Lecture 5: Ideal monatomic gas

Transcription

1 Lecture 5: Ideal monatomc gas Statstcal mechancs of the perfect gas Ams: Key new concepts and methods: Countng states Waves n a box. Demonstraton that β / kt Heat, work and Entropy n statstcal mechancs Proof of Boltzmann s conjecture Maxwell dstrbuton of veloctes. February 04 Lecture 5

2 Ideal monatomc gas Countng states: Ψ We need to quantse the atoms n the gas Waves n a box, a cube of sde a. Wavefuncton vanshes at the edges. r, t) Asn lπ x a) sn mπ y a) sn nπ z a) Wth l, m, n,,,4 Plane standng waves wth π k ε a ) l + m + n k π l, m, n + m ma ) l + m n States form a closely spaced) lattce of ponts. Calculate the mean energy of the gas usng partton functon. Z ε p Z d Z d β exp βε ) l, m, n l, m, n To calculate Z we need the densty of states. February 04 Lecture 5

3 Densty of states Exchange sum n Z for an ntegral Need the densty of states..e. no. of states n dε at energy ε, gε)dε. It s the number of states n /8th of a sphercal shell, wdth dk. Remember l, m, n > 0) state occupes a volume of of π/a) All states n n the shell have energy ε k /m States unformly dstrbuted n n k-space g 4πk d k 8 ε ) dε g k) d k. Volume of of shell February 04 Lecture 5 π a Vol. of of one state

4 Mean energy of a gas Replace k wth mε)/h a m g 4π ε ) dε ε dε Aε dε Z Partton functon 0 0 β ε g ε ) exp βε ) Aε 0 exp Aexp d Z Z d β dε βε ) ) x β dε Knetc theory shows <ε> kt/. Thus, x d x x βε Integral s s smply a constant β kt n agreement wth our earler guess. February 04 Lecture 5 4

5 Note: Adabatc changes We can derve the law of adabatc compresson from the densty of states. ε l, m, n V a Now compress box. Energy levels all rase. BUT Partcles reman n the same states. pε ) does not change. So, or ε V const T V const Quantum theory and Statstcal mechancs confrm the results of classcal thermodynamcs. February 04 Lecture 5 5

6 Heat and Work System of non-nteractng partcles. Energy levels ε, and occupaton numbers n. Internal energy U. U du Change n n populatons, wth energy levels fxed dq n ε ε d n + n dε ) T d S p dv We, thus, have a consstent vew of heat and work n statstcal mechancs. Heat: changes the occupaton of energy levels no change n the levels). Work: changes the energy levels themselves no change n the occupatons). Change n n energy levels, populatons fxed dw February 04 Lecture 5 6

7 A mnor) problem wth entropy Entropy We avoded calculatng the entropy, whch s S k lng) Number of of confguratons If U s defned exactly then g s a small number, e.g. 0,, An unreasonable value! If U s slghtly ll-defned, say by du, the number of confguratons s gu)du. BUT, what value of du do we use? Answer: t doesn t matter much), as the followng rough calculaton shows: Typcal entropy of a macroscopc body s JK -. Compare lngu) du) wth du Joule and du 0-00 J 0-8 ev). Dfference n S s k ln0-00 ) -0 k.x0 - J K -. Dfference s rrelevant! A pure number February 04 Lecture 5 7

8 Maxwell dstrbuton The Boltzmann dstrbuton gves the mean energy and the dstrbuton of speeds: exp ) ) ε kt exp m u + v + w The velocty dstrbuton, a -D Gaussan. To get the dstrbuton of speeds we add all probabltes lyng nsde a sphercal-shell of thckness dc. p c)d c p c)d c 4πc Standard ntegral 4πc 0 d c exp 4πc d c exp d c exp ) mc kt ) mc kt ) mc kt T T T kt T 5 m p c)d c πkt 4πc d c exp ) mc kt February 04 Lecture 5 8

9 Correctons Slde Formula for ε corrected to ε k /m February 04 Lecture 5 9