Physics 115. Molecular motion and temperature Phase equilibrium, evaporation

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1 Physcs 115 General Physcs II Sesson 9 Molecular moton and temperature Phase equlbrum, evaporaton R. J. Wlkes Emal: phy115a@u.washngton.edu Home page: 4/14/14 Physcs 115 1

2 Lecture Schedule (up to exam 1) Just joned the class? See course home page courses.washngton.edu/phy115a/ for course nfo, and sldes from prevous sessons 4/14/14 Physcs 115 Today 2

3 Announcements Prof. Jm Red s standng n for RJW ths week Exam 1 ths Frday 4/18, n class, formula sheet provded YOU brng a bubble sheet, pencl, calculator (NO laptops or phones; NO personal notes allowed.) We wll post sample questons tomorrow, and go over them n class Thursday Clcker responses from last week are posted, so you can check f your clcker s beng detected. See lnk on class home page, 4/14/14 Physcs 115 3

4 Last tme Gas Law: Avogadro s number and R Countng molecules to get N s dffcult, so t s convenent to use Avagadro s number N A, the number of carbon atoms n exactly 12 g (1 mole) of carbon. 1 mol {molecular mass, A} grams of gas (For elements, what you see on the Perodc Table s A averaged over sotopes) N A x molecules/mole and N nn A, where n number of moles of gas PV nn A kt nrt Notce PV energy: N-m R N A k J/(mol K) Notce: for real gases, PV /nt 8.3J/(mol K) only at low P PV nrt Ideal Gas Law, n moles R Unversal gas constant Good approx at low P for real gases Non-deal gases 4/14/14 Physcs 115 4

5 Isotherm plots PVNkT results from many dfferent observatons: Hold N, T constant and see how P, V vary: fnd PV const P const V (wth T and N fxed) Boyle s Law For dfferent T s we get a set of (1/V)-shaped curves 4/11/14 Physcs 115A 5

6 Isobar plots Hold N, P constant and see how V, T vary: fnd V (const)t ( constant N, P) Charles & Gay-Lussac Law For dfferent P s we get a set of lnear plots 4/11/14 Physcs 115A 6

7 Example: Volume of an deal gas What volume s occuped by 1.00 mol of an deal gas f t s at T 0.00 C and P 1.00 atm? PV V nrt so nrt P (1.00 mol) L atm/(mol K) ( K) L [ ] (1.00 atm) If we ncrease the V avalable, wth same T: P must drop If we ncrease the T, wth V kept the same: P must rse Standard Temperature and Pressure (STP) 0 C, 1 atm At STP, one mole of any deal gas occupes 22.4 lters 4/14/14 Physcs 115 7

8 Example: heatng and compressng a gas An deal gas ntally has a volume 2.00 L, temperature 30.0 C, and pressure 1.00 atm. The gas s heated to 60.0 C and compressed to a volume of 1.50 L what s ts new pressure? PV T PV T so VT (2.0 L)(60.0 C C) (1.00 atm) (1.5 L)( ) 1 2 P2 P1 VT 2 1 C + C 1.47 atm Notce: we must use Kelvn temperatures when applyng deal gas laws what would result have been f we use the rato (60/30)? 4/14/14 Physcs 115 8

9 Quz 5 Two contaners wth equal V and P each hold samples of the same deal gas. Contaner A has twce as many molecules as contaner B. Whch s the correct statement about the absolute temperatures n contaners A and B, respectvely? A. T A T B B. T A 2 T B C. T A (1/2)T B D. T A (1/4) T B E. T A (1/ 2)T B 4/14/14 Physcs 115 9

10 Quz 5 Two contaners wth equal V and P each hold samples of the same deal gas. Contaner A has twce as many molecules as contaner B. Whch s the correct statement about the absolute temperatures n contaners A and B, respectvely? A. T A T B B. T A 2 T B C. T A (1/2)T B PV nrt so T ( PV / nr) ( 1/ n) D. T A (1/4) T B E. T A (1/ 2)T B 4/14/14 Physcs

11 Relatng gas laws to molecular moton P, V, T are macroscopc quanttes Human-scale quanttes, measurable on a table-top Molecular moton (x, v vs t ) mcroscopc quanttes Knetc theory of gases: connect mcro to macro Model for deal gas N s large, molecules are dentcal pont-partcles Molecules move randomly No nelastc nteractons: collsons are always elastc Recall: elastc means no loss of KE due to collson Elastc collson wth wall means momentum (so, v ) component perpendcular to wall gets reversed Speed unchanged Vertcal v unchanged Horzontal v reversed 4/14/14 Physcs

12 Calculate the pressure of a gas Change n horzontal momentum of molecule Δp x p x, f p x, mv x ( mv x ) 2mv x Change s due to force exerted by wall: FΔt Δp x, F ON WALL F BY WALL Average force exerted on wall by one molecule F AVG Δp x Δt where Δt tme between collsons round-trp tme Δt 2L / v x F AVG 2mv x mv 2 x 2L / v x L Assume symmetrcal contaner (LxLxL): (doesn't matter n the end) P AVG F AVG A 1 $ 2 mv ' x L 2 & % L ) mv 2 x ( V Addng up all molecules, PV N m ( v 2) x AVG 4/14/14 Physcs

13 Defnng temperature (agan): molecular scale Now we can connect macro to mcro: PV NkT 2N 1 mv 2 2 x ( ) av ( 1 mv 2) 1 kt 2 x av 2 Nothng specal about the x-drecton: random moton means ( v 2 ) ( 2 ) ( 2 ) and ( 2 ) ( 2 ) ( 2 ) ( 2 ) 3( 2 ) x vy vz v vx + vy + vz vx av av av av av av av av (because random v components are ndependent of one another) The average translatonal knetc energy of the molecules s: K translatonal av ( 1 ) 2 mv2 3 av 2 ( 2 ) trans 2 av 2 2 kt per molecule K N mv NkT nrt ( v 2 ) 3kT av m molecule Deep and fundamental! Avg KE of gas molecule s proportonal to T, wth Boltzmann constant as the factor Root-mean-square (RMS) - useful avg where quantty-squared s what matters: 3N A kt N A m molecule 3RT M MOLE and v RMS v 2 ( ) av 3RT M MOLE 4/14/14 Physcs

14 Example: RMS speed of gas molecules RMS means: take each molecule s speed and square t, then fnd the average of those numbers, and THEN take the square root. In practce: we fnd the statstcal speed dstrbuton of the molecules, and use that to estmate RMS speed Oxygen gas (O 2 ) has a molar mass* M of about 32.0 g/mol, and hydrogen gas (H2) has a molar mass of about 2.00 g/mol. Assumng deal-gas behavor, what s: (a) the RMS speed of an oxygen molecule when the temperature s 300K (27 C), and (b) RMS speed of a hydrogen molecule at the same temperature v O2 RMS 3RT M O 3(8.314 J/mol K)(300 K) ( kg/mol) Note: Walker says molecular mass for molar mass confusng. M X grams n 1 mole of X, m X mass (n kg) of one X molecule 485 m/s v H2 RMS 3RT M H 3(8.314 J/mol K)(300 K) ( kg/mol) 1,934 m/s 4/14/14 Physcs

$Probablty Dstrbutons We gve a 25 pont quz to N students, and plot the results as a hstogram, showng the number n of students, or fracton f n /N of students, for each possble score vs.$

15 Probablty Dstrbutons We gve a 25 pont quz to N students, and plot the results as a hstogram, showng the number n of students, or fracton f n /N of students, for each possble score vs. score, from 0 to 25. Such plots represent dstrbutons. For reasonably large N, we can use f n /N to estmate the probablty that a randomly selected student receved a score s. Notce, the fractons wll add to 1 for all possble scores, so that Σf 1. In that case the hstogram represents a normalzed dstrbuton functon. We have the followng relatons: It s not useful for class grades, but we could also calculate the average squared score: n N f 1 1 s n s f s av N 1 sav n s f s N 4/14/14 Physcs s s f s Peak or mode s wth max probablty 2 2 RMS av

Chapter 17 Temperature & Kinetic Theory of Gases 1. Thermal Equilibrium and Temperature

Chapter 17 Temperature & Kinetic Theory of Gases 1. Thermal Equilibrium and Temperature Any physical property that changes with temperature is called a thermometric property and can be used to measure