1.3 Saturation vaor ressure Increasing temerature of liquid (or any substance) enhances its evaoration that results in the increase of vaor ressure over the liquid. y lowering temerature of the vaor we can make it condense back to the liquid. These two hase transitions, evaoration and condensation, are accomanied by consuming/evolving enthaly of transition and by a change in entroy of the material. elow we will consider water vaor as an examle of the vaor hase. ll conclusions we will get for water vaor are true also for any vaor. We will use water vaor not only for the reason that it is the most familiar vaor hase but also that it lays a major role in the cloud formation and reciitations. 1.3.1 Vaor ressure bove the surface of liquid water there always exists some amount of gaseous water and consequently there exists a vaor ressure. When a container containing water is oen then the number of the escaing molecules is larger than the number of molecules coming back from the gaseous hase (Fig. 5.1). In this case vaor ressure is small and far from saturation. When the container is closed then the water vaor ressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig.5.2). fter some time, the number of molecules escaing the liquid and that coming back becomes equal. Such situation is called by dynamic equilibrium between the escaing and returning molecules (Fig. 5.3). In this case, it is said that the water vaor ressure over the liquid water is saturated. Fig.5. Evaoration and condensation of water molecules in oen and closed container. (1) In the oen container, when gas is under-saturated, number of the evaorating molecules is larger than the number of molecules coming back from the gaseous hase. In this case water vaor is unsaturated. (2) When the container is closed then the number of the returning molecules increases. (3) fter some time, when dynamical equilibrium between the escaing and returning molecules is established, the water vaor ressure becomes saturated. Saturated water vaor ressure is a function of temerature only and indeendent on the resence of other gases. The temerature deendence is exonential. For water vaor the semi emirical deendence reads as
+ + C lnt + Dt T, s =, (1.49) w e where temerature is in Kelvin and = 77.34, = -7235, C = - 8.2, D = 0.005711. Evaoration and condensation are the first order hase transitions which are accomanied by a change in the degree of molecular arrangement of water. The change in the degree of molecular arrangement is necessarily accomanied by a change in entroy, S. In the atmoshere, condensation (formation) of cloud and rain dros and their evaoration occur at constant ressure. Therefore seaking about the latent heat of evaoration/condensation we should remember that we deal with enthaly change, H va/cond, because Q = H at constant ressure. Evaoration is an endothermic rocess because heat (enthaly change) is consumed i.e., H va > 0. Condensation is an exothermal rocess because heat is evolved i.e. H cond < 0. The entroy change during the evaoration can be calculated as S va = H va /T tr, where T tr is temerature of transition (evaoration). ecause H va > 0, we obtain that the S va > 0. This means that during evaoration the entroy increases, because the degree of molecular order decreases when liquid water transforms to gaseous hase. During condensation, the entroy decreases, S va < 0, because the degree of molecular order increases. For any liquid, vaor ressure and enthaly of evaoration H va at temerature T tr are connected by the Claeryron equation, d dt Sva H va = =, (1.50) V T V va tr va where we used the definition of the entroy change of S va = H va /T tr (see above). ecause molar volume of a vaor V m (v) is much larger than that of a liquid V m (l), we can assume that volume change during evaoration V va = V m (v) - V m (l) is mainly due to the change of volume of the vaor itself i.e., V va V m (v). Moreover, if the vaor behaves erfectly, then for one mole of the vaor we can write V = RT/ (here we omitted a subscrit m and tr). Substituting in (1.48) the volume change V va by RT/ and sssuming that the enthaly of evaoration H va is indeendent of temerature, the exact Claeyron equation (1.50) can be rearranged into the Clausius-Claeryron equation d ln dt H va =. (1.51) 2 RT The equation (1.51) integrates to give H va 1 1 ( ) R T T = e, (1.52) where * is vaor ressure at temerature T* and vaor ressure at temerature T.
For water, the assumtion that the enthaly of evaoration H va is indeendent of temerature is not true. For ure water the H va deends on T exonentially and instead of (1.50) we have the equation (1.47). 1.3.2 ir humidity mount of water vaor in the air can be exressed by several different ways: mh 2 O Secific humidity: Mass of water vaor in unit mass of humid air: mair bsolute humidity: Mass of water vaor er unit volume of humid air (kg/m 3 ) Relative humidity (RH): Ratio of water vaor ressure w to the saturated water vaor ressure at that temerature multilied by 100%, w RH = 100%. (1.53) ure ( T ) s, w Saturation ratio, S: Ratio S = w ure. (1.54) s, w ( T ) From the last two definitions we see that RH = S 100% i.e., their hysical meanings are almost the same. Suersaturation: S - 1 > 0. Dew oint (kasteoiste) is temerature at which relative humidity of cooling air becomes 100% (Fig. 2.3). ure Fig. 2.3 Curve SS` deicts the saturation ure water vaor ressure w, s as a function of temerature T. If we take an air samle at temerature T o with vaor ressure corresonding to
oint X then its due oint is T d. Relative humidity at oint X is XW 100%. The value of YW ressure deicted by XY is called by saturation vaor ressure deficit. When a warm humid air and a cold dry air arcels are mixed then the vaor ressure w and temerature of the resulting air arcel can be reresented as a linear combination of their initial values. Since the saturation water vaor ressure varies exonentially with temerature, the mixed air may have humidity above 100%, although the initial arcels were under-saturated, Fig. 2.4. s a result a mixing cloud is formed. This exlains the formation of a cloud of a fog after breathing in cold air. Exhaled air may reach RH u to 200 %., ure w s Fig. 2.4 When two air arcels, warm humid N and cold dry S, are mixed then so called mixing cloud is formed in the region NL, where water vaor ressure is above the saturation curve. When two air arcels S and N` are mixed there is not mixing clouds. 1.4 Vaor ressure over solutions esides the laboratory samles, ure liquids are rarely encountered in nature. Usually liquids are mixed (if they do not react together) or contain some dissolved substances, for examle, salts. dissolved substance is called solute and the liquid solvent. The mixture itself is called solution. Vaor ressure of a comonent of the solution (for examle, solvent) over the surface of the solution is smaller than that over the ure comonent. This is due to the fact that the solute molecules at the surface of the solution revent escaing solvent molecules into the vaor hase, but do not hinder their return. elow we will consider the solution that is in equilibrium with its vaor. This means that chemical otentials of the liquids and vaors are similar i.e, µ vaor = µ liquid. (1.55) We will consider a two comonent (binary) solution of a liquid and a liquid. The chemical otentials of the ure liquids will be denoted by µ* and µ*, and the chemical otentials of liquids in the solution by µ and µ. The vaor ressures over the
ure liquid will be denoted by * and * and the artial vaor ressures over the solution by and. Since at equilibrium, the chemical otentials of the liquid and its vaor are similar we can write (see Eq. 1.40) for the chemical otential µ* of the ure liquid, µ* = µ + RT ln ο, (1.56) and for the chemical otential µ of the liquid in the solution, µ = µ + RT ln ο. (1.57) ο In the Eqs. (1.56) and (1.57), the ressure is the standard ressure of 1 bar over the ure liquid (see 1.41 and 1.42). Combining (1.56) and (1.57) to eliminate the standard chemical otential µ and o we can write that the chemical otential µ of the liquid in the solution is µ = µ* + RT ln. (1.58) Similar result can be written for the chemical otential µ of the liquid in the solution, µ = µ* + RT ln. (1.59) 1.4.1 Raoult s law and solvent activity a) Ideal solution: The exression (1.58) shows that the chemical otential of the liquid in the solution increases with increasing the artial vaor ressure and becomes equal to the chemical otential of ure liquid when = *. The French chemist F. Raoult exerimentally found that the ratio of the artial vaor ressure of each comonent to its vaor ressure as a ure liquid is aroximately equal to the mole fraction X in the solution i.e., = X or = X *. (1.60) The Eq. (1.60) is known as the Raoult s law. Linear deendence of the artial vaor ressures and are shown by the dashed lines in Fig. 6.
Fig. 6. Equilibrium artial ressures of the comonents of ideal and nonideal binary solution as a function of the mole fraction X. For the real solution, relationshis between the,, and the mole fractions X, X are not linear as it is in the case of ideal solution where the relationshis are linear what is shown by the straight dashed lines LR and KQ. When X 1, then we have a dilute solution of in. In this region the Raoult s law = X * is alied for the comonent. For the comonent the Henry s law = K X is alied (see below). In the region where X 0, we have the Raoult s law = X * for comonent and the Henry s law = K X for comonent. The straight lines LM and KN deict the Henry s law (Seinfeld and Pandis, 1998). Using (1.60) we can write for the chemical otential µ of the liquid in the solution (instead of 1.59) the exression µ = µ* + RT ln X. (1.61) The solutions, which obey the Raoult s law (1.60) throughout the whole comosition range from ure to ure, are called ideal solutions. The solutions, which comonents are structurally similar, obey the Raoult s law very well. In Fig. 6, the equilibrium artial ressures of the comonents and in the ideal solution are shown by the straight dashed lines LR and KQ. This shows that solution is only ideal if (1.61) is satisfied for each comonent. Dissimilar liquids (large difference in the liquid structures) significantly deart from the Raoult s law. Nevertheless, even for these mixtures, the law is obeyed closely for the comonent in excess as it aroaches urity i.e., when X 1 or X 0 (Fig. 6). b) Real solution: The Eq. (1.61) is the exression for the chemical otential of the solvent in the ideal solution. It is reasonable to reserve the form of the Eq. (1.61) also for the real solution because then the deviations from the idealized behavior will be seen most simly. Similar to the case of the fugacity introduced for the real gas, the form of
Eq. (1.61) can be reserved also for the solution that do not obey the Raoult s law, if we introduce a new quantity called the solvent activity a. and write µ = µ* + RT ln a.. (1.62) In chemistry, activity is a measure of how different molecules in a non-ideal (real) gas or solution interact with each other. The solvent activity is a kind of effective mole fraction just as the fugacity is an effective ressure. ctivity is the mole fraction, corrected with a coefficient, described with the equation a = γ X, (1.63) where γ is the activity coefficient, which must be measured for different substances, and X is the mole fraction of solvent. The activity coefficient is a factor by which the value of a concentration of a solute must be multilied to determine its true thermodynamic activity. In solutions, the activity coefficient is a measure of how much the solution differes from an ideal solution. The activity coefficient of ions in solution can be estimated, for examle, with the Debye-Hückel or the Pitzer equation. ctivity effects are the result of interactions between ions both electrostatic and covalent. The activity of an ion is influenced by its surroundings. The reactivity of an ion in a cage of water molecules is different from that in the middle of a counter-ion cloud. The activity of the solvent aroaches the mole fraction as X 1 i.e., and the activity coefficient a X as X 1, γ 1 as X 1, at all temeratures and ressures. t this conditions (when X 1) the solvent activity can be exressed as a =, (1.64) i.e., when X 1 the solvent activity can be determined exerimentally by measuring the artial vaor ressure over the solution and then using Eq. (1.64). If the values of the X, γ, and * are known then one can calculate the artial vaor ressure of the solvent = * γ X. (1.65) Using (1.63) the chemical otential of the solvent can be written as µ = µ* + RT ln X + RT ln γ. (1.66)
1.4.2 Henry s law and solute activity In ideal solutions, both the solvent and solute obey the Raoult s law. ut it had been found exerimentally that for a real low concentration solution (X 0), although the artial vaor ressure of the solute is roortional to its mole fraction X the constant of roortionality is not the vaor ressure of the ure solute * as it is in the case of the Raoult law but a some constant K i.e., = K X. (1.67) This exression is known as the Henry s law. The coefficient of roortionality K is called by the Henry s constant. The Henry s constant is emirical with the dimension of ressure. Value of the K is chosen so that the straight line deicted by (1.67) is tangent to the exerimental curve at X = 0. In Fig. 6, the straight lines LM and KN deict the Henry s law. The Henry s constant is a material characteristic and deends only on temerature. ecause amount of solute is very small the distance between the molecules is very large and, therefore, interaction between the solute molecules can be neglected. Vaor ressure of solute above the solution is determined mainly by the interaction of the solvent molecules with solute molecules. From (1.64) one can obtain molar fraction of solute in solution if vaor ressure and Henry s constant are known, X =. (1.68) K The exression (1.68) can be considered as definition of solubility of solute. It can be treated as follows: If we somehow increase the artial vaor ressure of solute over the solution then amount of solute in the solution increases too i.e., increases its solubility. Dimension of the Henry s law constant can be different. One has to be careful with this. For examle, from the Eq. (1.67), its dimension is [K ] = [Pa]. Dimensionless Henry s law constant is obtained when we, instead of values used in (1.67), will do a relacement: c g (concentration of solute vaor in gaseous hase, [mole/m 3 ]), X c l (concentration of solute in solution, [mole/m 3 ], or molar concentration [mole/l]). When is in Pa and X c l with [mole/m 3 ] then [K ] = [mol/m 3 Pa]. Liquid mixtures in which the solvent obeys the Raoult s law and the solute obeys the Henry s law are called ideal-dilute solutions. Physically the different behavior of the solvent and solute is exlained by their different environment. Environment of the solvent is only slightly modified by the very small number of solute molecules. In contrast, the environment of the solute is very strongly erturbed because the solute molecules are very far from each another. Therefore, the solvent behaves like a slightly modified ure liquid, whereas the solute behaves totally differently from its ure state.
Only when the structure of solute molecules is similar to that of the solvent then the solute also obeys the Raoult s law. The solute activity: In contrast to the solvent activity a which is defined when X 1, the solute activity a is defined when X 0 i.e, when the solution aroaches ideal-dilute behavior. The chemical otential of in the dilute solution obeying the Henry s law = K X can be written as µ = µ* + RT ln K = µ* + RT ln + RT ln X. (1.69) Since both * and K are characteristics of the solute, the first two terms can be combined to get a new standard chemical otential, Then the exression (1.69) can be written as K µ ө = µ* + RT ln. (1.70) µ = µ ө + RT ln X. (1.71) s in the case of the solvent, all deviations from the ideal-dilute behavior of the real solute is most simly exressed by introducing the solute activity a instead of X in the Eq. (1.71) i.e., µ = µ ө + RT ln a. (1.72) ut in contrast to the solvent activity, which can be exressed as a = instead of * we should use the Henry s constant K, when X 1, a =. (1.73) K The solute activity coefficient is introduce in the same way as in the case of solvent activity i.e., a = γ X. (1.74) ecause the solute obeys the Henry s law, = K X, when X 0 we have a X and γ 1 as X 0 (1.75) at all temeratures and ressures. Combining (1.73) and (1.74) we get an exression
= γ K X, (1.76) which allows to calculate the vaor ressure of the solute,, if the values on the right hand side are known. (Comare with (1.65)). Electrolyte solution: The above consideration has been erformed for the substances that do not dissociate into ions during the dissolution. When strong acid, base or salt (HM) dissolve in water they dissociate into ions, H + and M -. Such solution is called electrolyte solution. For examle, in water, salt NaCl dissociates in Na + and Cl -, whereas sugar does not. Fig. 7 shows how dissociation affects water vaor ressure over salt and sugar solutions. In order for the dissociated substance could escae from the electrolyte solution into the gaseous hase, the ions should link together to form the initial substance because evaoration of the ions is of little robability (due to additional attractive electrical force). If in the electrolyte solution, the comonents totally dissociate into ions, then the exression for vaor ressure can be written as s X(H + ) X(M - ) f ± 2, (1.77) where f ± 2 is mean rational activity coefficient, X(H + ) and X(M - ) molar concentration of the totally dissociated substance. Fig. 7. Equilibrium RH above the mixture (sugar in water) and salt solution as a function of Percent Solute Molecules. If the Raoult s low were true then the decreasing of RH would has followed the straight lines (dotted and dashed). Interaction between water and solute molecules decreases the RH even more (solid lines). Salt reduces RH in larger extent than sugar, because it dissociates into ions but sugar does not.