chemical principle1 revbited
Edited by MURIELBOYDBISHOP Clernson Unlvenlty Clemson. SC 29631
I. The Thermodynamic Equilibrium Constant Adon A. Gordw The University of Michigan, Ann Arbor, MI 48109
In this of chemical .... ~series ~ ~ - - eieht . - ~on ~ - - eouilibrium we " .articles examine how various assumptions, sometimes unstated, result in the simolified eouilihrium eauations found in most introductory texts. In some cases we begin with rigorous thermodvnamic exnressions and show the conditions under which the approximations are valid. In other cases we introduce new methods of evaluating the approximations. This series of articles consists of the following! I. The ThermodynamicEquilibrium Constant 11. Deriving an Exact Equilibrium Equation III. A Few Math Tricks IV. Solutions of Weak Acids or Bases V. Seeing an Endpoint in Acid-Base Titrations VI. Buffer Solutions VII. pH Approximations in Acid-Base Titrations VIII. Precipitates ~
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In this article we consider the general nature of the equilibrium constant: K, Le Chatelier's principle,' and the effect of temperature on K. The Equlllbrlum Constant For the specific case of a chemical reaction consisthg of ideal gases the thermodynamic equations for the Gihbs free energy result in a simple expression for chemical equilibrium (the equilibrium constant). The equation is:
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AGO -RT lnK
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Journal of Chemlcal Education
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AG"
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-RT InK + 2bN0,(1.00atm - PNOJ + ~ N ~ O , ( P-N1.M) ~ O aim) * (3)
where K is the same as ineq 2 if the standard state is defined as 1.00atm. If the eases ohevthe van der Wads eouation. the final AGO equation is eveLmessier. And, for substances or mixtures that deviate greatly from nonideality (compared to gases, ionic solutions are extremely nonideal), the equilibrium expression for AGO is even more of an algehraic mess. Therefore, to maintain the simple form of eq 1, a new term, activity, assigned the symbol a, is defined? and the thermodynamic equilibrium constant is written in terms of activities. For example, the equilibrium constant for the reaction: AgCl(s) Agt + C1- would be written as K =
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a~~+ac~-la~,ct.
The next step is to specify how to calculate activities. We consider four types of chemical species: gases, nonionic solutes, solids (e.g., precipitates), and ionic species.
(1)
where AGO is the standard-state Gihhs free energy change for the ideal gas reaction at the chosen absolute temperature, T,with R the gas constant and K the thermodynamic eouilihrium constant. for which values are listed in handbooks and texts. The thermodvnamic eauilibrium constant for an ideal gas reaction involvis terms ihat are ratios of pressures: PiPO where PO is the standard-state pressure, defined2as PO 1.00 atm. For instance, K for the idea-gas reaction: 2N02(g) NnOa(g) is: PNpjl.OOatm K (PNo,ll.OOatm)2 Because the equilibrium constant is dimensionless, pressures2 must be given in the same units as the standard state, which is usually chosen as atmospheres. If the reaction involves real (i.e., nonideal) gases or, for example, ionic species, the equation relating AGO to measured equilibrium properties (pressures of gases or concen-
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trations of ions) becomes very complicated (if it can even be solved) and the simple form of eq 1does not apply. For example, if the gases are only slightly nonideal so that they obey the equation: P(V nb) = nRT, where b is the volume per mole of gas molecules, then the Gihbs free energy expression for the 2NOz N204equilibrium becomes:
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There is some ]ustiflcationIn avoldlng dlscusslon of the vaguely worded Le Chateller prlnclple, as has been discussed by various authors (see for example: Kemp, H. R. J. Chem. Educ. 1987, 84,482484, and references therein), and Instead use a thermodynamic approach. We have done both because most texts still refer to the Le Chateller prlnclple. The standard state in Si pressure unltsof Pascal isusually deflned as 1.00 bar = lo5 Pa and the pressures are also glven In bar. The differenceIs mlnor because 1 atm = 1.01325 bar. 3 AS is pointed out in some physical chemistry texts, the Important step In the derlvatlon of eq 1 involves evaluating the Integral of VdP for each specles. Because V = nRT/Pfor an Ideal gas, the derivative becomes nRTd In P. To malntaln the simple form of eq 1 for nonideal substance8 there Is defined VdPF nRTd In at0 parallel the nRTd In P expression for an ideal gas. In lntegratlng this equation, an activlty ratlo, aleo, Is obtained, where the actlvlty In the denomlnator, aO,is called the standard-state actlvlty and is deflned as = 1.00. Thls Is slmllar to the ldeal-gas pressure ratlo: PIP = PI 1.OO atm. For Ions, a1 eO= [XI fxll.OOwhere f, is the activity coefficient for Ion X and 8 Is a hypothetically Ideal 1.00 M solutlon of the ion. The number 1.00 is always chosen because It, In effect,drops out of the equatlons.
Gases Most gases around room temperature and a t pressures up to a few atmospheres deviate only slightly from the ideal gas law so that th& pressure (in atm) can be used in K. Nonionic Solutes Most nonionic solutes such as undissociated weak acids or bases. water-soluble organic comoounds such as suear, ethvl alcohol, etc., behave n&ly ideaify, often at conce&r&ok u n t o 0.1 or even 1.0 M, so that their molar concentration can bk used directly in K. Pure Sollds or Liquids Pure solids (precipitates, for instance) or pure liquids are a special category for which i t is generally possible to calculate the activity. If the volume per mole, V, of the pure solid or liquid is assumed to be independent of pressure, then:
V(P- 1.00 atm) = R T Ln (aho)
For instance, consider a solution that is both 0.015 M Na9HPOAand 0.010 M NaqPOA.The nrincioal eauilibrium ;&be described in terms ofthe third step in 'the di'ssociation HIPOa . . for which the thermodv" . (.H P O i - H+ POL3-) namic K value is:
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For this solution, the extended Debye-Hiickel equation 2 -0.345, SO results in f ~ =+0.824, f p o p = 0.0911, and f ~ p 0 ~ = that = 0.218 and K3< = K3/K3/ = 4.80 X 10-13/0.218 = 2.20 X 10-12. Although there is some degree of cancellation of the f terms in Kt, in general, Ki # 1.00 for most ionic solutions so that precise equilibrium calculations mlcst take into account nonideal ionic effects. Certain equations, however, include coefficients how nonideal do -.not - - - ~ - activitv ~ ~ ~ no matter ~ ~ ~ solutions may he. Mass balance and charge balance expresr sions are two examoles. The reason is that activitv is defined to maintain the simplicity of the ideal-gas thermodynamic eouilibrium-constant eauation and. therefore, actiuitv . coef. ficienrs occur only in e&ilibriurn constant eipressions. Almost all veoeral chemistrv texts consider solutions to be ideal and the concentrationi calculated from equilibrium ex~ressionscan often be in error bv as much as 10-100% or mire. In subsequent articles we will usually assume that solutions are ideal, although we will occasionally show how nonideality can beincluded in equilibrium calculations. ~~~~
(4)
where a0 is the standard-state activity of the pure solid or pure liquid and is defined as = 1.00; the standard-state nressure on the solid or liauid is defined as 1.00 atm. The pressure independence of volume is usually a very eood aooroximation. At room temDerature and 100 atm the ;olume bf a typical solid or liquid differs by less than 1% from the volume a t one atm; water, for instance, is about as incompressible as is steel. As an example, consider the case of AgCl(s). Its density is 5.56 g/mL, and its molecular weight is 143.3; therefore V = 25.8 mL/mol. Shown in the table are the values of the activity of AgCl(s) for various pressures calculated using eq 4. Adlvlty of Solld AgCI at 288 K and Varlous Pressures
As seen in the table, the calculated activity of AgCl(s), which would be typical of other pure solids or pure liquids, begins to differ markedly from 1.00 only for very high pressures. As a result, the activity of a pure solid or liquid, for pressures up to a few atmospheres, can be approximated as equal to 1.00 in the K expressions. For this reason, the terms corresponding to solids are omitted from solubility product equations. They are present but equal to 1.00. However, for the K equation to he valid, there plust be some pure solid present. (Of course, appreciable impurities in the solid will affect the activity of the solid.) ionic Species Ionic solutions show the greatest deviations from ideality, even a t concentrations as low as 0.001 M. For any ion, X, the activity is defined as a x [Xlfx, where the concentration is given by molarity and fx is called an activity coefficienk4 i t is . that dimensionless. Thus, for H+ ions, aH+ = [ H f ] f ~ +[Note ] a result, the the precise definition of pH = -log a ~ + . As activity terms in the thermodynamic equilibrium constant can be separated into a quotient, Q,based on the concentration terms and another quotient, Ki, based on activity coefficients so that: If a solution is assumed to be ideal. then all f terms as well as Kl equal 1.00 and K = Kc, which is implicitly assumed in most introductorv texts. There are various theoretical expressions for f v&es of ions and these include the extended Debye-Hiickel equation and the Davies e q ~ a t i o nIn . ~general, the greater the charge on an ion or the higher the ionic concentration of the solution, the greater the degree of nonideality of the ion.
Lo Chateller's hlnclple
The principle of LeChatelierl states that when a chemical system a t equilibrium is subjected to a stress, the reaction will reequilibrate in the direction that will relieve that stress. The principle is most easily understood by examining the reaction quotient, Q, for a chemical reaction. The quotient has the same form as the eauilibrium constant with concentrations or pressures of readants (terms in the denominator) and nroducts (terms in the numerator) each raised to a pow& based on the stoichiometry of the balanced equation, as thev are in K. However, Q is not necessarily numerically e q u a l t o K since Q represents the actual, not necessarily equilibrium, concentrations and pressures."f Q = K, then the reaction is a t equilibrium. If Q < K, reequilihration must occur in the direction that will cause Q to become equal to K. This will happen when the numerator (product terms) increases numerically while the denominator (reactant terms) decreases: this corresoonds t o a reeauilibration wherebv some of the reactants are converted to products. ~ o n v e r s e l j , if Q > K, the numerator (product terms) must decrease while the denominator (reactant terms) must increase; this corresponds to a reequilibration whereby some of the products are converted toreactants. The only terms that are uneffected, if present in Q and K, are activities for solids or pure liquids because their activities equal 1.00 as long as there is some solid or pure liquid present. 4Unfortunately,there Is no way experimentally to measure single f values for ions. The reason is that you cannot create a solution having only a single ionic species. All ionic solutions are electrically neutral and have both positive and negative ions. Thus, any measure of the deoree of nonidealitv of an ionic solution will aive information onlv on " the (weighted)average activihl coettic~ents01 the variom ons in the solution. These act vity coefficients are ohen listed tor use with concentrations in molaiity: they can be convened to f va Jes tor Lse with moiarities. These equations are usually described in undergraduate physical chemistry texts. For example, K = 7.01 at 25 OC for the 2N02(g) N,O,(g) ,, equilibriumgiven by eq 2. If amixture is prepared sothat inltially & = 0.200 atm and hOI = 0.300 atm, then Q = PN20,1(PNoJ2 = 0.2001 (0.300)2= 2.22.
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Volume 68 Number 2 February 1991
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For example, in the ionization reaction of acetic acid: HAc H+ Ac-. the addition of some NaAc to an eouilihrium mixture resuits in excess [Ac-I, so that temporarily Q > K and some of the products are converted to the reactant. The addition of NaOH, which will react with H+ and thus decrease its concentration, will result temporarily in Q < K so that, t o reestablish equilibrium, some of the reactant must dissociate and produce products. The exception to this general rule exists for the case in which there could be precipitation. If Q is less than the solubility product, K,, and if there is no solid present, then reactant cannot be converted into products because there is no reactant available. However, if Q is greater than K,, then products can be converted into reactant and some solid precipitate will form.
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Volume Changes A "stress" can also he applied by changing the volume of the reaction mixture, and the effect can also be determined by comparing the values of Q and K. The net effect in reestablishing equilibrium is always in the direction toward the side having the larger numbeiof free particles when the volume is increased or, conversely, toward the side havine the smaller number of free particles when the volume i;J decreased. For instance, if an equilibrium solution of HAc is diluted with water (i.e., the v o l b e increased), the reaction H+ Ac- will proceed in the forward direction HAc because there are two particles as products, H+ and Ac-, but only one as reactant, HAc. We can show this by defining the original equilibrium concentrations as [H+]., [Ac-I., and [HAc],. When the solution is diluted, the "temporary concentrations" are x[H+],, x[Ac-],, and x[HAc]., where x is a number less than 1.00. Thus, Q = (x[H+].)(x[Ac-].)lx[HAc], = xK. Because x is less than 1.00, Q < K and the direction of the reequilibration is from reactants to products. (If the number of free particles is the same on both sides of the equation, the x terms cancel, Q = K, and there is no need to reestablish equilihrium because of a change in volume.) This effect of a chanee in volume also avolies .~ .. to reactions involvinggases. An increase in volume of the reaction vessel results in the reeauilibration ~roceedinein the direction toward the side having the larger number ofgaseous molecules, and vice versa. This direction of shift in the acetic acid equilibrium may appear t o contradict what would be suggested intuitively. After all, dilution of an acid with water decreases the acidity,
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'There are possible exceptionsto this general rule for the effect of heat, but there are no common examples. The exceptions would occur in cases where A@ is a very, very small value, so that a temDerature increase would result in onlv a vew small chanae -- in Kbut woild be more than compensated for by (1) fbr lone reactfons,the d 1111 on effectof the volume expansion of the solurlon or (2)for gasphase reactions. the pressure Increases that result fromthe temperature increase. An "apparent" inconsistency in this rule would appear to exist for the case of the addltion of a substance such as solid NaOH to water in which heat is generated when the NaOH dissolves. This suggeststhat heat is a product and that the amount of NaOH that dissolves should decrease as the temoerature is raised. As evervone knows. - ~ ~the-- . solubility of most sub&nces, including N ~ O Hincreases ~ with temperature. The inconsistency in the reasoning is related to the factthat two separateprocessesare being muddled inthe "soiub1lity"description given here. What one needs to do is to examine the process whereby solid NaOH is in contact with a saturated solution of NaOH that is in eguilibrium with Naf and OH-. This reaction is endothermic. andan increase in temperature does result in an increase in solubility. The soiubilitv of NaOH is discussed bv Bodner. G. M. J. Chem. Edoc. 1980, 57, ti7-119, whereas the general concept of the~effectof temperature on solubility Is discussed by Brice. L. K. J. Chem. Educ.
[H+],of the solution and eventually, with sufficient dilution, the solution simply takes on the properties of water and the pH approaches 7.0. This is true. What the Le Chatelier principle says is that i t is the total amounts of H+ and Ac-in solution that increase while the total amount of HAc decreases. The concentrations are another matter. When HAc is diluted, all three concentrations decrease, but after reequilibration the H A c concentration has decreased more than the H' and Ac- concentrations. Inother words, there is a greater degree of ionization of HAc. For example, 0.100 M HAc is about 1.3% ionized. whereas 1.00 x M HAc is about 34.4% ionized. Tmperatwe Changes
Another tvoe of stress that can be applied to a chemical reaction is change in temperature. a e general rule that can be followed in evaluating the Le Chatelier stress is to think of heat as a reactant (ifihe reaction is endothermic) or a product (if the reaction is exothermic). For instance, if the teinerature of an endothermic reaction initiallv a t eouilibrium is increased, then some of the reactants will be converted t o ~roducts.This nhenomenon can be justified bv examining the thermodynamic equation that relates the equilibrium constant to the absolute temperature. T. and the heat of ~hV reaction, AH? The nnintegrated-form is: d ( k ~ ) / =d A R P or, by approximating derivatives as difference terms, A(ln K)IAT w AHQIRP. If the reaction is endothermic, AHQ is positive, and the derivative is also positive; therefore, an increase in T (i.e., AT is positive) results in A(ln K) = a positive value, which is the same as an increase in K, requiring a decrease in reactants and an increase in products. ~ f t h e reaction is exothermic, AHQ is negative, and the reverse
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Any reaction that involves the absorption or release of heat will have an eauilihrium constant that changes in value as the temperature is changed. If the (standard-state) heat of reaction, AHQ, is assumed constant in the temperature range of interest,then the integrated form of the expression relating log K to M and the ahsolute temperature T is:
Consider the case of the ionization of water, which is an endothermic reaction: heat H20 H + OH-. Using the handbook value of AHQ = 56.08 kJ/mol for the temperature range 28a313 K (15-40 *C) and the value of K, = 1.00 X 10-l4 a t 298 K, you can solve for the constant C in eq 7. The resulting equation is:
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Because the ionic strength of pure water is very low, the solution can be considered ideal and, because [H+] = [OH-], we have pH = pOH = (1/2)pKw = -(1/2)log K, so that the analogous expression for the temperature dependence of the K) is: pH of pure water (in the range 2%3-13
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1983, 60.387-389.
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Journal of Chemical Education
Because the water ionization reaction is endothermic, an increase in temperature causes the reaction to proceed to the right, as predicted by Le Chatelier's principle, and results in higher H+ and OH- concentrations. The pH of a neutral solution, therefore, decreases as the temuerature is increased. For instance, at 37 "C = 310 K (hod; temperature), K , = 2.4 X 10-14 according.toea 8, and the pH of pure water, according to eq 9, is 6.81.
