Mathematical Intermezzo 3: More Mathematical Relationships Among Partial Derivatives (PD s)

Size: px

Start display at page:

Download "Mathematical Intermezzo 3: More Mathematical Relationships Among Partial Derivatives (PD s)"

Melina Newton
6 years ago
Views:

1 Mathematical Intermeo 3: More Mathematical Relationships Among artial Derivatives (D s) 1. Eact Differential Epression (total differential in terms of partial derivatves) If =(,) is a state function, then when and change b d and d, changes b: d d d 3. Second artial Derivatives Ma Be aken In An Order 2 2 = 4. est For Eactness 2. Creating And Relating Similar artial Derivatives Divide b d and set f constant for new D s, where f=f(,) also: f f V U U and e.g.: N M Nd Md d d and, eact, where: is If N M hen (continued) 1 rof. Zvi C. Koren

2 5. Inversion Rule 1 cclic permutation: 6. Chain Rule, Epression, 1. EactDifferential From d d d If =(,) and =(,f), then =(,f). f f divide b df and set constant for new D s 7. Euler s Ccle Rule 1 d, d d divide b d and set constant for new D s, and use Inversion Rule: (Applications on net slide) [Leonhard Euler, 1707 (Basel, Switerland) 1783 (St etersburg, Russia): From 1. Eact Differential Equation, Nike roblem: Just Do It 2 rof. Zvi C. Koren

3 Some Applications of the Mathematical Relationships: Relating Similar artial Derivatives - Eamples V C course: of ; and 2. Strateg is to transform to known measurable coefficients: 1. Begin with 2 nd, Create the 1 st 2. Convert /* to 2 V/* D s in order to introduce coefficients and/or, through Ccle Rule and Inversion Rule, etc. 3. Convert /* to C and /* (giving a measurable change in propert), through Ccle Rule and Inversion Rule, etc. Interim Results: Vπ C U U U V V and 1. μc Final Result: κ αμ C V 1 Nike roblem: Just Do It But we dit I (cont d) κ α V Joule-homson Coefficient μ Nike roblem 3 rof. Zvi C. Koren

James rescott Joule (1818-1889) English phsicist Joule-homson Coefficient μ = ( i, i ) Denotes the abilit of an epanding gas to undergo cooling or heating(!

submarine cable between Ireland and Newfoundland. For his work on the transatlantic cable homson was created Baron Kelvin of Largs in 1866 b the British government.

4 James rescott Joule ( ) English phsicist Joule-homson Coefficient μ = ( i, i ) Denotes the abilit of an epanding gas to undergo cooling or heating(!) at constant Signs of +, cooling, heating William homson (Lord Kelvin) 1824 (Belfast, Ireland) 1907 (Netherhall, near Largs, Arshire, Scotland) homson designed and implemented man new devices, including the mirrorgalvanometer that was used in the first successful sustained telegraph transmissions in transatlantic submarine cable between Ireland and Newfoundland. For his work on the transatlantic cable homson was created Baron Kelvin of Largs in 1866 b the British government. he Kelvin is the river which runs through the grounds of Glasgow Universit and Largs is the town on the Scottish coast where homson built his house. Note: All gases, ecept e and 2, undergo cooling from 1 atm and 25 o C Intermolecular forces for a substance at a certain state of matter = f(,) = 0 at the inversion temperature, I Analtical roblems: d = d = d 4 rof. Zvi C. Koren

Apparatus for Measuring the Joule-homson Effect: Cooling b Adiabatic Epansion Gas epands through the porous barrier, which acts as a :( מ ש נ ק, צ ו אר ( throttle Isenthalpic epansion (stead stream)

5 Apparatus for Measuring the Joule-homson Effect: Cooling b Adiabatic Epansion Gas epands through the porous barrier, which acts as a :( מ ש נ ק, צ ו אר ( throttle Isenthalpic epansion (stead stream) his one is maintained at a constant low pressue. his one is maintained at a constant high pressue. Whole apparatus is thermall insulated: Adiabatic. (ceramic) (stead stream) 5 rof. Zvi C. Koren

n these 2 parts are reall on the same side; so, of course, et p i is the same as int i.

6 hermodnamic Analsis of the Joule-homson Effect: et virtual piston Consider the passage of a given amount of gas through the throttle from high i to low f. n these 2 parts are reall on the same side; so, of course, et p i is the same as int i. he pistons represent the upstream and downstream gases, which maintain constant pressures either side of the throttle. these 2 parts are reall on the same side (continued) n et 6 rof. Zvi C. Koren

Gas is compressed isothermall b the upstream gas acting as a piston with pressure p i : Gas epands isothermall against p f provided b the downstream gas acting as a piston to be driven out: w left =

7 Gas is compressed isothermall b the upstream gas acting as a piston with pressure p i : Gas epands isothermall against p f provided b the downstream gas acting as a piston to be driven out: w left = p e dv = i (0 V i ) = + i V i w total = w left + w right = i V i f V f w right = p e dv = f (V f 0) = f V f U total = U f U i = w total, q adiabatic =0 = i V i f V f U f + f V f = U i + i V i f = i Joule-homson Effect is an Isenthalpic rocess (continued) 7 rof. Zvi C. Koren

8 Direct (as in the previous set-up): μ Measurements of Indirect (modern method): lim 0 i (tpicall: i > f ) f lim 0 Recall: μc known known Isothermal J- coefficient (I AE IS NAME) q = electrical heating required to offset the cooling = q : [= f is constant in right compartment] (continued) 8 rof. Zvi C. Koren

9 μ μ(,) Notes: 1. is a slope 2. = 0 at I (inversion temperature), at an isenthalp maimum 3. here are 2 inversion temps. at a given, a higher I and a lower I. 9 rof. Zvi C. Koren

Selected Values at 1 atm Gas b /K I /K upper 298K / Katm -1 CO 2 194.7 1500 1.11 20.

10 Selected Values at 1 atm Gas b /K I /K upper 298K / Katm -1 CO e N O 2 10 rof. Zvi C. Koren

11 Joule-homson Coefficients (in deg/atm) for N 2 [able 7-1 of Maron & Lando] -150 C -100 C 0 C 100 C 200 C 300 C (atm) First-Law roblems: Joule-homson: rof. Zvi C. Koren

Linde Refrigerator: Liquefaction of Gases http://www.chemheritage.

deutsches-museum.de/ausstell/dauer/phsik/e_luft.

near Baden, Austria; devised process of liquefing air 1895.

12 Linde Refrigerator: Liquefaction of Gases Carl von Linde ( ), German engineer, born in Berndorf, near Baden, Austria; devised process of liquefing air cools gas low- cold gas cools epanding high- gas, which cools further upon epansion 12 rof. Zvi C. Koren

13 Final Notes Regarding J- Coefficient 1. For an ideal gas, = 0. Nike roblem: Just Do It So, is unchanged for an i.g. during Joule-homson epansion. owever, simple adiabatic epansion does cool an ideal gas because it does work. 2. So, wh isn t 0 as 0 for real gases? Because depends on derivatives and not directl on, V, and themselves. 13 rof. Zvi C. Koren

Chapter 3 The Importance of State Functions: Internal Energy and Enthalpy. Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid

Chapter 3 The Importance of State Functions: Internal Energy and Enthalpy. Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 3 he Importance of State Functions: Internal Energ and Enthalp homas Engel, hilip Reid Objectives Epress the infinitesimal quantities du and dh as eact differentials. Derive the change of U with