( )( )( )( )( ) University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2010

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1 Hoework Assignent 4: Due at 5 p.. 7/9/ Text Probles: 6.4, 6.9, 6, 6.3 University o Washington Departent o Cheistry Cheistry 45/456 Suer Quarter P6.9) Jaes Watt once observed that a hard-working horse can lit a 33. lb. weight. t. in one inute. Assuing the horse generates energy to accoplish this work C H O s + 6O g 6CO g + 6H O l.. Calculate by etabolizing glucose:. ( ) ( ) ( ) ( ) 6 6 how uch glucose a horse ust etabolize to sustain this rate o work or one hour. Assue T=98.5 Work perored by the horse in one inute: w = Mg h = 33lb.4536kglb 9.8s t.348t = 4476J. ( )( )( )( )( ) To sustain this rate o work or hour we require that the ollowing aount o

2 glucose be etabolizedby the horse: work M.86kgol = 6 in hr = 4476J in 6 in hr =.679kg glu cose glu cose ( ) ( )( ) 3 in Gglu cose 88 Jol 3 P6.) N ( g) + H ( g) NH3 ( g) 3 G = G ( NH ) G ( N ) G ( H ) = G ( NH ) = kjol rxn 3 3 ( ) ( ) 6.45 G NH3, T G NH3, T = H ( NH3) T T T T ( ( NH T ) G 3, ) G ( NH3, T ) = T + H ( NH3) T T T (( = 4K 46.kJol = 6.38kJol 98K 4K 98K (( 6.45kJol (, 4 ) * G NH3 K - * Jol - ) K4 = exp + / = exp + / = 6.68, + RT / +, ( 8.3JK ol )( 4K )/ 3 3 P6.3) N ( g) + H O( ) NH3 ( aq) + O ( g) 3 ( ) ( ) 3 G = G NH, aq + G O, g G ( N ) G ( H O, ) rxn ( 3 ) ( ) ( )( ) = G NH, aq G H O, = 8.3kJol 37.3kJol = 75.4kJol K ( ) G NH, aq 75.4 Jol = exp = exp = 5.8 ( 8.3 )( 98 ) ( RT ) ( JK ol K ) V RT = + where n P a B( T ) = b, where a and b are constants. The constant a relects RT attractive interactions and b relects repulsive interactions. Obtain H expressions or P and U. Possible ters in the inal T V T expressions are a, b, P, R, and T. ) Suppose a gas obeys the equation o state: B( T )

3 V RT = V = + B( T ) n P V R db R a T ( P dt P RT = + = + P H V R a = V ) T = V ) T + P ( T, T ( P RT ( RT a R a a = + b ) ) T + = b ) ( P RT P RT RT V = RT P + B ( T ) = RT P + b a RT RT P = V B T ( )P )T V = R V b + a ( )U )V T ( ) = RT V b + a = R V b + a = T )P )T P V = RT V b + a = a V b + a RT a + RT RT V b + a + a T V b + a + a V b + a P = a RT V b + a ) Wind turbines are now recognized as useul devices or power generation. Consider an ideal wind turbine. The blades o the wind turbine sweep out an area d R A = = where R is the length o the blade and d=r is the rotor diaeter. Assue the turbine blade rotates without riction. Assue the wind blows axially into the turbine blade (i.e. parallel to the rotor blade shat) with velocity v, and exists the rotor area with velocity v <v.

4 Figure; Idealized wind turbine with rictionless rotor blades o length R. The wind is assued to blow axially into the rotor blade with velocity v and exits the rotor area A with velocity v <v. The velocity o the wind at the rotor blade is assued to be the average v=(v +v )/. a) In the absence o a wind turbine rotor, the power generated by a ass o air 3 oving through an area A is given by P = Av where the density o air at sea 3 level is =.3kg and v is the velocity o the wind. Calculate the power generated when air with wind velocity 4s - oves through a circular area 6 eters in diaeter. P = Av 3 =.3kg3 6 ( ) =. * 7 Js ( 4s ) 3 =.6kg 3 ( )( 469 )( s ) 3 b) The rate at which a ass o air oves over the turbine rotor blade is deined as: d v + v = = Av, where the velocity o the wind at the rotor blades v = is dt the average o the incoing wind velocity v and the outgoing wind velocity v. I a wind turbine blade sweeps out an area A, the turbine generates power P = ( v v ). The largest wind turbine in the world has a rotor diaeter o 6. I v =4 s - and v =s -. Calculate the power that this turbine generates. ( ) = 4 A ( v + v ) v ( v ) ( )( 469 )( 4s )( 96 s ) = Js P = v v =.3kg 3

5 P c) The eiciency o the wind turbine is deined as =. Show that in general the P eiciency o a wind turbine is given by = + v v v v ( v 3 v ( ( (. This eans the eiciency with which an ideal wind turbine operates depends only on the ratio v /v. = P = A ( v + v ) v ( v ) v 3 = + v v v v 3 v = + v v 3 3 P 4Av v v v ) v 3 ( v ) ) ( ( ) d) Deterine the axiu eiciency o a wind turbine ax and the velocity ratio v /v at which this eiciency occurs. Repeat the power calculation in part b assuing axiu eiciency. = + v v v v ( v 3 v ( ( ( = + x ( x x ) 3 ) d dx = = ( x 3x ) * x 3x = * x = v P ax = 4 A v + v 3 v ( v 3 = (.3kg )( (3 469 )( 4s ) ( 3 4 v = 3 = 4 Av ( 3 * P ax = Av = 6 7 P =.593P = + P ax *+ ax = =.6 ) 7 Js ( e) The U.S. consues,68, barrels o crude oil per day. There are 59 L in a barrel o oil. Crude oil has an average density o 9 kg -3 and the average heat o cobustion ΔH cob =4.47x 4 kj/kg. Suppose we burn on average 9 o the oil that we consue per day. How any wind turbines operating at axiu eiciency as in part d would we require to atch the energy obtained by burning oil?

6 ( )( 59Lbrl )(. 3 L )( 9kg ) 3 =.96 9 kgday ( )(.96 9 kgday ) ( day / 864s) =.53 9 kjs ( )(.53 9 kjs ) =.38 9 kjs oil,68,brlday P oil,total = kjkg P oil,burn =.9 P wind,ax = 4. 3 kjs P oil P wind,ax =.38 9 kjs.6 4 kjs =. 5 turbines Note: The largest wind turbine in the world is the Enercon E6 in Gerany. The E6 has a rotor diaeter o 6, and generates about 6 MW, about hal the power generated by the ideal turbine described in part d. So we would need closer to, E6 s to replace oil consuption. The United States recently surpassed Gerany as the world s largest producer o wind power with a total national wind power capacity o 35, MW=3.5x 7 kjs -. 3) The shells o arine organiss contain calciu carbonate CaCO 3, largely in a crystalline or known as calcite. There is a second crystalline or o calciu carbonate known as aragonite. Physical and therodynaic properties o calcite and aragonite are given below. Properties (T=98K, P=at) Calcite Aragonite (kj/ole) H G (kj/ole) S (J/ole K) C P (J/ole K) Density (g/l).7.93 a) Based on the therodynaic data given, would you expect an isolated saple o calcite at T=98K and P= at to convert to aragonite, given suicient tie. Explain. G = G aragonite G calcite = kjol =.kjol ( ) ( ) ( ( )) G > so the process will not occur. b) Suppose the pressure applied to an isolated saple o calcite is increased. Can the pressure be increased to the point that isolated calcite will be converted to aragonite? Explain. By La Chatelier s Principle, an increase in pressure will avor the crystalline or with the higher density, i.e. the saller olar volue. This would be aragonite.

7 c) What pressure ust be achieved to induce the conversion o calcite to aragonite at T=98K. Assue both calcite and aragonite are incopressible at T=98K. V calc = M gol = d arag.7gl = 36.9L ol = ol V arag = M d calc = gol.93gl = 34.L ol = ol V = V arag V calc = ol ol = ol ( ) = = G( at,98.k ) + V P G( at,98.k ) =. 3 Jol = Pa = 393bars V G P,98.5K P = d) Can isolated calcite be converted to aragonite at P= at i the teperature is increased? Explain.. S = S ( arag) S ( calc) = 88.7JK ol 9.88JK ol = 4.JK ol Assue the enthalpy change and entropy changes are constant with T, then G = H T S. Because S <, G will becoe ore positive as T increases. 4) Many biological acroolecules undergo a transition called denaturation. Denaturation is a process whereby a structured, biological active olecule, called the native or, unolds or becoes unstructured and biologically inactive. The equilibriu is unolds native denatured olds For the enzye chyotrypsin at ph= the enthalpy change associated with denaturation is H = 48kJ ol and the entropy change is S =.3kJ K ol a) Calculate the Gibbs energy change or the denaturation o chyotrypsin at ph= and T=33K. Assue the enthalpy and entropy are teperature independent between 98K and 33K. G den = H den T S den = 48 3 Jol 33K = 48kJol 4kJol = 8.kJol ( )( 3JK ol ) b) Calculate the equilibriu constant or the denaturation o chyotrypsin at ph and T=33K. K 33 = exp G den RT ( = exp 8Jol = 7.86 ) 8.3JK ol 4 ( ) ( ( ) 33K

8 c) Based on your answer or parts a and b, is chyotrypsin structurally stable at ph and T=33K. D K = so at 33K D N. The protein is structurally stable. N d) The elting teperature is deined as the teperature at which the equilibriu constant or denaturation K=. Assuing that the enthalpy o denaturation is teperature independent, calculate the elting teperature o chyotrypsin at ph. G = H T S so = H den T Sden. Thereore den den den T = H den = 48 3 Jol S den 3JK ol = 37K 5) The technology to build sall icroscopic-sized otors (e.g. olecular otors) exists. In principle, i a icroscopic otor can be built, whose action is based on luctuations in the environent, the liit iposed on the eiciency o an engine by the Second Law can be circuvented Or can it? Consider the ollowing icroscopic engine, held at constant teperature: Assue the otor is so sall that a gas olecule (rando otion a luctuation) hits the paddle wheel causing it to turn clockwise or counterclockwise. When the paddle turns the axle turns liting a one cell organis which is tethered to the axle in soe way. Now because the otion o gas olecules is rando, as any olecules hit the paddle causing it to turn clockwise, as hit it in the opposite direction causing it to turn counterclockwise, so the cell is randoly raised up and down with no net work accoplished. But it is proposed that a ratchet, held down with a spring against a gear, can prevent counterclockwise otions o the paddle. Then the cell will be raised by clockwise otions o the paddle. Net work is done (ass o the cell ties the change in gravitational potential) and because the orce causing the work originates in luctuations, the eiciency o this engine is not governed by Second Law liits. a) Will the otor work? Hint: consider and discuss the proposed action o the ratchet in ters o rando otions iposed by luctuations.

9 The engine does not work. The reason it does not work is that the teperature is assued unior over this tiny achine. Thereore the sae nuber o olecules that bobard a unit area o the paddle wheel per unit tie also bobard the ratchet per unit tie. Insoar as the ratchet and the wheel are both tiny, they are both perturbed by olecular collisions. The ratchet cannot work as advertised unless there is soe way to abate the olecular collisions that all upon it. b) I the otor will not work as shown, can you think o a siple change that ight ake it work? One way to abate olecular collisions against the ratchet would be to place it in an enclosure uch colder than the area around the wheel. But then we would have created a teperature gradient, and this achine would then just operate as any other achine. We have to provide uel to pup heat out o the enclosure around the ratchet, so the achine requires uel and will have a inite eiciency.