Partitio Fuctios ad Ideal Gases PFIG-1 You ve leared about partitio fuctios ad some uses, ow we ll explore them i more depth usig ideal moatomic, diatomic ad polyatomic gases! efore we start, remember: Q( N,, ) (, N! ) N What are N,, ad? We ow apply this to the ideal gas where: 1. he molecules are idepedet.. he umber of states greatly exceeds the umber of molecules (assumptio of low pressure).
Ideal moatomic gases PFIG- Where ca we put eergy ito a moatomic gas? ε ε + ε atomic Oly ito latioal ad troic modes! he total partitio fuctio is the product of the partitio fuctios from each degree of freedom: (, ) (, ) (, ) otal atomic partitio fuctio raslatioal atomic partitio fuctio Electroic atomic partitio fuctio We ll cosider both separately
raslatios of Ideal Gas: (, ) PFIG-3 Geeral form of partitio fuctio: βε e states z Recall from QM slides ε h 8ma ( x + c y b + a z ) x x, y, z 1,,..., So what is?
PFIG-4 Let s simplify ( ) + + 1 1 1 1,, 8 exp,, z y x z y x z y x z y x ma h e β βε c b a c b a e e e e + + 1 1 1 8 exp 8 exp 8 exp z y x z y x ma h ma h ma h β β β Recall: All three sums are the same because x, y, z have same form! We ca simplify expressio to:
is early cotiuous PFIG-5 We d like to solve this expressio, but there is o aalytical solutio for the sum! 3 (, ) βh exp 1 8 ma No fears there is somethig we ca do! Sice latioal eergy levels are spaced very close together, the sum is early cotiuous fuctio ad we ca approximate the sum as a itegral which we ca solve! 3 βh (, ) d exp 8 ma 0 Work the itegral Note limit chage oly way to solve but adds very little error to result πmk (, ) h 3/ a 3
raslatioal eergy, ε PFIG-6 With we ca calculate ay thermodyamic uatity!! I Ch 17 (Z otes) we showed this for the average eergy ε k l here πmk (, ) h 3/ ε k l 3/ πmk h 3/ As we foud i Z otes! (Recall: this is per atom.)
Ideal moatomic gas: (, ) PFIG-7 Next cosider the troic cotributio to : Agai, start from the geeral form of, but this time sum over levels rather tha states: levels g ei e Degeeracy of level i βε ei Eergy of level i We choose to set the lowest troic eergy state at zero, such that all higher eergy states are relative to the groud state. ε 1 0 For moatomic gases!
Ca we simplify? PFIG-8 e βε e3 ( ) g + g e + g e βε (18.1) e1 e e3 +... terms are gettig small rapidly he troic eergy levels are spaced far apart, ad therefore we typically oly eed to cosider the first term or two i the series Geeral rule of thumb: At 300 K, you oly eed to keep terms where ε ej < 10 3 cm -1 (e -βε > 0.008)
A closer look at troic levels PFIG-9 βε e βε e3 ( ) g + g e + g e e1 e e3 +... (18.1) Geeral treds, 1. Nobel gas atoms: ε e 10 5 cm -1 (at 300K, keep term(s)). Alkali metal gas atoms: ε e 10 4 cm -1 (at 300K, keep term(s)) 3. Haloge gas atoms: ε e 10 cm -1 (at 300K, keep term(s))
Fial look at PFIG-10 I geeral it is sufficiet to keep oly the first two terms for ( ) g + g e e e1 e βε However, you should always keep i mid that for very high temperatures (like o the su) or smaller values of ε ej (like i F) that additioal terms may cotribute. If you fid that the secod term is of reasoable magitude (>1% of first term), the you must check to see that the third term ca be eglected.
Fially we ca solve for Q! PFIG-11 For a moatomic ideal gas we have: Q( N,, ) ( (, ) (, )) N! N with πmk (, ) h 3/ βε e ( ) g + g e e1 e
Fidig thermodyamic parameters U PFIG-1 We ca ow calculate the average eergy, U E U E U k Plug i ad Nk l Q l Nk Nk l U 3 Nk πmk h + Ng 3/ e ε e l ( βε ) e g + g e e e1 βε e e Electroic cotributio typically small (i.e., egligible) So U or U molar eergy
PFIG-13 Fidig thermodyamic parameters C N d du C, C Nk 3 R d R d C N 3 3, Molar heat capacity for a moatomic ideal gas: Could also fid heat capacity:
Fidig thermodyamic parameters P PFIG-14 P k l Q N, Nk l Nk l( ) Plug i ad P Nk l πmk h 3/ ( βε ) e g + g e e1 Oly fuctio of e So P or P Look familiar?? molar pressure
Ideal Moatomic Gas: A Summary Partitio Fuctio: From Q get Eergy Q( N,, ) mk (, ) h 3/ (, ) N! N PFIG-15 π βε e ( ) g + g e U 3 Nk e1 (molar) 3 U R e Heat Capacity Pressure
Addig complexity diatomic molecules PFIG-16 I additio to. ad. degrees of freedom, we eed to cosider: 1. Rotatios. ibratios ε ε + ε + ε + diatomic rot vib ε otal Eergy z raslatioal Eergy Rotatioal Eergy ibratioal Eergy Electroic Eergy (( (( c b a x
Diatomic Partitio Fuctio PFIG-17 Q. What will the form of the molecular diatomic partitio fuctio be give: ε ε + ε + ε + ε diatomic rot vib? As. Q. How will this give us the diatomic partitio fuctio? As. Now all we eed to kow is the form of rot,,, ad. vib Start with : his is the same as i the moatomic case but with m m 1 +m!
Diatomic Gases: PFIG-18 We defie the zero of the troic eergy to be separated atoms at rest i their groud troic eergy states. With this defiitio, ε e D e 1 Ad Note the slight differece i betwee moatomic ad diatomic gases! Figure 18.