( ) 1/ 2. ( P SO2 )( P O2 ) 1/ 2.

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1 Chemstry 360 Dr. Jean M. Standard Problem Set 9 Solutons. The followng chemcal reacton converts sulfur doxde to sulfur troxde. SO ( g) + O ( g) SO 3 ( l). (a.) Wrte the expresson for K eq for ths reacton. In terms of actvtes, the equlbrum constant s ( a SO3 ) ( a SO ) a O /. The gases can be treated as deal gases, so the actvty s equal to the pressure dvded by the standard state pressure. Snce SO 3 s a pure lqud, ts actvty s equal to. Substtutng yelds, Settng the standard pressure equal to bar leads to ( a SO3 ) ( a SO ) a O P SO P / P / O P ( P ) 3/ ( P SO ) P O /. ( P SO )( P O ) /. (b.) Calculate the value of the standard molar Gbbs free energy of reacton, ΔG, for ths reacton at 98 K usng standard molar Gbbs free energes of formaton found n the appendx of your textbook. Usng the values gven for the standard molar Gbbs free energes of formaton at 98 K from the appendx, we have ΔG ΔG ΔG f ( SO 3 ) ΔG f SO 374. kj/mol 300. kj/mol 74. kj/mol. ΔG f ( O ) 0 kj/mol

2 . Contnued (c.) Calculate the value of K eq for ths reacton. Usng the equaton ΔG T ln K eq, the equlbrum constant s ln K eq ΔG T, or K eq exp ΔG T. Substtutng, K eq exp ΔG T ( J/mol) exp 8.34Jmol K e K eq K (d.) If.0 bar of SO and.0 bar of O are enclosed n a system n the presence of some SO 3 lqud, n whch drecton wll the system move n order to reach equlbrum? In order to determne the drecton that the system would move, t s helpful to determne the reacton quotent, Q. Q Q ( a SO3 ) ( a SO ) a O P SO P / ( a SO3 ) P / O P 3/ ( a SO3 ) P ( P SO ) P O /. Assumng that the actvty of lqud SO 3 equals and that the standard pressure s bar, ths smplfes to Q Q 3/ ( a SO3 ) P ( P SO ) P O / ( P SO )( P O ) /.

3 3 (d.) Contnued Substtutng the ntal partal pressures of SO and O, the reacton quotent becomes Q ( P SO )( P O ) / (.0 bar).0 bar Q.0. / Note that Q s untless (because of the standard pressures cancelng the unts of the partal pressures). When we compare Q to the equlbrum constant, , we see that Q s much, much smaller. To make the quotent larger, more products and fewer reactants are requred. Thus, the reacton must shft to the rght (very far to the rght snce t s so far from equlbrum).

4 4. Calculate the equlbrum constant and standard molar Gbbs free energy of reacton at 0 C for the reacton CuSO 4 4NH 3 ( s) CuSO 4 NH 3 ( s) + NH 3 ( g). The equlbrum pressure of ammona gas at 0 C s 8.6 kpa. In terms of actvtes, the equlbrum constant s ( a CuSO4 H O) a NH3 ( a CuSO4 4H O). For the solds, we can always assume that the actvty equals. The gases can be treated as deal gases to yeld ( a CuSO4 H O) a NH3 ( a CuSO4 4H O) ( ) P NH 3 P P NH 3 P. Usng the equlbrum pressure of ammona, P NH 3 be calculated as Pa bar, the equlbrum constant may P NH 3 P bar bar The standard molar Gbbs free energy of reacton at 0 C s then ΔG ΔG T ln K eq 93.5K 8.34Jmol K 00 J/mol or. kj/mol. ln

5 5 3. At 500 K, the equlbrum constant s for the reacton PCl 3 ( g) + Cl g PCl 5 ( g). The standard molar enthalpy of reacton s 69.8 kj/mol. Determne the equlbrum constant at 700 K, assumng the standard molar enthalpy of reacton s ndependent of temperature over ths range. The equlbrum constant at 700 K may be determned usng the ntegrated form of the van't Hoff equaton, ln K K. T T Solvng for the natural log of the equlbrum constant K yelds ln K K or ln( K ) ln( K ) +, T T. T T Substtutng 69.8 kj/mol J/mol, K , T 500 K, and T 700 K, we have ln( K ) ln( K ) + ln K ln( ) ln K The equlbrum constant at 700 K s therefore T T J mol 8.34Jmol K 500 K 700 K ln( K ) or K e K From ths result, we see that the equlbrum constant at 700 K s much smaller than the equlbrum constant at 500 K, whch was gven as Ths s the expected result from the van't Hoff Equaton (a decrease n equlbrum constant at hgher temperatures) for an exothermc reacton.

6 6 4. The dssocaton of mercurc oxde can be descrbed by the reacton HgO (s) Hg (g) + O (g). The total pressure at equlbrum s Pa at 40 C and Pa at 450 C. By expressng the equlbrum constant n terms of partal pressures, determne the equlbrum constant at each temperature. Also determne the standard enthalpy of reacton, assumng t s ndependent of temperature over the range. In terms of actvtes, the equlbrum constant s ( a HgO ). ( a Hg ) a O For the sold, HgO, the actvty can be assumed to be. The gases can be treated as deal gases to yeld ( a HgO ) ( a Hg ) a O P Hg P P Hg P PO P ( ) PO P. The partal pressures and mole fractons can be determned n terms of the extent of reacton, moles nt. 0 0 moles equl. ξ ξ ξ x ( gases) HgO (s) Hg (g) + O (g). ξ 3ξ 3 ξ 3ξ 3 Note that the total number of moles ncludes the number of moles of gases only, snce the mole fractons are gas phase mole fractons.

7 7 4. Contnued At 40 C, the partal pressures are P Hg x Hg P Pa 3 P Hg Pa or bar, and P O x O P Pa 3 P O Pa or 0.7 bar. Calculatng the equlbrum constant at 40 C, we have At 450 C, the partal pressures are and P Hg P PO P bar bar P Hg x Hg P Pa bar bar P Hg Pa or 0.70 bar, P O x O P Pa 3 P O Pa or bar. Calculatng the equlbrum constant at 450 C, we have P Hg P PO P 0.7 bar bar bar bar

8 8 4. Contnued The enthalpy of reacton can now be calculated usng the ntegrated form of the van't Hoff equaton, ln K K. T T Solvng for the enthalpy of reacton, we get ln K / K T T ( 8.34 J mol K )ln K 73.5K J/mol or 307.8kJ/mol.

9 9 5. Consder the followng reacton, 3 O ( g) O 3 ( g). (a.) Wthout performng a calculaton, predct n whch drecton the equlbrum wll shft as the pressure s ncreased. We showed n class that n order to analyze the behavor of a chemcal equlbrum wth respect to pressure, t s helpful to dvde the expresson for the equlbrum constant nto a pressure-ndependent porton and a pressure-dependent porton, K eq K x P ν P, where K x s the equlbrum expresson n terms of mole fractons, K x ν x, and ν s the sum of the stochometrc coeffcents, ν ν. For ths partcular example, the sum of the stochometrc coeffcents ν s ν ν O +ν O3 3+ ν. Snce the sum of the stochometrc coeffcents s negatve, f the pressure ncreases then the factor ( P / P ) ν decreases. In order for the overall equlbrum constant K eq to mantan a fxed (constant) value, the expresson nvolvng mole fractons, K x, must therefore ncrease. Ths means that products wll ncrease and reactants decrease, and therefore the reacton wll shft to the rght. Therefore, the extent of reacton wll ncrease. Alternately, the analyss may be carred out usng the deas from LeChateler's Prncple. In ths case, there are more moles of gas on the reactant sde than on the product sde. If the pressure s ncreased, the system wll shft to reduce the number of moles present. Therefore, n ths case, the reacton wll shft to the rght, toward the product sde, snce there are fewer moles of gas on the product sde.

10 0 5. Contnued (b.) Usng data for standard molar enthalpes found n the appendx of your textbook, predct n whch drecton the equlbrum wll shft as the temperature s ncreased. From the appendx, the standard molar enthalpy of formaton of O s 0 kj/mol and for O 3 t s 4.7 kj/mol at 98 K. The standard molar enthalpy of reacton s therefore ( O 3 ) 3 ΔH f ( O ) 3( 0 kj/mol) ΔH f 4.7 kj/mol 85.4 kj/mol. Accordng to the van't Hoff equaton, ln K K, T T for an endothermc reacton, ncreasng the temperature leads to an ncrease n the equlbrum constant and therefore a shft n the reacton to the rght. The endothermc reacton may be wrtten n the followng form n ths case, 3 O ( g) + heat O 3 ( g). For such a reacton, LeChateler's Prncple predcts that f the temperature ncreases, the system must shft to the rght n order to allevate the stress of addtonal heat. A shft to the rght leads to an ncrease n products and a decrease n reactants, whch ncreases the equlbrum constant as predcted by the van't Hoff equaton. (c.) Usng data for standard molar Gbbs energes found n the appendx of your textbook, calculate the equlbrum constant for the reacton at 98 K. Then, calculate the equlbrum constant for the reacton at 600 K, assumng that s ndependent of temperature. epeat the calculaton at 700 K. Compare your results wth your answer from part (b). From the appendx, the standard Gbbs energy of formaton of O s 0 kj/mol and for O 3 t s 63. kj/mol at 98 K. The standard molar Gbbs energy of reacton s therefore ΔG ΔG ( O 3 ) 3 ΔG f ( O ) 3( 0 kj/mol) ΔG f 63. kj/mol 36.4 kj/mol. The equlbrum constant at 98 K s ΔG T ln K eq, or K eq exp ΔG T.

11 5 (c.) Contnued Substtutng, K 98 exp ΔG T exp e 3.7 K ( J/mol) 8.34 J mol K 98K The equlbrum constant at 600 K may be determned usng the ntegrated form of the van't Hoff equaton, Substtutng, ln K 600 K 98 or ln( K 600 ) ln( K 98 ) + ln( K 600 ) ln( K 98 ) + ln( ) + ln( K 600 ) ln( K 600 ) The equlbrum constant at 600 K s therefore 98 K, 600 K 98 K 600 K J mol 8.34Jmol K ln( K 600 ) or K 600 e K K. 600 K 98 K 600 K Smlarly, the equlbrum constant at 700 K may be determned usng the ntegrated form of the van't Hoff equaton, ln K 700 K 98 or ln( K 700 ) ln( K 98 ) + 98 K, 700 K 98 K. 700 K

12 5 (c.) Contnued Substtutng, ln( K 700 ) ln( K 98 ) + ln( ) + ln( K 700 ) ln( K 700 ) The equlbrum constant at 700 K s therefore 98 K 700 K J mol 8.34Jmol K ln( K 700 ) or K 700 e K K 700 K A summary of the calculated equlbrum constants s gven n the table below. T (K) K eq Here we see that as the temperature ncreases, so does the equlbrum constant for ths reacton. These results are n accord wth the predcton from LeChateler's Prncple for an endothermc reacton.

13 3 6. At 000 C and a total pressure of atm, water s % dssocated nto oxygen and hydrogen gas accordng to the reacton H O ( g) H ( g) + O ( g). (a.) Calculate the equlbrum constant for the reacton. The equlbrum constant can be expressed n terms of the extent of reacton, H O ( g) H ( g) + O ( g). moles nt. 0 0 moles equl. ξ ξ x ξ ξ ξ ξ + ξ + ξ + ξ The equlbrum constant expressed n terms of the mole fractons s P O P / PH P P H O P x O P P x H OP P / xh P P. Wrtng the equlbrum constant n terms of the extent of reacton leads to the expresson, / xh P x O P P x H O P P / P ξ ξ ( +ξ /) +ξ / P P ξ +ξ / P P ξ 3/ P P ( ξ) ( +ξ /) / /. 3/

14 4 6 (a.) Contnued Usng P atm.03 bar, P bar, and ξ 0.00 (snce water s % dssocated), the equlbrum constant s ξ 3/ P ( ξ) ( +ξ /) / ( 0.00) 3/ ( 0.00) / P.03bar / bar / (b.) Determne whether the extent of reacton ncreases or decreases f the pressure s reduced. To analyze the behavor of a chemcal equlbrum wth respect to pressure, t s helpful to dvde the expresson for the equlbrum constant nto a pressure-ndependent porton and a pressure-dependent porton, P K x, where K x s the equlbrum expresson n terms of mole fractons, P v K x v x, and v s the sum of the stochometrc coeffcents, v v. For ths partcular example, the sum of the stochometrc coeffcents v s v v H O + v H + v O ++ v. Snce the sum of the stochometrc coeffcents s postve, f the pressure decreases then the factor ( P /P ) v also decreases. In order for the overall equlbrum constant K eq to mantan a constant value, the expresson nvolvng mole fractons, K x, must therefore ncrease. Ths means that products wll ncrease and reactants decrease, and therefore the reacton wll shft to the rght. Therefore, the extent of reacton wll ncrease.

15 5 6. Contnued (c.) If the total pressure s fxed at atm, determne whether the extent of reacton ncreases or decreases f argon gas s added. In the case of addton of a non-reactve gas, t s helpful once agan to dvde the expresson for the equlbrum constant nto a pressure-ndependent porton and a pressure-dependent porton, P K x, where K x s the equlbrum expresson n terms of mole fractons, P v K x v x, and v s the sum of the stochometrc coeffcents, v v. Addton of an nert gas mpacts the mole fractons, so t s further helpful to consder that the mole fractons x may be wrtten as x n n tot. Here, n s the number of moles of speces and n tot s the total number of gas phase moles, ncludng those of the added nert gas. Substtutng, the equlbrum constant expresson becomes P P v v n. n tot Snce the total moles s ndependent of the ndex, t may be moved outsde the product notaton to gve the equlbrum constant expresson as P n tot P v v n. Agan for ths partcular example, the sum of the stochometrc coeffcents v s v v H O + v H + v O ++ v. Snce the sum of the stochometrc coeffcents s postve, at constant pressure P, f an nert gas s added v P to the reacton vessel, the term n tot P must decrease because the total number of gas phase moles n tot ncreases.

16 6 6 (c.) Contnued In order for the overall equlbrum constant K eq to mantan a constant value, the expresson nvolvng moles of each speces, n v, must therefore ncrease. Ths means that products wll ncrease and reactants decrease, and the reacton wll shft to the rght. Therefore, the extent of reacton wll ncrease. (d.) Determne whether the extent of reacton ncreases or decreases f oxygen gas s added and the total pressure remans constant at atm. The reacton quotent for ths reacton s Q P H P P O P P H O P /. Intally, f O gas s added to the reacton mxture, the partal pressure of O, P O, wll ncrease. Snce ths factor appears n the numerator of the reacton quotent Q, ths means that Q has ncreased relatve to the equlbrum constant; that s, the reacton quotent wll be larger than the equlbrum constant just after addton of O and before the system has returned to equlbrum Q > K eq. To make the quotent smaller, more reactants and fewer products are requred. Thus, the reacton must shft to the left n order for the system to return to equlbrum. Thus, the extent of reacton decreases.