Le Chatelier's Principle: The Effect of Temperature on the Solubility of Solids in Liquids L. K. Brice Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 The purpose of this article is to provide a rigorous but s t r a i g h t f o m d thermodynamic treatment of the temperature dependence of the soluhility of solids in liquids that is suitable for presentation a t the undermaduate level. This subiect is given only cursury treatment in rurrenrly available physical chemistry texts. though detailed disrussims can br: found in more advanced treatisis (1,2) and in an article by Williamson (3). hut in a form too sophisticated for presentation at the &nentary level. The preient discussion may also suggest how to approach better the qualitative aspects of the subiect for freshmen. We first consider the qualitative application of Le Chatelier's Principle t o the effect of temperature on the solubility of NaN03, NaCzH302, LizS04 and NaOH in water. Solubility curves for these compounds are shown in Figure 1 (4). For NaN03 and NaC2H302, the solid phase in equilibrium with the saturated solution is the anhydrous salt. Sodium nitrate forms no hydrates. Sodium acetate forms a hydrate, NaC2H30y3Hz0, that is thermodynamically more stable than the anhydrous salt below 58°C. However, sodium acetate solutions remain supersaturated with respect to the hydrate even in the presence of anhydrous NaCzH302, so that the solubility of the latter can be measured between O°C and 58OC (5,651.For lithium sulfate and for sodium hydroxide, the solid vhase in eauilibrium with the saturated solution is the monohydrate. Molal heats of solution (AH,) of NaNO-. NaC7H107. in Li2S04.H20,and NaOH.H20 at %"cand 1a& are Table 1 and are shown graphically in Figure 2. (The AH, values were obtained by interpolation of molal heats of solution calculated from the heats of formation of the-corresponding solutions (7): AH, = m(AHof(,,l,ti,,,,) - AHof(,.lt) ~ A F f ~ f ( ~ ~where o , l ) )u, is the number of water molecules of hydration per molecule of salt.) The molalities of the saturated
solutions ( m , d are indicated by a n arrow. For each solute, the slope of the curve, ( a A H , l a m ) ~ , ~at, mmt is equal to the differential heat of solution of the solid solute (component 2) in the standard solution, Aff~(,,~), which determines the sign of (a in mlaT)p. The limiting slope of the curve at m = 0 is equal to the standard heat of solution of solid solute in the infinitely dilute solution, AR02, which determines the sign of (a h K , l aT)p, where K, is the thermodvnamic equilibrium constant for the prtaess(srt. beluw).'l'he valueof ; A H , ml at m,,,, is the iotrrral hcilt d s o l u r i m uf the solute in thesdturatrd fiw lution, Table 2 shows a comparison of the algebraic sign of the temperature coefficient of solubility with those of various enthalpy changes described above for water solutions of the compounds listed earlier. Table 3 gives the values of these Table 1. Molal Heats of Solution in Water at 25% in kJ mol-' (AH-)
+
I
0 0
20
30
40
50
60
70
TEMPERATURE(T1 Figure 1. Solubility curves for water solutions. rn = molality of the saturated solution. The value of "a" designated for each compound must be added to the numbers along the vertical axis to obtain In m.
i Figure 2. Molal heats at solution in water at 25% solutions.
Volume 60 Number 5
Arrows indicate saturated
May 1983
387
Table 2. Comparison of Algebraic Slgn of Temperature Coefllcient of Solubility with Those of Various Enthalpy Changes For Water Solutions at 25'C solute
I.+
(a miaqp
AM^^,,
+
+-
+-
-
-
NaNO3 NaC2H302
-
Li2S04-H20
+
NaOH-H.0
4
A%lnaf]
+ + -
+
macsatj
= E2 - E,, the partial molar (or differential) where heat of solution of B in the saturated solution. The coefficient, ( a p 2 / a m ) ~can ~ , be related to the vapor pressure of the solution by use of the Gibbs-Duhem equation. We have at constant T and P, nidfil+ nzdfi2 = 0 (5) where n l and nz are the numbers of moles of A (the solvent) and of B in the saturated solution, respectively. If nz = m, n l becomes 1000lM1, where M1 is the molecular weight of the solvent. Division of eqn. ( 5 ) hy dm together with these suhstitutions yields
=--(-Iapl 1000
Table 3. Enthalpy Changes for Aqueous Sodlum Hydroxide Solution at 25°C NaOH.H20(s)+ 0.95 HzO(l) NaOH(x, = 0.339)' + 1.95 HIO(x, = 0.661) (2) NaOH.H20(s) NaOH(x2 = 0) + H2O(a, = 1) (3) NaOH.HzO(s) NaOH(x2= 0.339)* (1)
-
-
+ H,O(xl
-
am
T,P
mM, am
(6)
T,P
For the solvent
AH,,,,^ = -2.11 kJ mal-' A*, = -22.8 kJ mol-' = 8-58kJ mol-'
= 0.661)
me mole traction of ~aoHinan aquaavs solution of sodium hydroxidematis saturated w i n respect to NaOH.H20 at 25'C is equal to 0.339.
= pa, + RT inpdp'l (7) where p l and p o l are the vapor pressures' (or more precisely, the fugacities) of A over the solution and over pure A, respectively. combination of eqns. (31, (4), (61,and (7) yields a in m MI ARZ,~,~, (8) = - l o o n p / ~ , pn i z If the saturated solution is dilute, p l = p o i x l , where x l is the mole fraction of A in the solution, so that (a l n p l l a m ) ~ , p = -l/(lOOO/M1 m) -M1/1000. Equation (8) then reduces = m.2 and a 2 = m for a dilute solut o (2) since LG~~,,~, tion. If the solution is not dilute, eqn. (8) must be used. The term (a In p l l a m ) ~is, ~always negative since the vapor pressure of the solvent decreases as the solute concentration increases at constant T and total pressure P (8).The sign of (a in m/aT).~ is therefore determined by the sign of the differential heat of solution, in accordance with Le Chatelier's Principle. As seen in example~bove,the sign of AH2(,,t, sometimes differs from those of AH02 and AHI(,,~).
1 (MP
(
+ --
changes together with the corresponding thermochemical equations for aqueous sodium hydroxide solutions. From Table 3 we may conclude that the thermochemical equation describing the equilibrium between NaOH.HzO(s) and its saturated water solution at 25'C is NaOH .HzO(s)+ 8.58 kJ
NaOH, Hz0 (sat. saln.)
~1
Unsolvated Solutes To obtain the quantitative relationship between (a in m l aT)p and AE2(,,t, for unsolvated solutes, we consider the equilibrium between pure solid B (component 2) and its saturated solution, B(s) a B(a2 = ~ 2 m )
(1)
For this equilibrium, K , = a2 = yzm, where a2 is the activity, y 2 the activity coefficient and m the molality of B in the saturated solution. The temperature coefficient of K , (or a2) is given by van't Hoff's equation,
Solvated Solutes If the solid phase that is in equilibrium with a saturated solution of B in A is a solvate, B.vA, we must consider the equilibrium, B .uA(s) s B(a2 = ~ m+ )uA(ad We have K , = alPa2and
-
(9)
where r n - 2 is the enthalpy change for the process, B.vA(s) Bim = 0) vA(a, = 1).i.e.. the heat chance that occurs
mo2
where is the enthalpy change for the process, B(s) B(m = 0). T o obtain (a in m l a T ) ~we , uroceed as follows. If small rewr.il)le ihiinyc. uccur in t h tvmperarorc ~ d t h e equiIil)rl~m 116 a.rihed I)\, r l I at ,un
