. At 25 °C, the equilibrium pressure of ammonia vapor above a 0.500 M aqueous ammonia solution is 6.8 torr. Calculate the Henry s law constant and determine the equilibrium pressure of ammonia vapor above a 2.5 M solution. 

The equilibrium pressure (vapor pressure) above a liquid is independent of the amount of liquid, because the ratio of evaporating to condensing molecules is independent of amount. 

The problem asks for the equilibrium pressures and a molecular view illustrating the equilibrium conditions. 

Now we apply Equation to obtain expressions for the equilibrium pressures ... 

To evaluate x, we substitute the equilibrium pressures into the equilibrium constant expression ... 

H2(g)+Br2(g) 2HBr(g) Find the equilibrium pressures ofall gases if 10.0 bar of HBr is introduced into a sealed container at 1024 K. 

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