Physical Chemistry I for Biochemists. Chem340. Lecture 12 (2/09/11) Yoshitaka Ishii. Announcement

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1 hsical hemistr I for Biochemists hem34 Lecture 1 /9/11 h4.1- Review for Eam 1 Yoshitaka Ishii Announcement lease submit W4! he area to be covered is h1-3 3including the lecture materials Eam 1 will be held this Frida 1.5 h You need a calculator for the eam I will be available for our last minute questions tomorrow at 5:-6: pm. he Lecture Note 13, which ma contain some additional hint on Eam 1, will be uploaded tomorrow ~4 pm. 1

2 h 4.1 Energ Stored in hemical Bonds is Related or aken Up in hemical Reactions Significant amount of internal energ or enthalp is stored in the form of chemical bond. 1/N g + q47 J Natom, g 1/ g + q18 J atom, g Assume that q is the mmum heat needed at 1 bar at 98.15K. he energ is described as a standard formation enthalp f for Ng and g as f for N = 47 Jmol -1 f for = 18 Jmol -1 Q. What is the formation energ for N 3? +

3 Reaction Enthalp For a reaction that takes place at constant and, the heat flow needed d for a chemical reaction is reaction. A A + B B X X + Y Y reaction = X fx + Y fy - A fa - B fb Be sure to read tet p66. Assignment for Eam Solve Eample roblem 4.1 in page 67! Reaction Enthalp For a reaction that takes place at constant and, the heat flow needed for a chemical reaction is reaction. A A + B B X X + Y Y reaction = X fx + Y fy - A fa - B [Q1] fb In general, or r oduct Reac tant reaction i f, i k f, k ix k A i f, i i reaction Note use I > for products and I < for reactants 3

4 Overall Stud Guide ~6% from the W or modified W ompletel understand W questions Stud Quiz 1 Multiple hoices 5-6% - Deition 15- % check vocabular list at the end of each chapter in the tet - onceptual Question 15- % - alculations -3 % Long alculation ~4% Derivations ~1 % practice until ou can derive it b ourself Long alculation ~4% 4 Questions We will pick questions perhaps 1 more out of the following W questions. W1: 1., 1.9, 1.15, 1.7, 1.3 W: Q4, Q7, Q8 we give ou n a n! e d n1 W3:.7,.19,.,.3,.6,.9, Q3 W4: 3., 3.3, 3.16, 3.19, 3.3, Q1 a 4

5 3.15 An 8.-g piece of gold at 65. K is dropped into 1. g of Ol at 98 K in an insulated container at 1 bar pressure. alculate the temperature of the sstem once equilibrium has been reached. Assume that,m for Au and O is constant at their values for 98 K throughout the temperature range of interest. eat from gold to O gold pm n65 eat to O from gold O pm m-98 Or Use q = n pm f i & = q gold + q O 3.16 A mass of 35. g of Os at 73 K is dropped into 18. g of Ol at 35 K in an insulated container at 1 bar of pressure. alculate the temperature of the sstem once equilibrium i has been reached. Assume that t,m for O is constant at its values for 98 K throughout the temperature range of interest. eat to Os: n + water m fusion pm n 73K eat from Ol to ice: water pm m35 - Or = q ice73k water + q water35k = n m fusion + water pm n 73K + water pm m - 35 For 3.16 Use enthalp of fusion for Ice 68 Jmol -1 5

6 3.16 Mod A mass of 35. g of Os at 53 K is dropped into 18. g of Ol at 35 K in an insulated container at 1 bar of pressure. alculate the temperature of the sstem once equilibrium i has been reached. Assume that t,m for O is constant at its values for 98 K throughout the temperature range of interest. Ice53K Ice73K Water73K Water p,ice fusion p,water 73K r this! W Because / = J, the change in enthalp of a gas epanded at constant temperature can be calculated. o do so, the functional dependence of J on must be known. Q1. reating Ar as a van der Waals gas, calculate when 1 mol of Ar is epanded from 4. to 1. bar at 3. K. Assume that µ J is independent of pressure and is given b µ J = [a/r b]/,m, and,m = 5/R for Ar. Q. What value would have if the gas ehibited ideal gas behavior? Q1. d = / d + p d What is d in this case? d = [Q1] ow do ou obtain? [Q] d For an ideal gas, J- =. his means / = [Q3] 6

7 h3. ow to Obtain artial Derivatives tet p44 d, d d Step 1: Rewrite as a function of and. nr / Step : ut the constants outside the derivative. / nr 1 is considered to be a constant t for this partial derivative. Step 3: erform the derivative with respect to 1 nr? artial Differentials for,, here are 6 possible partial differentials m m m m m m Q. ow man are independent functions? 1 In general, m Z =1/ m Z clic Rule: z m 1 z X = -1 Y m m Z An two of them having different colors are independent. 7

8 8 h3. otal Differential, Eact Differential, & the est for Eactness Z Z dz is called total differential of Z d g d f d Z d Z dz,,,,, dz is called an eact differential if Z Z,, g f,, or When dz is eact, Z is a state function. Namel, the change in Z does not depend on a path of,. Z =Z, - Z, W Starting with the van der Waals equation of state, d an epression for the total differential d in terms of d and d. B calculating the mied partial derivatives // and // determine if // and //, determine if d is an eact differential. an der Waals equation: m m a b R Note: he second term vanishes ou should show wh. is a state function for vdw gas nb nr nb nr b R m nb nr nb nr b R m nb nr nb nr nb nr nb nr

9 Is d, eact for ideal gas? f, d, nr d 1 nr nr d g, 1 nr f nr Same nr nr g is a state function for an ideal gas nr 3. Dependence of U on and U varies b changing and as U du, U d Using du = q + w = Dq - et d Dq et When d = U U d Dq d lease memorize this Dq d U d d U d U v o be derived in Sec. 5.3 You do not have to memorize 9

10 1 eat apacit & U/ v d Dq U 1 1 mv v d n d U fi U d U In general, d d U v v 1 Zero for ideal gas Small for real gas Joule s Ep gas vdw for a U m / lease memorize this 3.4 as a function of & varies b changing and as d d d When d = & = et d d d, d d Dq p p About Derivation See Sec. 3.6 d Dq About Derivation See Sec. 3.6 When = k his term is zero Q.What is this / value for an ideal gas? problem 3.7 p57

11 11 eat apacit & / d Dq 1 1 m d n d fi d In general, d d p p 1 lease memorize this J p p Zero for ideal gas Small for real gas J- Ep h1 Deitions W1, Lec. 1- Sstem, Surrounding, losed Sstem, Isolated Sstem Open Sstems Sstem, Open Sstems Equilibrium, Zero-th law of thermodnamics Equation of State for ideal gas artial ressure i / = n i /n = i an der Waals Equation of State Meaning of a and b for = nr/-nb-n a/ Isotherm, Isobar, Isochore SI Units m, kg, s, A, K, mol, cd J,

12 Equations to be Memorized = nr, = R - ow to choose appropriate R - onvert units L = 1-3 m 3 - is in K : K = artial ressures, Molar fractions i / = n i R//nR/ = n i /n = i = ; n = n 1 + n + n 3 + orrect for an ideal gas 1 Equilibrium 1, 1, n 1,..,, n,.. wo isolated sstems characterized b sstem variables such as,,.. he echange of energ and/or matters through the boundar A sstem variable reaches a constant over time in an part of the sstems Equilibrium hermodnamics equilibrium: Equilibrium with respect to,, and concentration or n/ 1, 1, n 1,..,, n,.. 1

13 Zeroth Law of hermodnamics p4 wo sstems that are separatel in thermal equilibrium i with a third sstem are also in thermal equilibrium hermal Equilibrium & -3 hermal Equilibrium 1- hermal Equilibrium Sample question on isothermal process 8 6 a / 1/m3 =nr/ Q1. he 4 lines shows isothermal plot representing 1/ dependence of for the same ideal gas at = K, 3K, 4K, 5K. Which graph shows that for 5 K? Q. redict how man moles of gas eists? n = /R ~ 6 1/1.4/8.35 =.1 mol 13

14 Non-ideal Gas wo assumptions for the ideal gas 1 No interactions between gas molecules he gas molecule can be treated as a point mass no volume is considered he assumption is correct when gas densit is low. roughl speaking < 1 bar and > 1K van der Waals Equation of State nr nb n a b: ite size of 1 mol of the molecules a: attractive force constant R b a m m I will give ou the vdw equation, but understand the meaning of nb & -n a/ m / n 14

15 Deitions W, Lec 3-5 Isothermal - lot for Real Gas Deviation from ideal gas, Liquidationid - lot for an der Waals Equation ritical arameters c, c, mc c mc = 3/8R c Boltzmann distribution Average speed <c>, rms speed <c > 1/, most probable speed c mp Equations to be Memorized ritical arameters: mc = 3b, c = a/7b, c mc = 3/8R c Boltzmann Distribution p j ep E j / kb / Z Kinetic Energ of Gas for one molecule/one mole E mv / 3k / trans mv X mv Y mv B Z k B Average speed <c>, rms speed <c > 1/, most probable speed c mp see the net slides for equations Q. ow much is the energ for one mole? 15

16 <>, < > 1/, Most probable Raff Average speed, : ome work 8/ =.6 3 / 1 / m 3 8R m 4 ep d k k M Root-Mean-Square speed rms : 1/ 3 / 1/ m m ep d k k R 4 ep M 1/ Most probable : mp 1/ R d/d = mp M Q. Which is greater? rms > > mp Isothermal - lots for Real Gas O 5 c: ritical temperature Liquid O ondensation from gas to liquid below c c X denotes inflection point in isotherm d/d =, d /d = & at this point are named c and c. Raff p

17 bar van der Waals Equation for O Well reproduces the eperimental curves above c! 14 bar ideal gas at 3K 1 O at 9K 1 O 34.3K 8 O 315K 6 O 35K 4 Mawell construction reproduces G-L transition Lmol-1 ritical temp obtained from d/d m = and d /d m = alculate c from an der Waals Eq. R b a 6 4 m m m m At c, the following relationships are epected. m R m c b a 3 m R 6 a c 3 4 b m 4 m m R b b 3 m 6a R a 4 m m b 3 mc = 3b, c = 8a/7bR, c = a/7b c mc 3 R / 8 c m 17

18 Mawell-Bolztmann Distribution Molecule Molecule Dependence of of Speed Speed Distribution Distribution opulation m/s 39 m/s O at 3K 115 m/s M A B M mp M B M A R M A B 1/ O MW ~3 m/s Which molecules represent the red line? a e b N c Ar d l Derivation - Kinetic heor of Gas Atkins p An ideal gas of N molecules mass m is enclosed in a cubic bo of length L L -mv mv Ever time a molecule hit this wall, it gives a moment of p = mv. It takes t= L/v to hit the wall once. X Force b one molecule: F = mdv/dt = dp/dt ~ p/t = mv /L/v = mv /L 18

19 Derivation - Kinetic heor of Gas An ideal gas of N molecules mass m is enclosed in a cubic bo of length L m<v X >/ = k B / Force b one molecule F = mdv/dt = dp/dt ~ p/t = mv /L/v = mv /L L = F/L -mv mv B N molecules, N F mv k 1 1 mn N mn v k N k 1 v / L k / L / L = mn<v >/L 3 = Nk B /L 3 = nr/ Internal Energ of ranslation, Rotation, ibration et h3 p6 ranslational Energ for N molecules 3 NkB 3 nr U Internal Energ of Ideal Gas Rotational Energ U R = monatomic U R = Nk linear U R = 3Nk/ non-linear ow much is E Rot or E vib ibrational Energ for ideal gas, or novel ~ Nhc gas e, Ar? U ep hc ~ / k 1 19

20 v for an ideal gas Assuming that U = U trans + U rot for an ideal gas v U U U trans rot alues for n- mol of gas Monatomic e, Ar Linear O Non-linear O, 6 6 U trans 3nR/ 3nR/ 3nR/ U rot nr 3nR/ U trans + U rot 3nR/ 5nR/ 3nR 3nR/ 5nR/ 3nR Deitions h Lec 6-8 First law of hermodnamics Internal energ U eat, work Reversible & Irreversible process, Equilibrium Work for reversible process, irreversible process eat capacit, p,, pm, vm State function, ath function Integral over du, U, clic integral Enthalp

21 h. First law of hermodnamics: he internal energ, U, of an isolated sstem is constant. In other words, if the surrounding echanges energ with the sstem, the total energ of the surrounding and the sstem should not change. = U otal = U sstem + U surrounding U sstem = -U surrounding In a closed sstem: -U surrounding = qheat + wwork U = q + w Work b pressure ressure: = F/A, where A is area F = eternal A he work done b the sstem: w F dl eternal A dz d eternal eternal dz Adz A e Note: In general, eternal sstem 1

22 Work Equations to be memorized w F dl eternal A dz eternal d Equations related to epansion of gas see the net slide eat capacit & heat lim q Dq d q d When is a constant For ideal gas nr State function U U U du U U Integral & differentials see the following 3 pages Also, memorize equations needed for W3. w Math in Epansion of Gas d eternal We assume // d ase [3] Isothermal Reversible Epansion. eternal = int = nr/ & = const see p7 w eternal d ase [1] If eternal = const p4 w eternal d eternal eternal ase [] Reversible Epansion eternal = int w, d int w nr nr / d nr {ln nr {ln 1/ d nr [ln ] ln / } }

23 1 1/ 1/ n n 1 n Some math Lec. 3 n n a n a 1 n n n n 1 1/ n n 1 Math Summar of integral df f { } d d In Lecture 3, we learned d 1 d d d d 1/ 1/ d n n 1 n Y =epx X = lny lna-lnb = lna/b {1 } d 1/ d 1 / d d / n 11 1 nn / n n d Ecept for n = d ln 3

24 w & q in various process for ideal gas pe of work w q U Epansion for et = const isotherm - et -w adiabatic - et - et U/ Reversible epansion/ compression isotherm -nr ln / -w adiabatic v v { / a -1} a=1- / =1- Derivation -.8 Determng U & Introduction of Enthalp : a New State Function 1 Relationship between U and q under a constant If w= d =, U = q U can be determined b heat under a constant as U = q v What about it if = const? U = q + w = U = q - et q = U + et 3 Let us dee as U + int 4 Relationship between and q under a constant int = U + = q p for reversible process 4

25 Derivation -.9 alculating q, w, U, for Ideal Gas Note: all the equations in this section of the tet are valid onl for Ideal Gas U q U nr UnR U nr d p nr q d Derivation -.1 Reversible Adiabatic Epansion for an ideal gas Adiabatic process q = U = w d = - et d For a reversible process et = int d = - int d d = -nr/d 1/d = -nr1/d f in 1/ d nr 1/ d ln / f in nr ln / 5

26 ln ln / nr ln / p nr / ln ln 1 1 ln 1 Derivation - h 3.6 From deition = U + d =du +d + [Q1] d pd [Q] U d d d d d U d For isothermal process d =, U d U d d d d [ Q3 ] d d [ Q4] 6

27 7 Derivation Joule-hompson Ep Improved version of Joule s ep to d U/ L et = 1 R et = Work applied w = w left + w right 1 > et 1 et left right d d R L Adiabatic: q = U =U U 1 = w = [1] Eq [1] can be rearranged to Eq. [1] can be rearranged to U + = U = 1 : Isenthalpic epansion and in J- eperiment ields Joule-hompson oefficient J lim ow is J- related to / and U/? d d d p I J l h i t d [ p = / ] Using eq. [3.43] in h 3.6 In Joule-hompson eperiment d =. d d p J p p For an ideal gas, / = J- = g q [ ] d d U d d U J p

28 3.1 he Joule coefficient is deed b / U = 1/ [ / ]. alculate the Joule coefficient for an ideal gas and for a van der Waals gas. For an ideal gas: = nr/ = k For a van der Waals gas: p n R n b n a Onl the second term matters. 8