Lecture 25: Heat and The 1st Law of Thermodynamics Prof. WAN, Xin

Transcription

1 General Physics I Lecture 5: Heat and he 1st Law o hermodynamics Pro. WAN, Xin xinwan@zju.edu.cn

2 Latent Heat in Phase Changes

3 Latent Heat he latent heat o vaporization or a given substance is usually somewhat higher than the latent heat o usion. Why?

4 Mechanical Equivalence o Heat he amount o energy transer necessary to raise the temperature o 1 g o water rom 14.5 o C to 15.5 o C.

5 Heat Heat is deined as the transer o energy across the boundary o a system due to a temperature dierence between the system and its surroundings. Heat transer Conduction, convection, radiation

6 Heat Conduction: Microscopic U L, N L U R, N R Q l l

7 Heat Conduction: Macroscopic Fourier heat conduction law Q t t A d dx Remind you o Ohm s law?

8 Energy ranser hrough wo Slabs

9 Mean Free Path d d d d v v Average distance between two collisions

10 Mean Free Path During time interval t, a molecule sweeps a cylinder o diameter d and length vt. Average number o collisions z n d vt olume o the cylinder d vt Mean ree path l n vt d vt n 1 d kb d p p n k n p / B k B

11 Mean Free Path During time interval t, a molecule sweeps a cylinder o diameter d and length vt. Average number o collisions z n d vt Mean ree path l n d vt 1 kb ( v) t n d d Relative motion v v p

12 Q&A: Collision Frequency Consider air at room temperature. How ar does a typical molecule (with a diameter o m) move beore it collides with another molecule?

13 Q&A: Collision Frequency Consider air at room temperature. How ar does a typical molecule (with a diameter o m) move beore it collides with another molecule?

14 Q&A: Collision Frequency Consider air at room temperature. Average molecular separation:

15 Q&A: Collision Frequency Consider air at room temperature. On average, how requently does one molecule collide with another? 8k v ~ m v l k m Expect ~ 500 m/s Expect ~ 10 9 /s ry yoursel!

16 Kinetic heory Energy exchange across plane A Q 1 U U C R L 1 U L, N L U R, N R Q t 1 N P lv th Al kb l d dx l l Q t d ~ m t A 1 m l d dx

17 or Air at Room emperature t From earlier lecture l m v 500 m/s 1 P t lv th N/m 300K W/(m K) m 500m/s A actor less than larger than the measured value o Not bad ater so many crude approximations.

18 ransport in Comparison Phenomena Imbalance hings being transported Experimental observation Unit o Coeicient hermal conduction temperature energy W/m K iscosity velocity momentum N s/m Diusion density particle m /s Charge conduction Q d t A t dx dv F A dy ( I n ) x ( I e ) x dn DA dx d A dx voltage charge -1 m -1

19 Internal Energy, Heat & Work Heat is deined as the transer o energy across the boundary o a system due to a temperature dierence between the system and its surroundings. Energy can also be transerred to or rom the system by work. Internal energy is all the energy o a system that is associated with its microscopic components atoms and molecules when viewed rom a reerence rame at rest with respect to the object.

20 Work in hermodynamic Processes Quasi-static assumption: the gas expands slowly enough to allow the system to remain essentially in thermal equilibrium at all times. Work done by the gas dw Fdy PAdy Pd W i Pd

21 Work in hermodynamic Processes Work done by the gas dw Fdy PAdy Pd W i Pd he work done by a gas in the expansion rom an initial state to a inal state is the area under the curve connecting the states in a P diagram.

22 Warning: Sign Convention Historically, people are interested in the amount o work done by the expansion o gas, say, to drive a steam engine. he common treatment is Positive work: gas expands Negative work: gas compressed In mechanics we use the opposite sign, unortunately. But some books ollow the same convention in thermal physics as in mechanics. rust your common sense!

23 Work Depends on the Path Work Depends on the Path ) ( ) ( i a P W ) ( ) ( i i b P W ) ( ) ( ) ( b c a W W W he work done by a system depends on the initial, inal, and intermediate states o the system.

24 Ideal Gases Experiments ound p Nk B Kinetic theory ound p N 3 1 mv 1 mv Generalized equipartition theorem (can be proved based on statistical principles) U U N kb C Nk 3 k B ixed B

25 Isothermal vs Free Expansion An adiabatic process is one during which no energy enters or leaves the system by heat. An energy reservoir is a source o energy that is considered to be so great that a inite transer o energy rom the reservoir does not change its temperature.

26 Isothermal Expansion W i Pd i Nk B d Nk B ln i U 0 at ixed Q W Nk B ln i

27 (Adiabatic) Free Expansion U 0 Q W 0 Energy transer by heat, like work done, depends on the initial, inal, and intermediate states o the system. - Is it possible to show the process on the P diagram? - Is the process reversible?

28 he 1st Law o hermodynamics Although Q and W both depend on the path, the quantity Q-W is independent o the path change. he change in the internal energy U o the system can be expressed as: U Q W he ininitesimal change: du dq Pd reminding you that it is path dependent

29 Discussion on the 1st Law he 1st law is a statement o energy conservation (now with the internal energy included). he internal energy o an isolated system remains constant. In a cyclic process, U 0, Q W he net work done by the system per cycle equals the area enclosed by the path representing the process on a P diagram.

30 Discussion on the 1st Law On a microscopic scale, no distinction exists between the result o heat and that o work. he internal energy unction is thereore called a state unction, whose value is determined by the state o the system. In general, U U (, )

31 Digression on Multivariate Calculus I we take energy and volume as parameters, how comes heat is path dependent? dq du Pd In mathematical language, du + pd is an inexact dierential. In multivariate calculus, a dierential is said to be exact (or perect), as contrasted with an inexact dierential, i it is o the orm dq, or some dierentiable unction Q.

32 Inexact Dierential Assume (,1) (1,1) dg (,) (,1) dx x y dx dy x y dy 1 ln (1,) (1,1) (,) (1,) dx x y dy ln 1 Note: d dg x dx x dy y is an exact dierential. Integrating actor ( x, y) ln x ln y 0

33 Isobaric Processes Isobaric Processes Nk C B Nk P B B P Nk C Q C ixed P C U ) ( i P W P C W U Q isobaric C C Nk C P B /

34 Isobaric vs Isovolumetric Processes isobaric W P ( i ) U C Q U W CP isovolumetric W 0 U C Q U C Molar speciic heat: C P C R

35 Degrees o Freedom, Again C C Nk B P Nk B C P C