Kinetic Molecular Theory of Ideal Gases
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1 Kinetic Molecular Theory of Ideal Gases Bernoulli et al. (178) Theoretical deelopment of ideal gas laws that were determined empirically Boyle Charles Gay-Lussac Aogadro ( ) ( ) ( ) ( ) PV = k 1, [n,t] V/T = k, [n,p] V/n = k, [P,T] PV = nrt Main Postulates 1. Gas molecules in ceaseless chaotic motion.. Pressure (P = f/a) eerted on the container walls is due to the bombardment of the container by the gas molecules.. All molecular collisions are elastic, i.e., no energy loss due to friction. 4. No intermolecular forces. 5. Molecules are point masses, i.e., infinitesimally small molecular olumes. 6. Absolute T is proportional to the aerage kinetic energy of all the molecules. 1 Gas Problems: Ideal Gases: 1-1. Prof. Zi C. Koren
2 N identical gas molecules m = mass of each molecule i i molecule elocity ector of k j i iz iy iz iy i. Mean Square Velocity z y N Mean or Aerage N i For random motion : z y Mean Square Velocity : Note z y Mean (or Aerage) Velocities Prof. Zi C. Koren
3 The Model A l For simplicity, first consider: 1. Only one molecule (i) is present.. Molecule s motion is only in -direction: only component. & molecule i will be hitting wall A with elocity. Then, we ll consider N molecules. Prof. Zi C. Koren
4 f A l So, for one molecule i: f = force eerted by molecule i on wall (m) (Newton) ma m t t Rate of change of momentum Note: force leads to a change of elocity (and momentum) upon collision with the wall For each collision cycle: f p t m / m p t p = m(+ ) m(- ) = m t = d/ = l/ For N molecules (continued): 4 Prof. Zi C. Koren A
5 So, for one molecule i (from before): f p t m / m For N molecules: f total, N f i m m N f total 1 Nm N i N ( correction of the assumption that all moement is only in -direction) P total f total A 1 Nm A 1 Nm V 1 PV Nm (continued) 5 Prof. Zi C. Koren
6 (continued) 1 PV N m N = # of molecules = n N Aogadro, N A = molecules/mol M = molar mass [g/mol] = N A m m total = N m = n M PV KMT (Note the units) PV 1 n M 1/ ep PV RT M rms = root mean square elocity Graham s Law of Diffusion and Effusion (for gases at the same T): t t R R 1 1 N (½) M M 1/ m 1/ nrt KE k T B k B nn KE A Boltzmann constant R, N R = 8.1 J/molK A Prof. Zi C. Koren N KE PV KMT KE total 1 kt (Energy of translation) ½kT = basic unit of molecular energy for each independent motion
7 From KMT PV N KE T All the empirical gas laws can be deried: Boyle: Charles & Gay-Lussac: Aogadro: PV = k 1, [N,T] V/T = k, [N,P] V/N = k, [P,T] Recall: At what conditions of T or P, does a real gas behae as if it were ideal? 7 Prof. Zi C. Koren
8 Mawell-Boltzmann Distribution of Molecular Velocities MW Effects: = f (T, MW) Why do molecules, all at the same T, hae such a wide span of elocities? > > (continued) 8 Prof. Zi C. Koren
9 Temperature Effects 1: (continued) 9 Prof. Zi C. Koren
10 Temperature Effects : 10 Prof. Zi C. Koren
11 11 Prof. Zi C. Koren
12 Most-Probable, Mean, and Root-Mean-Square Velocities Mawell s Distribution of Speeds: f( ) M 4 RT / e M / RT # mp 4/π mp 1/ / mp RT/M RT/M rms mp 1 Prof. Zi C. Koren
13 The mean speed with which one molecule approaches another identical molecule (eact deriation is too cumbersome) Qualitatie rendition: Relatie Mean Speed: One etreme rel Typical Another etreme from before rel : For two dissimilar molecules approaching each other: rel 8kT/ μ m m A mb m A B reduced mass 1 Prof. Zi C. Koren
14 Collision Frequency (z) & Collision Diameter (d) z = Aerage # of collisions per second made by one molecule in a system of N molecules in a olume V: d z reln/v = d = collision cross-section z relp/ kt (for an ideal gas), recall :k R/N A At constant T, z P. Logical? (target area that a molecule presents to an incoming molecule) A hit occurs when the centers of two molecules come within a distance d of each other, where d is the diameter of impenetrable hard sphere molecules. For a sample held at constant olume, as T increases, z inc. bec. rel inc. Collision Cross-Sections Gas /(nm) C 6 H Eample: For an N molecule at 1 atm and 5 o C, z s -1 CO He N 0.4 Prof. Zi C. Koren
15 Mean Free Path, = The aerage distance a molecule traels between collisions t free = Time in free flight between collisions = 1/z, = Aerage distance traeled in free flight = λ /z kt/( σ P) For N at 1 atm: = 70 nm 10 molecular diameters In a container of fed olume, is dependent on T? Summary t free z = collision frequency A typical ideal gas molecule (N or O ) at 1 atm and 5 o C traels at a mean speed of 50 m/s; Each molecule collides within 1 ns, Between collisions it traels molecular diameters. If d <<, gas is nearly ideal. Why? Gas Problems: KMT: 4, Prof. Zi C. Koren
16 Lecture Problem #1 (to hand in net week to the Recitation instructor): Fill in the table below at 1 atm and 5 o C: Gas mp /(m/s) /(m/s) rms /(m/s) d /(nm) /(nm) /(m/s) z /(s 1 ) t free /(ns) λ /(nm) λ/d C 6 H 6 CO He N (see net slide) 16 Prof. Zi C. Koren
17 Notes About the Preious Table: Careful with all the units. Don t forget to conert atm to Pascal and that (nm) is m. Gie the full reference for d (author s last name & first name initial, book title, publisher, city of publication, year, page number; and eact website address, if from internet). Eplain all the alues and interpret them. הנתונים לגבי קוטר מולקולרי d בשני הספרים האלה: e=bl&ots=_jaairauo&sig=9y8rvid9n7f5efgkhlpfazjpso#=onepage&q=benzene%0collision%0diameter&f=false יש גם טבלה בספר הקלאסי הבא שחישב את הקוטר המולקולרי לפי שתי שיטות )לפי פאראמטר B ולפי הצמיגות )η) של הגז), אבל תזהרו מהטבלה מכיוון שהוא מסמל את הקוטר עם אות σ שהיא משהו אחר במשוואות שלנו. תראו תנסו ותהנו. 4.html 17 Prof. Zi C. Koren
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