NY Times 11/25/03 Physics L 22 Frank Sciulli slide 1

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Bryce Mitchell
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1 NY Times /5/03 slide

2 Thermodynamics and Gases Last Time specific heats phase transitions Heat and Work st law of thermodynamics heat transfer conduction convection radiation Today Kinetic Theory of Gases slide

3 review Empirical Behavior of Ideal Gases in P, T, 7 8 th Centuries Experiments giving empirical behavior of gases in terms of volume, pressure, temperature, and mass of gas Keep other quantities fixed and P Boyle's Law T Charles Law P T Gay-Lussac Law m where m We put them together and express as the Ideal Gas Law n#of moles R gas constant p mass of gas nrt slide 3

Isothermal T is constant Essential, to predict general behavior, to know

4 Isothermal Expansion and Compression Note in general, the following can vary volume ppressure nnumber of moles (amount) of gas Ttemperature Isothermal T is constant Essential, to predict general behavior, to know what varies, what is constant, and more rules recall possible processes slide 4

5 Expansion at Constant Temperature p Units R 8.3 J/(mol-K) kn A Work (constant temperature) done obtained from integral in p- (see sample prob 0-) Expansion (or compression) at constant T follows isothermal contours with pconstant/ i nrt f f nrt f ln Ø Œ ø œ i W pd d nrt i Isotherms pnrt/ º ß slide 5

6 Sample Problem 0- Isothermal (T30K) expansion of mole (with p atm) of oxygen from liters to 9 liters How much work done? Final pressure? W nrt ln( /) f ()( 83. )( 30)ln( 9/ ) 80 Joules i p f i pi f (.) atm slide 6

7 review equivalent Gas Law from Atomic Perspective p nrt R p nrt nn T A ŁNA ł p NkT N # molecules Units k Boltzmann constant R 8.3 J/(mol-K) kn A k J/K (Boltzmann constant) slide 7

8 We will soon evaluate mean free path, l, between collisions! Gases and Atoms (molecules) We recognize now that gases consist of free (from each other) atoms or molecules Ideal Gas : interactions between atoms are elastic Interatomic forces can be neglected except at the instant of collision Most gases behave in a nearly ideal manner Interatomic forces (an der Waal forces) make only small modifications to the Ideal Gas Laws Monatomic Gas simple behaves like a billiard ball We consider this first and generalize slide 8

Kinetic Theory Atoms in Gases mass proton ~ mass neutron m N ~.7 0-4 grams A # protons + # neutrons From whence Avogadro s Number (N A )?

9 Kinetic Theory Atoms in Gases mass proton ~ mass neutron m N ~ grams A # protons + # neutrons From whence Avogadro s Number (N A )? Atom contains nucleus and electrons nucleus has neutrons (no charge) and protons (+e charge) Essentially all mass is in the nucleus atomic wt. A Molecule: use molecular weight A + A + slide 9

10 ) Define Derive Avogadro s Number (from the nucleon mass) ) But we hypothesize that the mole contains a fixed number of molecules 3) Mass of a single molecule is 4) Follows: 5) For () M mole mole A (grams) ( ) and (4) be consistent requires Can also do experiments to measure atomic velocities!! m N A molecule M Nm NmA A molecule A N Nm or N / m A N A ma 3 6) NA or NA atoms N N slide 0

11 elocity Distribution experiment q wt x vt v x w q Number molecules in fixed time interval. Measure and plot number vs speed this is velocity distribution or spectrum Peak of distribution moves higher if oven temperature is increased v slide

12 Probability (frequency) distributions x-component of velocity probability to have N in dv v v vpv ( ) dv x N x x x x v v v x y z v 0 x x 0 Relative probability velocity x-component (m/s) N v v vpv ( ) dv x N x x x x v v v x y z 0 v v + v + v x y z v 3 v x slide

13 Ideal Gas Law Derived from atoms! Average time between collisions on shaded wall is DtL/v x v x reversed other components the same F Consider dilute system of N moving, marble-like atoms in a (cubic) box pressure comes from impacts of atoms on walls Equal and opposite forces on atoms Calculate average force by wall on atom Infer pressure (F/A) on the wall F atom x x x ( ) mdv m v x x mv Dt L/ v L mv Fx L p A L N m p v with L 3 ŁL ł N p m v 3 x x x slide 3

14 N p m v x v v v x y z Ideal Gas 0 N x N x y z v v v v v v v v + v + v x y z rms x N N 3 rms N v N vx v 3 v It follows that p p Nm 3 v NkT rms Compare with slide 4

15 p p kt v rms Nm 3 3 mv Molecules in Motion rms v Check NkT rms 3kT m K mv kt atom 3 rms 3 int atom for gas of E N K NkT "billiard ball" atoms table 0- for typical molecular speeds eg, oxygen at room temp has v ~ 483m/s slide 5

16 Collisions between Gas Molecules l average distance free before collision Rough derivation in text (section 0-6) Should depend on atomic density (N/) and cross sectional area of atom Note dimensional analysis and intuition give above dependence up to factor.4 Sample problem 0-4: gas evaluated at STP (standard temp 300K and pressure atmosphere) d ~ m l ~ 0-7 m ~ 0-4 mm ~ 300 d t ~ l/v rms ~ sec is average time between collisions (for v rms ~450m/s) Mean Free Path l pdn / slide 6

17 Bouncing Molecules l ~ 0-7 m ~ 400 d means they collide 0 7 times per meter traveled (at STP) Forces between atoms (an der Waals) are very weak until they are essentially in contact Then they bounce energy of collisions much smaller than excitation energies of insides makes collisions elastic Small deviations ideal gas law due to an der Waals forces slide 7

18 Specific Heat Measures the Internal Energy of a Gas Useful: Also tells us how the heat may be transformed to useful (mechanical) energy Does this happen? Sure lots of examples Hero s engine steam engine slide 8

19 Internal Energy from Atomic Nature pnrt understood from atomic nature of matter pnkt is equivalent form Both are generally applicable (up to small van der Waals corrections) for all gases p kinetic energy of atoms Internal energy of the gas is a sum of all the energy forms (including kinetic energy) of the molecules simplest is monatomic gas (one atom in the molecule, rotationally symmetric) -> energy all translational real world: coefficient, 3/, only applies to noble gases ANY gas (prove soon!) p NkT nrt K atom atom 3 3 kt E N K NkT int monatomic gas E int nrt 3 p NkT nrt E atom C R Eint NEatom N kt E int nct kt C R slide 9

Specific Heats of Gas For simplicity of notation, we will use molar specific heats [instead of specific heat in J/(kg K)] Q n C DT defines C in J/(mol K) Two ways of

20 Specific Heats of Gas For simplicity of notation, we will use molar specific heats [instead of specific heat in J/(kg K)] Q n C DT defines C in J/(mol K) Two ways of adding heat with different answers: Keep volume of system fixed (no work done), so that the pressure must change Keep pressure fixed, vary volume (work done) C C P slide 0

21 Specific Heat at Constant olume (isochoric) st Law of Therm. de dq -dw int monatomic gas E int 3 nrt No change in volume implies no work done: dw 0 Heat introduced proportional to temperature change when no work Q n C DT Since dw0, then the heat added must equal the change in internal energy DE int Q n C DT And we predict: Monatomic (billiard ball) gases have C 3R/ any D E nc DT int gas slide

E any int Change in Internal Energy for Any Ideal Gas in Any Process gas nct Internal energy only a consequence of temperature (and mass) of system.

22 E any int Change in Internal Energy for Any Ideal Gas in Any Process gas nct Internal energy only a consequence of temperature (and mass) of system. E int depends on T only, not how it got there Hence any process (ifif ) resulting in a change in temperature produces the same change in internal energy any ideal gas and any process D E nc DT int slide

23 Specific Heat at Constant Pressure (isobaric process) Here, as expansion occurs with p constant, work is done and internal energy increases For process (n fixed) Q n C P DT And W pd nrdt Q D E + W P nc D T + p D nc DT int ( ) C C + R + nrdt Q nc + R DT slide 3

24 Sample Prob 0-8 Warm up cold cabin by turning on the heat (T i fi T f, say 70K fi 300K) What happens to gas (air) in cabin? Temperature First, what in cabin stays increases. fixed? olume of cabin is fixed. p What changes? If n were fixed, E int p nrt then... pressure must change 5 from normal atm (0 Pa). Dp p DT T ~ 0. Force on (eg, picture window) 5 of m m area ~ ( 0. ) ( 0 )( ) 4 ~ 0 N ~ 300lbs NO WAY!! p n Answer: fixed at atm., changes by 0%. (Air exits through cracks.) This implies that for this case E int nct also stays fixed! slide 4

25 Thermodynamics and Gases Today Kinetic Theory of Gases for simple gases Atomic nature of matter Demonstrate ideal gas law Atomic kinetic energy internal energy Mean free path and velocity distributions From formula for E int, can get specific heats Next Time Discuss further the specific heats of Simplest Gases Constant olume Constant Pressure Specific Heats for more complex gases Adiabatic Expansion Entropy slide 5

Ch. 19: The Kinetic Theory of Gases

Ch. 19: The Kinetic Theory of Gases In this chapter we consider the physics of gases. If the atoms or molecules that make up a gas collide with the walls of their container, they exert a pressure p on