Announcement. Physical Chemistry I for Biochemists. Chem340. Lecture 9 (1/31/11) Yoshitaka Ishii. Homework 4 is uploaded at the web site

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1 hsical Chemistr I or Biochemists artial Derivatives Ch HW3 Continued Chem34 Lecture 9 /3/ Yoshitaka Ishii Announcement Homework 4 is uploaded at the web site Monda ep = e in case that ou do not know the detion o ep.

2 Derivatives d d d d d d d d d d 5 =^ a-+ d d d d d d artial Derivatives d d d Slope along the X ais d d d d

3 3 How to Obtain artial Derivatives? tet p44 d d d Step : Rewrite as a unction o and. nr / p Step : ut the constants outside the derivative. nr / is considered to be a t t thi ti l? nr constant or this partial derivative. Step 3: erorm the derivative with respect to nr Sample Question et p44 Calculate or an ideal gas. Step : Rewrite as a unction o and. nr / p Step : ut constants outside the derivatives. nr Step 3: erorm the derivative with respect to nr

4 Eample rom A.6 = HW3. 5. Calculate C v = U/ and C p = { U + / } or n mole o ideal gas assuming that U = 5nR/ and n is a constant. What kind o molecular properties do the depend on? In general U can be given b U or U. For an ideal gas U = 5nR/. U/ = 5nR// = 5nR/ / C p = { U + / } = { 3nR/ + nr/ } = { 5nR// } =? Change this into a unction o and 4

5 5 Ch3. Math roperties o State Function d d d d d d d I is a state unction otal Dierential Eact Dierential & the est or Eactness d is called total dierential o d g d d d d d is called an eact dierential i g or When d is eact is a state unction. Namel the change in does not depend on a path o. = in in -

6 Sample Question est o ineactness o Dw =-d HW4 #. We deine work as Dw = -d. Show Dw is an ineact dierential with respect to and or an ideal gas. Dw = -d + d.? nr / nr / artial Dierentials or here are 6 possible partial dierentials m m m m m m Q. How man are independent unctions? In general m =/ m Cclic Rule: z m z X = - Y m m An two o them having dierent colors are independent. 6

7 7 Sample Question HW4 3.3 For ideal gas show the cclic relationship i t is correct. Use =nr/ =nr/ =/nr nr nr nr / / /? nr nr nr / / / Some o the artial Derivatives Have Special Meaning : olumetric thermal epansion coeicient : Isothermal Compressibilit

8 8 How to Use hem? d d d / d d d in in d d in in ln Sample Question p46 tet

9 3. Dependence o 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 Dq C v d U d d U d U o be derived in Sec. 5.3 w & q in various process or ideal gas pe o work w q U Epansion or et = const isotherm - et -w adiabatic - et - et U/C Reversible epansion/ compression isotherm -nr ln in / -w adiabatic C v C v { in / a -} a=-c /C =- 9

10 HW 3.7 For. mol o an ideal gas eternal = =. 3 a. he temperature is changed rom. C to 5. C and C m = 3/R. Calculate q w U and H. U q C nr C C H C q p H in in Reversible Note U < H U U nr U w q et w U q moles o an ideal gas are epanded rom 45. K and an tial pressure o 5. bar to a inal pressure o. bar and C m = 5/R. Calculate w or the ollowing two cases: a. he epansion is isothermal and reversible.b. b he epansion is adiabatic and reversible. b. q = w = U = C in - C v = C nr So the question is how to get in? in / in in in in in in Derive this in [Q] / / in X a b =X ab

11 .6 One mole o an ideal gas or which C m = 3/R tiall at 98 K and. 5 a undergoes a reversible adiabatic abat compression. o At the end o the process the pressure is. 6 a. Calculate the inal temperature o the gas. Calculate q w U and H or this process. Adiabatic process q = [Q] & U = [Q] w = C v [Q3] i i i p p i p p i i i i.9 A clindrical vessel with rigid adiabatic walls is separated into two parts b a rictionless adiabatic piston. Each part contains 5. L o an ideal monatomic gas with C m = 3/R. Initiall i = 98 K and i =. bar in each part. Heat is slowl introduced into the let part using an electrical heater until the piston has moved suicientl to the right to result in a inal pressure = 7.5 bar in the right part. Consider the compression o the gas in the right part to be a reversible process. a Calculate the work done on the right part in this process and the inal temperature in the right part. b Calculate the inal temperature in the let part and the amount o heat that lowed into this part. a Right part is subject to adiabatic reversible compression. q R = Obtain in ; w R = U = C b Obtain the volume o the right part. L = total R. U = q L + w L ; Note: w L = -w R

12 Mawell-Boltzmann Distribution o Speed C 3/ M MC C 4 ep C R R C = {v +v Y +v } / M C mp 4 R / e O at 3K C mp R M / HW3 Q3. a Calculate dc/dc. b Show there two solutions or dc/dc = C and one solution gives the most probable speed C = R/M / where m is the weight o a gas molecule M is its molar mass and k is the Boltzmann constant. C 4 d C AC dc d dc M R d dc 3/ MC ep R MC ep C R MC {ep } R d du ep u du dc AC d dc MC ep R MC C ep R

13 . An ideal gas described b i = 3. K i =. bar and i =. L is heated at constant volume until =. bar. It then undergoes a reversible isothermal epansion until =. bar. It is then restored to its original state b the etraction o heat at constant pressure. Depict this closed- ccle process in a diagram. Calculate w or each step and or the total process. What values or w would ou calculate i the ccle were traversed in the opposite direction? = const = const = const bar L bar L bar L bar L Which process involves work? How much work is involved? For the reverse process: W reverse = -W or each step For the reverse ccle: W reverse ccle = -W ccle.43 One mole o N in a state deined b i = 3. K and i =.5 L undergoes an isothermal reversible epansion until = 3. L. Calculate w assuming b that the gas is described b the van der Waals equation o state. Isothermal process = but or a vdw gas ou cannot use U = C! For a reversible process et = int in in w d d et int R b What is the percent error in using the ideal gas law instead o the van der Waals equation? a int m m 3

14 .3 A pellet o n o mass. g is dropped d into a lask contang i dilute H SO 4 at a pressure o =. bar and temperature o = 98 K. What is the reaction that occurs? Calculate w or the process. he chemical equation is given b n s H SO 4 aq n aq SO 4 aq H g - W = - et H 4