An Introduction to Thermodynamics phần 6

measurements, the following statement is put forth as a postulate: For all solutes that do not dissociate upon dilution, the slope of fugacity against concentration, df/dN , at the origin is finite and non-zero. HENRY’S LAW. In 1803 William Henry proposed, on the basis of his measurements, that the solubility of a gas in a liquid increases in proportion to the pressure of the gas. This result can be easily derived from the dilute-solution postulate, which can be written  df2  dN N2 0 (14) The number k is the slope at the origin, which will depend on the solute and on the solvent, and the notation N2 !6 0 indicates that the solution is very dilute. (That is, as N approaches zero the equation must become valid.) A derivative is, by definition, the limiting value for the ratio of the changes in the dependent and independent variable or, in this case, the limiting value, as ΔN2 is made small, of the ratio Δf2/ΔN2. But Δf2 cn be written f2 - 0, for a small increment of fugacity at the origin, and ΔN2 can similarly be written N2 - 0. With these substitutions, we write  df   Δf   f dN2 N2®0 ΔN2 N2®0 N2 0  = f2  N2®0  2 N2®0 Combining this result with equation 14, we obtain (N2 ®0) f2 = k 2 or (N2 ®0) f2 =kN2 (15) For a very dilute solution the fugacity, and vapor pressure, of the solute are very small and therefore the fugacity is equal to the vapor pressure. (N2 ®0) P = kN2 (15a) This is Henry’s law. For dilute solutions the concentration, c2, expressed as molarity or molality, is proportional to the mole fraction, N2, so Henry’s law may be written in the more general form (c2 ®0) f2 =kc (15b) or 7/10/07 3- 66 (c2 ®0) P =kc (15c) The value of k depends on the solute, the solvent, the temperature, the units in which c2 is expressed (mole fraction, molarity, or molality), and the units in which f2 or P2 is expressed (usually atm, Pa, or torr). Small deviations from equation 15c may be observed at moderate concentrations because the vapor is not ideal; large deviations may occur because the solution may not obey equation 15b (or 15) in solutions that are not highly dilute. An example of Henry’s law is provided by the human respiratory-circulatory system. Blood entering the lungs is exposed to air containing approximately 0.20 atm oxygen, and it becomes saturated, at this pressure, with oxygen. When the blood reaches the capillaries, the pressure of oxygen in the surrounding tissues is less than 0.20 atm, so oxygen is given up by the blood to the surrounding tissues. Meanwhile, the blood picks up carbon dioxide at a comparatively high pressure in the tissues surrounding the capillaries and loses carbon dioxide in the lungs where the partial pressure of the CO2 is comparatively low. Both gases are bound chemically within the blood (with hemoglobin or as carbonates), but that equilibrium is controlled by the fugacity of the free gas in the liquid phase, which is controlled in turn, through Henry’s law, by the partial pressure of the gas in the surrounding medium (the air of the lungs or the fluids around the capillaries.). NERNST’S DISTRIBUTION LAW. When two solutions, containing the same solute but different, immiscible solvents, are brought to equilibrium, the final concentration of the solute will generally be higher in one solvent than in the other. However, the ratio of the concentrations remains unchanged if more solute is added, provided the solutions are dilute. Consider one solute distributed between two immiscible solvents, A and B (Figure 2). The concentration of the solute in solvent A is c2A and the concentration of the same solute in solvent B is c2B. From Henry’s law, cA ®0 f A =kAcA c2 ®0 f2 =kBc2 In order that the system be at equilibrium, the free energy of the solute must be the same in the two solutions. This requires that the fugacity of the solute be the same, and therefore or cA ® 0,cB ® 0) 7/10/07 f A = f B kAcA =kBcB c2 = kB = KD (16) 2 A 3- 67 The ratio of the two Henry’s law constants, kB/kA , is also a constant, which is known as the distribution constant, or distribution coefficient, KD. Nernst’s distribution law states that for dilute solutions, a solute will divide itself between two immiscible solvents to give a constant ratio of concentrations (over a range of concentrations for which kA and kB are constant). The distribution law is the basis of solvent extraction procedures. Distribution coefficients are also important, for example, in the storage of gasoline, which is a mixture of many hydrocarbons. The storage tanks must be vented to the air and consequently water vapor can condense in the tanks. The water is not significantly soluble and simply sinks to the bottom of the tank, where it might be expected to cause no trouble. However, the blending of gasolines, especially for winter driving, requires a carefully controlled percentage of the lighter hydrocarbons, such as butane and pentane. Because these hydrocarbons are more soluble than the heavier hydrocarbons in water, the water layer becomes richer in light hydrocarbons and the gasoline becomes slightly depleted in the lighter, more volatile compounds. The effect is to make the gasoline lack cold-weather starting properties if the fractionation is not properly compensated for. IDEAL SOLUTIONS. Henry’s law states that, over the concentration range for which it is valid, the fugacity, or vapor pressure, of a solute is proportional to the concentration of that solute. The proportionality constant for each solute-solvent system must be determined by experiment. In certain solutions the Henry’s law constant takes on a particular value — the value of the fugacity, or vapor pressure, of the pure component — when the concentration is expressed as a mole fraction. Then, for the ith component, (I. Soln.) fi = fio Ni (17) A solution for which each component obeys equation 17 is called an ideal solution. Equation 17 is called Raoult’s law. It will be shown later that Raoult’s law will necessarily apply for the solvent (the major component) when a solution is very dilute, just as Henry’s law must then apply to the solute (the minor component). One of the properties of an ideal solution is that there is no heat effect upon mixing the components. Another property is that the volumes are additive. (I.Soln ) VA+B =VA +VB Let vA and vB represent molar volumes of liquids A and B, and let V represent the volume of the solution of A and B. Then, for an ideal solution comprising nA mol of A and nB mol of B, 7/10/07 3- 68 (I. Soln.) V = nA vA + nB vB (18) PARTIAL MOLAL QUANTITIES. It is often convenient to have an equation, for non-ideal solutions, similar to equation 18, by which the total volume may be ascribed to the “effective” contributions of the two components. These effective molar volumes (further defined below) will be indicated by V A and V B . Then V =nAVA +nBVB (19) for any solution of two components, A and B, whether the solution is ideal or not. The equation may be extended to any number of components. If the number of moles of A is held constant and more B is added to the solution, how much will the total volume increase? Assume VA and VB are not significantly changed.4 Then dV = d (nAVA )+ d (nBVB ) but we have specified that nA is constant, so (nA) dV =VBdn and the partial molal volume is defined as  ¶V   B  n A = V B (20) The partial molal volume (or partial molar volume) is the effective volume per mole. A graphical interpretation of the partial molal volume is that it is the slope of the plot of total volume against moles of the component added, at constant temperature and pressure, with the amounts of all other components held fixed (Figure 3). Even for an ideal solution, we cannot write an equation for free energy comparable to equation 18. G … n1G1 + n2G2 + n3G3 + ... because the mixing process introduces an increase of entropy that decreases the free energy. It is particularly important, therefore, to define an effective, or a partial molal free energy. For the ith 4 It will be shown later that even this restriction is not important, because the changes would be compensating. 7/10/07 3- 69 component of a solution, G =¶  (21)  i T,P,n This definition shows that the effective free energy per mole of the ith component is the rate of change of the total free energy of the solution upon addition of a small additional amount of the ith component, holding temperature, pressure, and the amounts of all other substances constant. As in equation 19, one can write, for any solution, G = n1G1 + n2G2 + n3G3 +××× (22) The partial molal free energy includes correction terms for the heat of mixing as well as the entropy of mixing. One should therefore employ the partial molal free energy, or the “effective free energy per mole”, in all equations. For a pure substance the partial molal free energy is identical with the molal free energy. The significance of the partial molal free energy in chemical thermodynamics cannot be overemphasized; it is the necessary key to the treatment of all solutions, whether solid, liquid, or gaseous. And, inasmuch as it includes pure materials as a special case, it provides a single function for the thermodynamic treatment of all substances. A necessary condition for physical equilibrium is that, for each substance, the partial molal free energy must be the same in every phase. Furthermore, chemical reactions can occur only in a manner that will minimize the total free energy (for the isothermal process), which is a sum of partial molal free energies, each multiplied by the number of moles (equation 22). Thus Gi is a measure of the potential reactivity of the ith species. Because of this particular importance, the partial molal free energy is often called the chemical potential, and given the symbol μ . Chemical potential Gi = mi The free energy of a solution depends on the number of moles of each of the constituents, on the temperature, and on the pressure. Therefore the change in free energy arising from changes in any of these quantities can be written as the sum dG=¶Gd 1 +¶Gd 2 +¶Gd 3 +×××+¶GdP+¶GdT 1 2 3 It is to be understood that each of the partial derivatives is evaluated with the other variables held constant. Substitution of equation 21 and of equations 22 and 23 of Chapter 2 puts this equation into the form dG =G dn +G2dn2 +G3dn3 +×××+VdP SdT (23) This is a more general form of equation 21, Chapter 2. 7/10/07 3- 70 ... - tailieumienphi.vn