Question

Use the Maxwell-Boltzmann distribution of speeds to estimate the fraction of N₂ molecules at 392 K that have speeds in the range 200. – 210. m·s⁻¹. Hint: The fraction of molecules with speeds in the range v to

(v + dv)

is equal to

f(v)dv,

Please show entire integration for boltzmann formula.

Answer 1

2 Maxwell -Boltzmann lig speed distribution in velocity distributio lawi- consider an ideal gas in a container. Assumptions: lin all assumptions of kinetic theory a) No mass motion * Take D = Inteny, z) Define P(x) = probability that a molecule has x-componand velocity ox similarly define Play), P1~2) PING), Plwy), P102) are independent ing Plu, vxtdux - probability that a molecule nas x-comp. velocity between - dnux No of favourable x and Nxtdax cases ~ Total no, of molecule a finaldox Similarly Plny, wytduy) = f(wy) any Puzztdoz) = f(uz) druz N) NO rotational or vibrational energy, Entire energy is translational wis Distribution of molecules isotrokic. will probability that on molecule has vel. comp. between moord utda د هډه ده رده) ۴ (لاله ره ره ) ؟ - (لاله ة ةم PNZ, NZ+82) dNurvyuz fwa) dua flwy) dry flvz) duz Inax vyvz = NS1wxfivag) f (2) dundry duz = NEN) duxdvydrz Scanned with Seanne

As, volume elements dux, duy, duz located at the tip of we take o fixed = fixed NF. (W) = fired idNF (~)Jeo [Nf (wxdfwy) + (W2)]=0 " fiwxgf (wy)f (₂) dux+ f(x) fing) f (~2) sv. + + + +(-4) 3 %) 1 - 1 Divide by flux)f(wy) f1z) to get, زده W2 - A1ss, 3 4 owx+ skvy) dwy + s'iwe? dvę = འདུལ དུ 3 རྣ ་ 5 = 12% 8% 1 4 3 % +22 - 3 redunt vydry &z duzzo.. @ Let us solved by Lagrange undeter- mined multiplier technique multiply equation by a constant a = Lagrange's undetermined multiplier x 4 % 5 4 3 2 = S - 3 x 44.8 x 4 + ③ = As, wo, wy, a Independent Each bracket term Independent. anned with anne

Jux taux=0 df (x) Const Integrate enfway=-& va tena -. in five cavab a flux) = te a flwy) =ée- five) - Ae- az = N(ANA", e-q (vett võs uz? )suad wydve - وه ر ب م ل A دهد به درد ه و - A = AKA" and No. set between Ward velocity distribution law, a speed distribution law To find molecule between a and di we integrale over (0,1 over askere in vel mm space. dnu s dnux ryuz

= NA e-Bv2 duxdry drez = NA e-Buz 0, NA? e-Bv² u2gu sine dede do 2 sirodos ad INV 41NA² e-Bu speed distribution law. ; No, between and the fuldne= Manwelliar speed distribution function. To find A and B N = dne = ANNA? Je B 2 2 2 as NiamNa? ) Again, Average Mean Square Speed <sy JNU u? ATINÁsé 2 do TINA? e-Buzdo we Boldre 62 - Buzdo

Now Average kinetic energy per molecule Boltzmann constant <tmay= 3 KT - Absolute temp. => { m<w2y = 2KT = m.BKT :. 0 $ - - - ZKT. In this problem, ME Mass of Namelewes = 28 g = 0.028kg : Aug, velocity = 200+210 = 205 mis. velocity range 200- 210mis dua (210.2oolmis omis. TE 392k. & Fraction of molecules vel. range 200-210 mis is flwian = 40 A e-ß i trgowa 40. Im 49. maT en het ~2 do KT vith ber

- 41. let) = AT. - ZRT 2TRT = 4*3.14 « la 10.028 kg moll 2x3.14x8.31434 mol (3926) $2 - 0,028kg mot 2x8.319 3 motax (392K)" liom = b 4x3.14 x (iex lo-10 m?s3 0.1805 * (205mellom 7.05 x 10-3 Scanned with CamScanner