CHEM2541 Physical Chemistry I. Assignment 2 Solution. Execises

Transcription

1 CHEM541 Physical Cheistry I Assignent Solution Execises A. 1(a) Use the equiartition theore to estiate the olar internal energy relative to U(0) of (i) I (ii) CH 4 (iii) C 6 H 6 in the gas hase at 5 o C According to equiartition theore, each translational and rotational degree of freedo contributes degree of freedo contributes to the olar internal energy, and each active vibrational to the olar internal energy. A vibrational ode is said to be active at a certain teerature if the energy required to excite the vibrational ode is coarable or saller than the k, i.e. where h is the Planck constant and is the vibrational frequency of the ode. At roo teerature =98K, we have wavenuber as, which converts to (i) I is a linear olecule, so ranslational degree of freedo = Rotational degree of freedo = ibrational degree of freedo =N-5 = 1 he vibrational frequency is 14c, so we shall consider it active 1 7 So U( ) U(0) ( 1) R R 8.67kJol (ii) CH 4 ranslational degree of freedo = Rotational degree of freedo = ibrational degree of freedo =N-6 = 9 But none of the vibrational odes are active since the vibrational frequency for the C-H stretching and bending odes are all higher than 1000c 1 So U( ) U(0) ( 0) R R 7.44kJol

2 (iii) C 6 H 6 ranslational degree of freedo = Rotational degree of freedo = ibrational degree of freedo =N-6 = 0 Estiating the nuber of active odes is hard in this case since any bending odes are in the range of 00c 1000c. Many of the ay have slight activity and contribute to the internal energy. Assue nuber of active odes 1 U( ) U(0) ( 4) R 7R 17.5kJol A. (a) Which of (i) ressure, (ii) teerature, (iii) work, (iv) enthaly are state functions? Pressure, teerature, enthaly A. (b) Which of (i) volue, (ii) heat, (iii) internal energy, (iv) density are state functions? olue, internal energy, density

3 A.4(a) A sale consisting of 1.00 ol Ar is exanded isotherally at 0 o C fro 10.0d to 00d (i) reversibly, (ii) against a constant external ressure equal to the final ressure of the gas and (iii) freely (against zero external ressure). For the three rocesses calculate q w, and. For all cases, since the internal energy of a erfect gas deends only on teerature. Fro the definition of enthaly,. So ( ) ( ) (erfect gas). Hence, as well, at constant teerature for all rocesses in a erfect gas.

4 A. 5(a) A sale consisting of 1.00ol of erfect gas atos, for which, initially at 1 = 1.00 at and 1 = 00 K, is heated reversibly to 400 K at constant volue. Calculate the final ressure,. For a erfect gas at constant volue, U nc, n R 400K 00K 1.00 ol 8.14 JK ol 100K w J 1.5 kj constant volue q U w 1.5kJ B.(a) he constant-ressure heat caacity of a sale of a erfect gas was found to vary with teerature according to the exression ( ) ( ). Calculate when the teerature is raised fro 5 o C to 100 o C (i) at constant ressure, (ii) at constant volue.

5 B. (a) When.0 ol O is heated at a constant ressure of.5 at, its teerature increases fro 60 K to 85 K. Given that the olar heat caacity of O at constant ressure is 9.4 J K ol, calculate.

6 C. (a) he standard enthaly of cobustion of cycloroane is -091 kjol at 5 o C. Fro this inforation and enthaly of foration data for CO (g) and H O(g). calculate the enthaly of foration cycloroane. he enthaly of foration of roene is +0.4 kjol. Calculate the enthaly of isoerization of cycloroane to roene. C.4(a) Given that the standard enthaly of foration of HCl(aq) is 67kJol, what is the value of ( )

7 C. 6(a) Given the reactions (1) and () below, deterine (i) and for reaction (), (ii) for both HCl(g) and H O(g) all at 98K (1) H (g) + Cl (g) HCl(g) () H (g) + O (g) H O(g) () 4HCl(g) + O (g) Cl (g) + H O(g)

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9 D. 1(a) Estiate the internal ressure,, of water vaour at 1.00 bar and 400 K, treating it as a van der Waals gas. Hint: Silify the aroach by estiating the olar volue by treating the gas as erfect. he internal ressure is defined as: U For a van der Waals gas which obey the equation of state n a nb nr a Its internal ressure is given by. ( Given in D. (a) ) R at 1.00 bar 1.01bar d at K 400K.6d ol 6 - a 5.464d at ol (.6d ol ) at 5.0bar

10 D.(a) For a van der Waals gas,,. Calculate for the isotheral exansion of nitrogen gas fro an initial volue of 1.00d to 0.00d at 98 K. What are the values of q and w? he internal energy is a function of teerature and volue, i.e. ( ) U U U du d d d d For isotheral exansion, d=0, U 0.00d 0.00d d 1.00d 1.00d 1.5d at ol 0.0d ol 1.84d at ol 6-1.5d at ol 1.00d ol d at ol 10.1Jol a 6 - a d 0.00d 1.00d 1015 Pa 10 d 1at R a w d where b 0.00d 1.00d R ln( b) U 8.14 JK ol K ln Jol 10.1Jol Jol q U w 7.51 R a d b d 0 Jol U

11 Probles A.1 Calculate the work done during the isotheral reversible exansion of a van der Waals gas. Plot on the sae grah the indicator diagras (grahs of ressure against volue) for the isotheral reversible exansion of (a) a erfect gas, (b) a van der Waals gas in which a=0 and b=5.11 x 10 - d ol, and (c) a =4. d 6 at ol - and b=0. he values selected exaggerate the ierfections but give rise to significant effects on the indicator diagras. ake =,, and. f ( a) wideal nr ln.0d 1.0ol 8.14ol 98Kln 1.0d 1.7kJ i For van der Waals gas, the work done is: ( See D. (a) ) w vdw f i f i d nr nb n a d f nb 1 1 nr ln n a i nb f i f nb ( b) wvdw nr ln nb.0 d d 1.0ol 8.14ol 98Kln 1.0d d 1.8kJ i 1 1 f ( c) wvdw nr ln n a i f i kj 1.0ol 4.d 6 at ol -.0d 1.0d 1.7 kj 0.1kJ 1.5 kj

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13 A.7 As a continuation of Proble A.6, (a) show that for sall extensions of the chain, when, the restoring force is given by vk nk F l Nl (b) Is the variation of the restoring force with extension of the chain given in art (a) different fro that redicted by Hooke s law? Exlain your answer. Fro Proble A.6, we have k 1 v k F ln ln(1 v) ln(1 v) l 1 v l For v 1, we can aly the aylor exansion for the natural log at v0 0, () (1) f ( v0 ) f ( v) f ( v0) f ( v0) v v 1 ln(1 v) ln(1 0) v 1 0 v 1 ln(1 v) ln(1 0) v 1 0 v k vk F v ( v) l l Hooke law: F kx vk x k k F l Nl l Nl x herefore, for sall dislaceents, the one-diensional chain odel obeys the Hooke s law.

14 B.1 he following data show how the standard olar constant-ressure heat caacity of sulfur dioxide varies with teerature. By how uch does the standard olar enthaly of SO (g) increase when the teerature is raised fro 98.15K to l500k? /K θ ( ) he change in the olar enthaly of foration of SO as the teerature increases fro 00K to 1500K is calculated as follows: 1500K C 98K, ( H f ) ( ) d C, ( ) can be obtained by fitting the heat caacity data to an equation of the for: ( ) c C a b, We find: a JK ol b JK ol - c JK ol hus K ( H ) C ( ) d f 98K 1500K 98K 1, a b 6.kJol c a b c 1500K 98K d

15 B. A sale consisting of.0 ol CO occuies a fixed volue of 15.0d at 00 K. When it is sulied with.5 kj of energy as heat its teerature increases to 41 K. Assue that CO is described by the van der Waals equation of state and calculate.

16 C.1 A sale of the sugar D-ribose (C 5 H 10 O 5 ) of ass 0.77 g was laced in a constantvolue calorieter and then ignited in the resence of excess oxygen. he teerature rose by K. In a searate exerient in the sae calorieter, the cobustion of 0.85g of benzoic acid, for which the internal energy of cobustion is -51 kj ol gave a teerature rise of K. Calculate the internal energy of cobustion of D-ribose and its enthaly of foration. Coound Mass Molecular ass Nuber of ole D-ribose 0.77g 150 g/ol x10 - ol Benzoic acid 0.85g 1 g/ol 6.76x10 - ol Heat caacity of calorieter: C U K 11.kJK 51kJol ol Internal energy of cobustion of D-ribose C 5 H 10 O 5 (s)+5o (g) 5CO (g)+5h O(l) U c c C n 11.kJK ol H U n R c K 17kJol g U (0) R 17 kj ol c H f [C5H10O 5(s)] 169.7kJol

17 C.9 An average huan roduces about 10 MJ of heat each day through etabolic activity. If a huan body were an isolated syste of ass 65kg with the heat caacity of water, what teerature rise would the body exerience? Huan bodies are actually oen systes, and the ain echanis of heat loss is through the evaoration of water. What ass of water should be evaorated each day to aintain constant teerature? he needed data are the secific heat caacity of water and enthaly of vaorization: D. 1 In 006, the Intergovernental Panel on Cliate Change (IPCC) considered a global average teerature rise of o C likely by the year 100 with.0 o C its best estiate. Predict the average rise in sea level due to theral exansion of sea water based on teerature rises of 1.0 o C,.0 o C and.5 o C given that the volue of the Earth s oceans is and their surface area is, and state the aroxiations that go into the estiates. Assutions: 1. Global water teerature is 5 o C at resent.. Melting of glaciers is neglected. herefore, (a) Rise in sea level is solely due to theral exansion (b) Mass of sea water is fixed

18 d d d (surface area) h K ( ) ( C) - (gc ) (k ) h(k) h() E E E-0.61 Alternatively, we can assue the coefficient of theral exansion is a constant for sall change in teerature. 1 P We sily use its value at 98K: [H O( l), 98 K] K 4 K ( ) (k ) h(k) h() , x , x ,1,85.41 x

19 D. Starting fro the exression ( ) ( ), use the aroriate relations between artial derivatives to show that C C / / Evaluate C -C v for a erfect gas. We ay use the Euler s chain relation we roved in Assignent Substitute in the given exression, we have C C For erfect gas, C C nr nr nr

20 D. 9 Use the fact that ( ) for a van der Waals gas to show that by using the definition of and aroriate relations between artial derivatives. Hint: Use the aroxiation P = R when it is justifiable to do so. Consider n=1 ol, H C, H U P H U ( ) H U a ( )( b) R 1 R R a b 1 b a n n( n 1) We aly binoial exansion 1 x 1 nx x and assue 1 a a a a R Ra R a b b R R a b R U a R Ra a R C, H a b R a End of Assignent