1.7.12
Intuition Badi picture (WHY yeh exist karta hai)Gas molecules sab ek hi speed se nahi chalte. Woh ek second mein billions baar collide karte hain, randomly energy trade karte hue. WHY koi distribution hona hi chahiye? Kyunki agar har molecule ki speed same hoti, toh ek collision usse change kar deta — chaos ek statistical steady state force karta hai. Maxwell–Boltzmann (MB) law batata hai ki kitne fraction molecules ki speed v sabse tez tail escape, reaction rates, aur exhaust velocity ko dominate karti hai.
Definition Speed distribution functionf ( v ) d v fraction of molecules jinki speed v v + d v normalized hai: ∫ 0 ∞ f ( v ) d v = 1 f ( v ) == (speed) − 1 == f ( v ) d v
KYA chahiye: f ( v ) v T m
Hum f ( v ) do independent ideas se build karte hain aur unhe multiply karte hain.
component Gaussian kyun hota hai
Teen components v x , v y , v z independent hain (koi preferred direction nahi — space isotropic hai). Ek velocity vector ki probability sirf speed v = v x 2 + v y 2 + v z 2
Maano g ( v x ) F ( v x , v y , v z ) = g ( v x ) g ( v y ) g ( v z ) .
Yeh step kyun? Independence ka matlab hai ki joint probability marginals ka product hai.
Lekin isotropy demand karta hai ki F v x 2 + v y 2 + v z 2 aur sirf sum of squares par depend kare, woh hai exponential of a quadratic :
g ( v x ) = A e − β v x 2 .
Yeh step kyun? Hume chahiye ki g ( v x ) g ( v y ) g ( v z ) = A 3 e − β ( v x 2 + v y 2 + v z 2 ) v 2 exponent mein sums ko bahar products mein badalta hai. Minus sign probability ko large speed par finite rakhta hai.
Yeh step kyun? Temperature define hoti hai mean kinetic energy se. Yahi T
∫ − ∞ ∞ A e − β v x 2 d v x = 1 ∫ − ∞ ∞ e − β x 2 d x = π / β A = β / π = 2 π k B T m
v 2 direction ki parwah nahi, sirf speed ki hai. Same speed v radius v par hote hain. Aisi states ki count surface area 4 π v 2 ke saath scale karti hai. Badi speed ⟹ bada shell ⟹ use paane ke zyada tarike.
Velocity space mein density hai F = A 3 e − β v 2 d v 4 π v 2 d v f ( v ) d v = shell area 4 π v 2 F ( v ) ( 2 π k B T m ) 3/2 e − 2 k B T m v 2 d v .
Har ek ko ek alag sawaal puchh ke derive karo.
Intuition Ordering aur WHY yeh propulsion ke liye matter karta haiHamesha == v p < v ˉ < v r m s == 2 : 8/ π : 3 ≈ 1.000 : 1.128 : 1.225 upar kheench leta hai. Ek rocket nozzle mein, exhaust velocity v r m s m Isliye hydrogen ek prized propellant hai.
Worked example Example 1 — 300 K par N₂ ki RMS speedm = 4.65 × 1 0 − 26 k B = 1.38 × 1 0 − 23 v r m s = 4.65 × 1 0 − 26 3 ( 1.38 × 1 0 − 23 ) ( 300 ) = 2.67 × 1 0 5 ≈ 517 Yeh step kyun? Humne v r m s = 3 k B T / m
Worked example Example 2 — Ratiov p : v ˉ : v r m s 2 : 8/ π : 3 = 1.414 : 1.596 : 1.732 1.414 1.000 : 1.128 : 1.225 Yeh step kyun? Ratios k B T / m universal hain — har gas ke liye har temperature par same.
Worked example Example 3 — Light vs heavy (propulsion punchline)H₂ (m = 3.3 × 1 0 − 27 m = 7.3 × 1 0 − 26 T v r m s ∝ 1/ m v C O 2 v H 2 = 73/3.3 ≈ 4.7 Yeh step kyun? Kam mass ⟹ broader, faster distribution ⟹ unit mass per better thrust.
f ( v ) Kyun sahi lagta hai: ek symmetric bell curve ke liye, peak = mean. Kami: f ( v ) v 2 skewed hai — long high-speed tail mean ko right mein kheenchti hai. Fix: v p < v ˉ < v r m s
4 π v 2 Kyun sahi lagta hai: Gaussian e − β v 2 dekhne mein poora answer lagta hai. Kami: yeh density velocity vector ke per hai; speed mein convert karne ke liye saari directions par sum karo ⟹ shell area 4 π v 2 Fix: speed distribution v = 0 v 2
Common mistake "Energy meinv r m s ⟨ v ⟩ 2 Kami: ⟨ v 2 ⟩ = ⟨ v ⟩ 2 ⟨ v 2 ⟩ Fix: ⟨ v 2 ⟩ − ⟨ v ⟩ 2 = > 0
Recall Compute karne se pehle predict karo
Agar T v p 2 ≈ 1.41 v p ∝ T
Kya T f ( v ) peak height badhti hai ya ghhatti hai? ::: Ghatti hai — curve spread hota hai, lekin area 1 rehta hai.
v = 0 f 0 v 2
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek badi bheed mein bumper cars hain jo hamesha ke liye ek doosre se takra rahi hain. Woh sab same speed maintain nahi kar sakti — har crash ek ko speed deta hai aur doosre ko slow karta hai. Thodi der baad ek pattern banta hai: bahut kam cars crawl kar rahi hain, bahut kam super fast zoom kar rahi hain, aur zyaatar beech mein hain lekin fast side ki taraf jhuki hain. Woh tedha "zyaatar medium hain, kuch rockets hain" pattern Maxwell–Boltzmann curve hai. Unhe heat karo (zyada temperature) aur puri bheed faster ki taraf shift hoti hai — isi tarah rockets hot gas ko fast bahar push karte hain taaki ud sakein.
Mnemonic Teen speeds yaad karo"2-8π-3" root ke neeche k B T / m v p = 2 v ˉ = 8/ π v r m s = 3 P eak, A verage, R MS" = chhota se bada. Mantra: "Peak sabse lazy hai, RMS sabse fast hai."
f ( v ) d v Un molecules ka fraction jinki speed v v + d v
f ( v ) v 2 Yeh velocity space mein speed-shell ka 4 π v 2
Har velocity component Gaussian kyun hota hai? Components ki independence + isotropy force karta hai ki g ∝ e − β v 2
β Equipartition ⟨ 2 1 m v x 2 ⟩ = 2 1 k B T β = m /2 k B T
Poora MB speed distribution formula? f ( v ) = 4 π ( m /2 π k B T ) 3/2 v 2 e − m v 2 /2 k B T
Most probable speed v p 2 k B T / m (
df / d v = 0 set karo).
Mean speed v ˉ RMS speed v r m s Teen speeds ka order? v p < v ˉ < v r m s 1 : 1.128 : 1.225
Curve skewed kyun hai, symmetric kyun nahi? v 2
f ( 0 ) = 0 v 2 v = 0
Propulsion: halka gas (H₂) prefer kyun karein? v r m s ∝ 1/ m , isliye halke molecules higher exhaust speeds aur specific impulse dete hain.
T Woh ghatti aur broad hoti hai (normalization total area = 1 rakhta hai).
only depends on v squared
convert velocity to speed
Joint velocity distribution
Gaussian g(vx)=A exp(-beta vx^2)
Propulsion and exhaust velocity
