Equilibrium Constants Revisited

Submitted by ChemPRIME Staff on Thu, 12/16/2010 - 15:24

In discussing entropyA thermodynamic state function, symbol S, that equals the reversible heat energy transfer divided by temperature; higher entropy corresponds to greater dispersal of energy on the molecular scale. See also standard entropy. and spontaneity, we have tended to treat chemical reactions as though they had no other alternatives than either going to completion or not occurring at all. It is only when both reactants and products are pure solids or pure liquids, however, that we find this all-or-nothing-at-all type of behavior. Reactions which involve gases or solutions are governed by an equilibrium constantThe value of the equilibrium constant expression when equilibrium concentrations are substituted; a value greater than one indicates the position of equilibrium lies toward products (product-favored), and a value less than one indicates the position of equilibrium lies toward reactants (reactant-favored)., and as we saw in the sections on chemical equilibrium, this means that there is always some reactant and some product in the equilibrium mixtureA combination of two or more substances in which the substances retain their chemical identity.. Consequently such reactions must always occur to some extent, no matterAnything that occupies space and has mass; contrasted with energy. how minute, and can never go quite to completion.

Figure 1 The variation of G with the extent of a reaction is a straight-line plot only when no gases or solutions are involved as in a. When gases or solutions are involved, the plot exhibits a minimum where the reaction has not fully progressed to completion, as in b.

Figure 1 illustrates how the free energyGibbs energy: a thermodynamic function corresponding to the tendency for spontaneous change in a system; represented by the symbol G. G varies as the reaction proceeds in the two cases. If only pure solids and liquids are involved, then a plot of G against the extent of the reaction is a straight line, as shown in part a of the figure for the reaction

Hg(l) + HgBr2(s) → Hg2Br2(s)      1 atmAbbreviation for atmosphere, a unit of pressure equal to 101.325 kPa or 760 mmHg., 298 K

In such a case, if ΔGm°, the free-energy difference between reactants and products, is negative, then the reaction will attain the lowest value of G possible by going to completion.

When gases and solutions are involved in the reaction, a plot of G against the extent of the reaction is no longer a straight line but exhibits a “sag,“ as shown in Fig. 1b for the reaction

2NO2(g) → N2O4(g)      1 atm      (1)

at two different temperatures. In such a case, even though ΔGm° is negative, the reaction will not go to completion but will end up at the lowest point of the curve. If the value of ΔGm° is quite small, below about 10 or 20 kJ mol–1, this results in an eventual equilibrium mixture containing an appreciable proportion of both reactants and products. This is the case for Eq. (1) at 298 K, when the reaction attains only 81 percent completion at equilibrium. A more usual situation is that shown for this same reaction at 200 K. Because the free-energy difference is larger at this temperature, the “sag” in the curve is much smaller, and as a result the minimum value of G lies extremely close to 100 percent completion (actually 99.9 percent). As a rough rule of thumb, therefore, we can say that if ΔGm° is negative and numerically greater than 20 kJ mol–1, the reaction will go virtually to completion; if ΔGm° is positive and larger than 20 kJ mol–1, the reaction will scarcely occur at all. Between the limits ΔGm° ± 20 kJ mol–1, we can expect measurable quantities of both reactant and product to be present at equilibrium. The above discussion suggests that there must be some relationship between the free-energy change ΔGm° and the equilibrium constant. The derivation of this relationship is too complexA central metal and the ligands surrounding it; also called coordination complex. to produce here, so that only the result will be given. For gases the relationship has the form

${\Delta G_{m}^{ o}}={-RT}{ln K^{o}}$       (2a)

or, if baseIn Arrhenius theory, a substance that increases the concentration of hydroxide ions in an aqueous solution. In Bronsted-Lowry theory, a hydrogen-ion (proton) acceptor. In Lewis theory, a species that donates a pair of electrons to form a covalent bond. 10 logarithms are used,

${\Delta G_{m}^{ o}}={-2.303 RT}{ln K^{o}}$      (2b)

Where K° (K standard) is called the standard equilibrium constant. K° is closely related to Kp and differs from it only by being a dimensionless number. Recall from the section on the equilibrium constant in terms of pressure that Kp is defined for the general chemical reaction

aA + bB + · · · $\rightleftharpoons$ cC + dD + · · ·

by the equation

$K_{p}=\frac{p_{\text{C}}^{c}\text{ }\times \text{ }p_{\text{D}}^{d}\text{ }\times \text{ }\cdots }{p_{\text{A}}^{a}\text{ }\times \text{ }p_{\text{B}}^{b}\text{ }\times \text{ }\cdots }$      (3)

etc., are partial pressures. The definition of K° is identical except that it involves pressure ratios (i.e., pure numbers) rather than pressures:

$K\text{ }\!\!{}^\circ\!\!\text{ }=\frac{\left( \frac{p_{\text{C}}}{p\text{ }\!\!{}^\circ\!\!\text{ }} \right)^{c}\text{ }\times \text{ }\left( \frac{p_{\text{D}}}{p\text{ }\!\!{}^\circ\!\!\text{ }} \right)^{d}\text{ }\times \text{ }\cdots }{\left( \frac{p_{\text{A}}}{p\text{ }\!\!{}^\circ\!\!\text{ }} \right)^{a}\text{ }\times \text{ }\left( \frac{p_{\text{B}}}{p\text{ }\!\!{}^\circ\!\!\text{ }} \right)^{b}\text{ }\times \text{ }\cdots }$      (4)

Where p° is a standard pressure—almost always 1 atm (101.325 kPa). Thus if the partial pressures pa, pb, etc., are expressed in atmospheres, K° involves the same number as Kp.

EXAMPLE 1 The equilibrium constant Kp for the reaction

2NO2 $\rightleftharpoons$ N2O4

is 0.0694 kPa–1 at 298 K. Use this value to find ΔGm°(298 K) for the reaction.

Solution We must first express Kp in atmospheres:

$K_{p}=\text{0}\text{.0694 kPa}^{-\text{1}}=\frac{\text{0}\text{.0694}}{\text{kPa}}\text{ }\times \text{ }\frac{\text{101}\text{.3 kPa}}{\text{1 atm}}=\text{7}\text{.03 atm}^{-\text{1}}$

Thus

K° = 7.03      a pure number

From Eq. (2b) we now have

ΔGm° = –2.303RT log K°

= –8.3143 J K–1 mol–1 × 298K × 2.303 × log 7.03 = –4833 J mol–1

= –4.833 kJ mol–1

Note: You may also use ln x as well as log x as long as you use Eq. (2a) directly:

ΔGm° = –RT ln K° = – RT × ln 7.03 = – RT × 1.950 = –4.833

In the section on the molecular view of equilibrium, we argued that the value of an equilibrium constant is the product of two factors:

${K}=\text{energy factor}\times\text{probability factor}$      (5)

The energy factor takes account of the fact that a higher-energy species is less likely to occur in an equilibrium mixture than a lower-energy species, especially at low temperatures. The probability factor reflects the fact that if there are a larger number of ways in which a moleculeA set of atoms joined by covalent bonds and having no net charge. can arrange itself in space, a molecule is more likely to occur in that state than in one for which a smaller number of spatial arrangements is possible.

We are now in a position to make quantitative the qualitative argument presented. Combining

${\Delta G_{m}^{o}}={\Delta H_{m}^{ o}}-{T\Delta S_{m}^{ o}}$

with                 ${\Delta G_{m}^{o}}=\text{RT}{ln{K^o}}$                              (2a)

we find      ${-}\text{RT}{ln{K^o}}={\Delta H_{m}^{ o}}-{T\Delta S_{m}^{ o}}$

giving us          ${ln{K^o}}=\frac{-\Delta H_{m}^{ o}}{RT}$ + $\frac{\Delta S_{m}^{ o}}{R}$               (6a)

or, in base 10 logarithms

${log{K^o}}=\frac{-\Delta H_{m}^{ o}}{\text{2}\text{.303}RT}$ + $\frac{\Delta S_{m}^{ o}}{\text{2}\text{.303}R}$       (6b)

If we now take the logarithm of each side of Eq. (5), we have

$log{K}=log(\text{energy factor})+{log(\text{probability factor})}\,$

This equation has the same form as Eq. (6b), and if we confine ourselves to the standard equilibrium constant K°, we can say that

$log (\text{energy factor})=\frac{-\Delta H_{m}^{ o}}{\text{2}\text{.303}RT}$      (7a)

and      $log (\text{probability factor})=\frac{\Delta S_{m}^{ o}}{\text{2}\text{.303}R}$      (7b)

Although we did not formally derive Eq. (2a), on which these results are based, we can examine the last two equalities [Eqs. (7a) and (7b)] to see that they agree with our qualitative expectations. If ΔHm° is negative, we know that the products of a reaction have a lower enthalpyA thermodynamic state function, symbol H, that equals internal energy plus pressure x volume; the change in enthalpy corresponds to the energy transferred as a result of a temperature difference (heat transfer) when a reaction occurs at constant pressure. (and hence lower energy) than the reactants. Thus we would predict that products would he favored by the energy factor, and this is what Eq. (7a) says—the more negative ΔHm°, the more positive the logarithm of the energy factor and the larger the standard equilibrium constant. Also, since T appears in the denominator of the right-hand side of Eq. (7a), the smaller the value of T, the larger (and hence more important) the energy factor for a given value of ΔHm°. We have also seen in the section on thermodynamic probability and entropy that an increase in entropy of a system corresponds to an increase in thermodynamic probability. This is precisely what Eq. (7b) says. The larger the value of ΔSm°, the larger the probability factor and hence the larger the standard equilibrium constant. Thus our qualitative description of the two factors affecting the equilibrium constant has been refined to the point where macroscopic quantities, ΔHm° and ΔSm°, can be related to what is happening on the microscopic level.

In addition to the feature we have just mentioned, Eq. (6b) is useful in another way. If we can measure or estimate ΔHm° and ΔSm° at temperatures other than the usual 298.15 K, we can obtain K° and calculate the extent of reaction as shown in the next example.

EXAMPLE 2 Find the concentrationA measure of the ratio of the quantity of a substance to the quantity of solvent, solution, or ore. Also, the process of making something more concentrated. of NO in equilibrium with air at 1000 K and 1 atm pressure.

N2(g) + O2(g) → 2NO(gHm° = 181 kJ mol–1

ΔSm° = 35.6 J K–1 mol–1

Solution We first find K° :

${log{K^o}}=\frac{-\Delta H_{m}^{ o}}{\text{2}\text{.303}RT}$ + $\frac{\Delta S_{m}^{ o}}{\text{2}\text{.303}R}= \text{–9.45 + 1.34}= \text{–8.11}$

Thus                                 K° = 10–8.11 = 7.71 × 10–9

However, since Δn = 0,      K° = Kp = 7.71 × 10–9

Assuming the air to be 80% N2 and 20% O2, we can write

pN2 = 0.8 atm      and      pO2 = 0.2 atm

Thus              ${K}_{p}=\frac{p_{\text{NO}_{\text{2}}}^{\text{2}}}{\text{(}p_{\text{N}_{\text{2}}}\text{)(}p_{\text{O}_{\text{2}}}\text{)}}$ = $\frac{p_{\text{NO}_{\text{2}}}^{\text{2}}}{\text{0}\text{.8 atm }\times \text{ 0}\text{.2 atm}} = {\text{7.71}}\times{\text{10}}^{-\text{9}}$

Giving      $p_{\text{NO}_{\text{2}}}^{\text{2}} = {\text{7.71}}\times{\text{10}}^{-\text{9}}\times {\text{0.16 atm}}^{\text{2}}={\text{1.23}}\times{\text{10}}^{-\text{9}}{\text{atm}}^{\text{2}}$

${p}_{NO2}=\sqrt{\text{1}\text{.23 }\times \text{ 10}^{-9}\text{ atm}^{\text{2}}}= {\text{3.5}}\times{\text{10}}^{- \text{5}}{\text{atm}} ={\text{3.5}}\times{\text{10}}^{- \text{3}}{\text{kPa}}$

But            ${c}_{NO2}=\frac{c_{\text{NO}_{\text{2}}}}{V}$ = $\frac{p_{\text{NO}_{\text{2}}}}{RT}$

Thus          ${c}_{NO2}=\frac{\text{3}\text{.5 }\times \text{ 10}^{-\text{3}}\text{ kPa}}{\text{8}\text{.3143 J K}^{-\text{1}}\text{ mol}^{-\text{1}}}\text{ }\times \text{ }\frac{\text{1}}{\text{1000 K}} = {\text{4.3}}\times{\text{10}}^{- \text{7}}{\text{dm}}^{- \text{3}}$

Almost all combustionVigorous combination of a material with oxygen gas, usually resulting in a flame. processes heatEnergy transferred as a result of a temperature difference; a form of energy stored in the movement of atomic-sized particles. N2 and O2 to high temperatures, producing small concentrations of NO. Under most circumstances this is not of great consequence, but in the presence of sunlight and partially burned gasoline, NO2 can initiate a form of air pollutionThe contamination of the air, water, and earth by personal, industrial, and farm waste. called photochemical smog. The presence of minute concentrations of NO in the upper atmosphere from high-flying supersonic jet airplanes can also deplete the ozone layer there.