# CHAPTER 14 (MOORE) CHEMICAL EQUILIBRIUM

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1 CHAPTER 14 (MOORE) CHEMICAL EQUILIBRIUM This chapter deals with chemical equilibrium, or how far chemical reactions proceed. Some reactions convert reactants to products with near 100% efficiency but others do not. Chemical equilibrium is a dynamic process, and occurs when the rates of two opposing processes (reactions) become the same. Unless conditions are somehow changed, the concentrations of both reactants and products remain constant after equilibrium is achieved. Consider the decomposition reaction in which N 2 O 4 (g) 2 NO 2 (g) What happens to the concentration of each, based upon the rate equations, as the reaction proceeds? Assuming reversible elementary processes, Rate (forward) = k (forward) x [N 2 O 4 ] and rate (reverse) = k (reverse) x [NO 2 ] 2 As [N 2 O 4 ] decreases, rate forward decreases, and, as [NO 2 ] increases, rate reverse increases, but more quickly why? Let s look at the graph of concentration vs. time the concentration lines converge at equilibrium. We can also graph changes in concentrations with time for 2 HI(g) H 2 (g) + I 2 (g) (see also Fig. 14.2, Moore, p. 676) Is there a mathematical relationship between [HI], [H 2 ] and [I 2 ]? Expressing Equilibrium Relationships 2 HI(g) H 2 (g) + I 2 (g), where rate(forward) = k f [HI] 2 and rate(reverse) = k r [H 2 ][I 2 ] or, rate(forward) = rate(reverse) k f [HI] 2 = k r [H 2 ][I 2 ] and, (k f /k r ) = [H 2 ][I 2 ]/[HI] 2 = constant = K C (the equilibrium constant) K C is called the concentration-based equilibrium constant! The product concentrations are on top and the reactant concentrations are on the bottom. Regardless of the starting concentrations, the value of the expression, with product concentrations in the numerator, and reactant concentrations in the denominator, where each concentration is raised to the power of its coefficient in the balanced chemical equation, is equal to a constant.

2 The Equilibrium Constant Expression (sometimes called the mass action expression) For the general reaction: aa + bb gg + hh, the equilibrium expression is: K c = {[G] g [H] h } / {[A] a [B] b } Products Reactants The Equilibrium Constant The equilibrium constant is constant regardless of the initial concentrations of reactants and products, as long as the temperature remains constant. The concentration equilibrium constant is K C Concentrations of the products appear in the numerator and concentrations of the reactants appear in the denominator. Recall that [A] means molar concentration of A. Important: the exponents of the concentrations are identical to the stoichiometric coefficients in the chemical equation!! Example 14.1 If the equilibrium concentrations of COCl 2 and Cl 2 are the same at 395 C, find the equilibrium concentration of CO in the reaction: CO(g) + Cl 2 (g) COCl 2 (g) K C = 1.2 x 10 3 at 395 C First, write the K C expression, where K C = 1.2 x 10 3 = [COCl 2 ] / [CO][Cl 2 ] Then, {COCl 2] = [Cl 2 ], so substitute in the equilibrium expression, and K C = 1.2 x 10 3 = [Cl 2 ] / [CO][Cl 2 ] = 1/[CO] [CO] = 8.3 x 10-4 M Reversing the Direction of the Chemical Equation How does this affect the form of K C? Consider the reaction: 2 NO(g) + O 2 (g) 2 NO 2 (g) Then, K C = [NO 2 ] 2 / [NO] 2 [O 2 ] Now consider 2 NO 2 (g) 2 NO(g) + O2(g), where K C = [NO] 2 [O 2 ] / [NO 2 ] 2 You should see that K C = 1/K C When the reaction direction is reversed, K C = 1/K C Modifying the Chemical Equation: Changing Coefficients Changing the coefficients in a balanced equation affects the form of the equilibrium expression!

3 Example The equilibrium constant at 718 K for the reaction below is ½ H 2 (g) + ½ I 2 (g) HI(g) (a) What is the value of K C at 718 K for the reaction, HI(g) ½ H 2 (g) + ½ I 2 (g) Solution. Reversing the reaction means taking the reciprocal of K C : K C = 1/7.07 = (b) What is the value of K C at 718 K for the reaction, H 2 (g) + I 2 (g) 2 HI(g) Solution. Doubling the coefficients in the original balanced equation means squaring K C : K C = (7.07) 2 = 50.0 Modifying the Chemical Equation: Changing Coefficients Another example If K C = 1.97 x for ½ N 2 (g) + 3/2 H 2 O(g) NH 3 (g) + ¾ O 2 (g) at 900 K, what is K C at the same temperature for the reaction 4 NH 3 (g) + 3 O 2 (g) 2 N 2 (g) + 6 H 2 O(g)? Solution. Reversing the reaction and quadrupling the coefficients means raising all exponents by a factor of 4 and taking the reciprocal of K C : K C = 1/(1.97 x ) 4 = 6.64 x Equilibrium Constants for Overall Reactions When reactions occur in two or more steps, we can use the K C values for each step to calculate K C for the overall process. Suppose we need: N 2 O(g) + 3/2 O 2 (g) 2 NO 2 (g) K C (1) =?? and we re given: N 2 O(g) + ½ O 2 (g) 2 NO(g) K C (2) = 1.7 x NO(g) + O 2 (g) 2 NO2(g) K C (3) = 4.67 x Adding the given equations gives the desired equation. Multiplying the given values of K gives the equilibrium constant for the overall reaction. Adding equations 2 and 3 gives: N 2 O(g) + 3/2 O 2 (g) 2 NO 2 (g) Multiplying K C (2) by K C( 3), we get K C (1) = (1.7 x )(4.67 x ) = 8.0 Equilibria Involving Gases - In reactions involving gases, we often use partial pressures in place of molarities. This means that we have to use a partial pressure equilibrium constant. For the reaction, aa(g) + bb(g) gg(g) + hh(g), K P = {(P G ) g (P H ) h } / {(P A ) a (P B ) b } where the P values represent partial pressures, typically in atm or mmhg K C and K P are related by: K P = K C (RT) Δn(gas)

4 where Δn(gas) is the change in the total number of moles of gas as the reaction occurs in the forward direction. More specifically, Δn(gas) = mol gaseous products mol gaseous reactants Example 14.3 Consider the equilibrium between dinitrogen tetroxide and nitrogen dioxide: N 2 O 4 (g) 2 NO 2 (g) K p = at 319 K (a) What is the value of Kc for this reaction? (b) What is the value of Kp for the reaction 2 NO 2 (g N 2 O 4 (g)? (c) If the equilibrium partial pressure of NO 2 (g) is atm, what is the equilibrium partial pressure of N 2 O 4 (g)? Solution. N 2 O 4 (g) 2 NO 2 (g) K p = atm at 319 K (a) What is the value of Kc for this reaction? K p = K C (RT) Δn(gas) where R = L atm mol-1 K-1 Here, Δn(gas) = 2 1 = +1 K C = atm/{( L atm mol -1 K -1 )(319 K)} +1 = mol L -1 (b) What is the value of K p for the reaction, 2 NO 2 (g) N 2 O 4 (g)? Now, the reaction direction is reversed recall that K P = 1 / K P So, K P = 1/(0.660 atm) = 1.52 atm -1 (c) If the equilibrium partial pressure of NO 2 (g) is atm, what is the equilibrium partial pressure of N 2 O 4 (g)? Begin by writing the K P equilibrium expression K p = (pno 2 ) 2 / (pn 2 O 4 ) 1 = atm and, then, substitute for p(no 2 ) = atm K p = (0.332 atm) 2 / (pn 2 O 4 ) 1 = atm p(n 2 O 4 ) = (0.332 atm) 2 / (0.660 atm) = atm Equilibria Involving Pure Solids and Liquids The equilibrium constant expression does not include terms for pure solid and liquid phases because their concentrations do not change in a reaction. Although the amounts of pure solid and liquid phases change during a reaction, these phases remain pure and their concentrations do not change. Example: CaCO 3 (s) CaO(s) + CO 2 (g) K C =??? K C = [CO 2 ] 1

5 Example 14.4 (see also Moore, Problem,-Solving Example 14.1 (b), p. 678) The reaction of steam and coke (a form of carbon) produces a mixture of carbon monoxide and hydrogen, called water-gas. This reaction has long been used to make combustible gases from coal: C(s) + H 2 O(g) CO(g) + H 2 (g) Write the equilibrium constant expression for K C for this reaction. K C = {[CO][H 2 ]} / [H 2 O] Why no [C]? Because C is a pure solid, C(s)! Equilibrium Constants: When Do We Need Them? A very large numerical value of K c or K p signifies that a reaction goes (essentially) to completion. A very small numerical value of K c or K p signifies that the forward reaction, as written, occurs only to a slight extent. An equilibrium constant expression applies only to a reversible reaction at equilibrium. Although a reaction may be thermodynamically favored, it may be kinetically controlled Thermodynamics tells us it s possible (or not). Kinetics tells us if it s practical (or not) Example 14.5 Is the reaction CaO(s) + CO 2 (g) CaCO 3 (s) likely to occur to any appreciable extent at 298 K? We know that for CaCO 3 (s) CaO(s) + CO 2 (g) KP = 1.9 x What does this tell us? That this reaction is reactant favored or that the equilibrium lies to the left. Lots of CaCO 3 with very little CO 2... where K P = p(co2) 1. Therefore, this reaction favors the formation of CaCO 3, so the answer is YES! The Reaction Quotient, Q For nonequilibrium conditions, the expression having the same form as K c or K p is called the reaction quotient, Q c or Q p. The reaction quotient is not constant for a reaction, but is useful for predicting the direction in which a net change must occur to establish equilibrium. To determine the direction of net change, we compare the magnitude of Q c to that of K c. If Q c >K c, then numerator is too large relative to the denominator and some product must be converted to reactant (i.e. equilibrium shifts to the left). When Q < K, the denominator of Q is too big; we have too much reactants. When Q = K, equilibrium has been reached. When Q > K, the numerator of Q is too big; we have too much products. Example 14.6 Predict the direction of net change for Experiment 3 in Table [HI] = M; [H 2 ] = M; [I 2 ] = M Q C = (0.000)(0.000)/(1.000)2 = K C = 1.84 x 10-2 Q C < K C, so equilibrium must shift to the right to form more product(s). See also Examples and Exercises, Moore, pp Le Châtelier s Principle When any change in concentration, temperature, pressure, or volume is imposed on a system at equilibrium, the system responds by attaining a new equilibrium condition that minimizes the impact of the imposed change.

6 When a system at equilibrium is stressed, it responds in such a way that it relieves the stress and attains a new state of equilibrium. Changing the Amounts of Reacting Species At equilibrium, Q C = K C. If the concentration of one of the reactants is increased, the denominator of the reaction quotient increases. Q C is now less than K c. This condition is only temporary because the concentrations of all species must change in such a way so as to make Q C = K c again. In order to do this, the concentrations of the products increase; the equilibrium is shifted to the right. Heterogeneous Equilibria and Le Chatelier s Principle Addition or removal of pure solids or pure liquids from a system at equilibrium does not affect the position of the equilibrium. Changing External Pressure or Volume in Gaseous Equilibria When the external pressure is increased (or system volume is reduced), an equilibrium shifts in the direction producing the smaller number of moles of gas. When the external pressure is decreased (or the system volume is increased), an equilibrium shifts in the direction producing the larger number of moles of gas. N 2 O 4 (g) 2 NO 2 (g) Increasing pressure shifts to the side with fewer molecules of gas, so 2 NO 2 combine to form 1 N 2 O 4 (equilibrium shifts to the left). If there is no change in the number of moles of gas in a reaction, changes in external pressure (or system volume) have no effect on an equilibrium. Example: H 2 (g) + I 2 (g) 2 HI Δn(gas) = 0 This equilibrium is unaffected by pressure changes. Example 14.8 An equilibrium mixture of O 2 (g), SO 2 (g), and SO 3 (g) is transferred from a 1.00-L flask to a 2.00-L flask. The balanced equation for the reaction is 2 SO 3 (g) 2 SO 2 (g) + O 2 (g) In which direction does a net reaction proceed in order to restore equilibrium? Volume increases (from 1.00 L to 2.00 L), so pressure decreases, favoring more molecules or the side with more moles of gas 2 mols on left, 3 mols on right, so equilibrium shift is left-toright. Temperature Changes Raising the temperature of an equilibrium mixture shifts equilibrium in the direction of the endothermic reaction; lowering the temperature shifts equilibrium in the direction of the exothermic reaction. We can treat heat as a product of an exothermic reaction or as a reactant of an endothermic reaction, and then apply Le Châtelier s principle. Heating an endothermic rxn shifts the equilibrium toward more products; heating an exothermic rxn shifts the equilibrium toward more reactants.

7 Example 14.9 Is the amount of NO(g) formed from given amounts of N 2 (g) and O 2 (g), N 2 (g) + O 2 (g) 2 NO(g) ΔH = kj greater at high or at low temperatures? From (+) sign of ΔH 0 we see that the reaction is endothermic as written. This means we can essentially write heat as a reactant, or, N 2 (g) + O 2 (g) + Heat 2 NO(g) Adding heat (higher temp.) means equilibrium should shift left to right and more product should form at higher temperatures. Effect of Catalysts and Catalysis A catalyst lowers the activation energy of both the forward and the reverse reactions. Adding a catalyst does not affect an equilibrium state. A catalyst merely causes equilibrium to be achieved more rapidly. Equilibrium Calculations 1. Finding K C from concentration data 2. Finding one equilibrium concentrations from K C and the remaining concentrations 3. Finding equilibrium concentrations using square roots 4. Finding equilibrium concentrations using the quadratic equation Determining Values of Equilibrium Constants from Experimental Data When initial amounts and equilibrium amounts of one or more species are given, the amounts of the remaining species in the equilibrium state and the equilibrium concentrations can be calculated. A useful general approach is to tabulate under the chemical equation: 1. concentrations of substances initially present (I) 2. changes in concentrations that occur (C) 3. the equilibrium concentrations (E) We use this procedure to make a table called an ICE table: Initial / Change /Equilibrium. Example In a 10.0-L vessel at 1000 K, mol SO 2 and mol O 2 react to form mol SO 3 at equilibrium. Calculate K c, at 1000 K, for the reaction 2 SO 2 (g) + O 2 (g) 2 SO 3 (g) [SO 2 ] = (0.250 mol/10.0 L) = M [O 2 ] = (0.200 mol/10.0 L) = M [SO 3 ] = (0.162 mol/10.0 L) = M Now set up the ICE table:

8 [SO 2 ] [O 2] [SO 3 ] I C - 2x - x + 2x E x x x = , so x = /2 = [SO 2 ] = ( ) = M [O 2 ] = = M K c = [SO 3 ]2 / ([SO 2 ]2[O 2 ]) = (0.0162)2 / ((0.0088)2(0.0119) = 2.8 x 10 2 Example Consider the reaction, H 2 (g) + I 2 (g) 2 HI(g) Kc = 54.3 at 698 K If we start with mol I 2 (g) and mol H 2 (g) in a 5.25-L vessel at 698 K, how many moles of each gas will be present at equilibrium? Initial [H 2 ] = (0.500 mol/5.25 L) = M = [I 2 ] [H 2 ] [I 2 ] [HI] I C - x - x + 2x E x x 2x K C = [HI] 2 / ([H 2 ][I 2 ]) = (2x) 2 / ( x) ( x) = 54.3 Now, solve for x 2x) 2 / ( x)( x) = 54.3 (2x) 2 /( x) 2 = 54.3 Taking the square root of each side, (2x)/( x) = (54.3)1/2 = x = (7.368)( x) = x x = [H 2 ] = [I 2 ] = x = = M [HI] = 2x = 2(0.0749) = M Now check by plugging back into the K C expression: K C = [HI] 2 / ([H 2 ][I 2 ]) = (0.150) 2 / ((0.0203)(0.0203)) K C = 54.6 (w/ round off error)

9 Calculating Equilibrium Quantities from K C and K P Values We can calculate the amount of substances present at equilibrium, starting with only initial reactants, no products and the known value of the equilibrium constant. 1. Set up an ICE table, using x to identify the changes in concentration that occur in reaching equilibrium. 2. All concentration changes are related to x, and the appropriate terms are substituted into the equilibrium constant expression. 3. Solve the equation for x. Example Suppose that in the reaction of Example 14.12, the initial amounts are mol H 2 and mol I 2. What will be the amounts of reactants and products when equilibrium is attained? H 2 (g) + I 2 (g) 2 HI(g) K C = 54.3 at 698 K If we start with mol I 2 (g) and mol H 2 (g) in a 5.25-L vessel at 698 K, how many moles of each gas will be present at equilibrium? K C = 54.3 = [HI] 2 / ([H 2 ][I 2 ]) Example Carbon monoxide and chlorine react to form phosgene, COCl 2, which is used in the manufacture of pesticides, herbicides, and plastics: CO (g) + Cl 2 (g) COCl 2 (g) K C = 1.2 x 10 3 at 668 K How much of each substance, in moles, will there be at equilibrium in a reaction mixture that initially has mol CO, mol Cl 2, and mol COCl 2 in a 10.0-L flask? Initial concentrations: [CO] = mol/10.0 L = M = [Cl 2 ] [COCl 2 ] = mol/10.0 L = M Calculate Q C and compare to K C to see the direction of the reaction. Q C = C(COCl 2 ) / C(CO) C(Cl 2 ) = (0.0100) /( ) 2 = 1 x 10 4 Since Q C >> K C, reaction goes to the left (more reactants) [CO] [Cl 2 ] [COCl 2 ] I C + x + x x E x x x Then, K C = [COCl 2 ]/{[CO] [Cl 2 ]} = ( x) / {( x)( x)} = 1.2 x 10 3

10 ( x) = (1.2 x 10 3 )( x) 2 = x + (1.2 x 10 3 x 2 Collecting terms, (1.2 x 10 3 x x Solve as a quadratic: a = 1.2 x 10 3 ; b = 3.4; c = Positive root: x = So, [CO] = [Cl 2 ] = x = = M and [COCl 2 ] = x = = M Then, mol CO = mol Cl 2 = ( mol/l)(10.0 L) = mol and mol COCl 2 = ( mol/l)(10.0 L) = mol

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