Foundations — Maxwell-Boltzmann speed distribution — derivation (key for propulsion)

1.7.12 · D1 · Physics › Thermodynamics › Maxwell-Boltzmann speed distribution — derivation (key for p

Yeh page kuch bhi assume nahi karta. Agar parent note Maxwell–Boltzmann derivation ne koi symbol bina explain kiye use kiya, toh hum usse yahan explain karte hain — order mein, har ek apne pehle waale pe lean karta hua.

0. Is topic ke har formula ko kaise padhein

Neeche har symbol ek chhoti si machine hai: kuch andar leta hai aur kuch bahar deta hai. Hum hamesha teen sawaal poochhenge: plain words mein iska kya matlab hai, yeh kaun si picture banata hai, aur topic ko yeh kyun chahiye.

1. Speed $v$ — asli hero

Picture: ek akele molecule ko ek dot ki tarah socho jisme ek arrow hai. Arrow ki length $v$ hai. Arrow ko kisi bhi direction mein ghuma lo — length (speed) nahi badlegi.

Topic ko yeh kyun chahiye: poora Maxwell–Boltzmann law ek statement hai kitne molecules ke paas har speed hai ke baare mein. Speed poori kahaani ka horizontal axis hai.

2. Velocity components $v_x,v_y,v_z$ — ek arrow ko teen mein todna

Picture (figure dekho): red arrow sachchi velocity hai. Floor pe uski shadow $v_x$ aur $v_y$ deti hai; wall pe upar uski shadow $v_z$ deti hai. Arrow teen shadows se bani ek 3D box ki hypotenuse hai.

Topic ko yeh kyun chahiye: derivation component land mein shuru hoti hai (jahan har direction simple aur independent hai) aur baad mein hi speed mein wapas collapse hoti hai. Parent ke Step A ko aap $v_x,v_y,v_z$ ke bina follow nahi kar sakte.

3. Fraction aur distribution function $f(v)$

Picture: ek histogram ki tarah socho. Speed axis ko $dv$ width ke pateele bars mein kato. Ek bar ki height $f(v)$ hai; ek bar ka area, height $\times$ width $=f(v)dv$, us slice mein molecules ka fraction hai.

4. Chhoti width $dv$ aur integral $\int$

Picture: poore $f(v)$ curve ke neeche ka total area exactly $1$ ke barabar hai. Agar gas ko heat karo, curve납납 납납 flatten aur spread hota hai — lekin neeche ka area $1$ pe pin rehta hai, kyunki molecules kabhi lose ya gain nahi hote.

Topic ko yeh kyun chahiye: normalization woh equation hai jo parent ki derivation mein leading constant $A$ fix karti hai. Integral nahi toh yeh nahin jaana ki curve kitni tall hai.

5. Mean / average $⟨\cdot ⟩$ — angle-bracket machine

Picture: ek weighted balance point. Har speed slice ek see-saw pe push karta hai; zyada molecules waali slice ($f$ bada) zyada push karti hai. $⟨Q⟩$ woh jagah hai jahan see-saw balance karta hai.

Do averages jo topic use karta hai:

• $⟨v⟩=\bar{v}$ : seedha mean speed ($Q=v$).
• $⟨v^2⟩$ : squares ka mean, energy ke liye use hota hai ($Q=v^2$).

6. Temperature $T$ aur Boltzmann's constant $k_B$

Picture: $k_B$ ek currency converter hai. Temperature kelvin mein price hoti hai; energy joules mein price hoti hai; $k_{B}T$ har molecule ka motion ki ek direction per "energy budget" hai.

Topic ko yeh kyun chahiye: har characteristic speed ki form $\sqrt{(\text{number})k_{B}T/m}$ hoti hai. Combination $k_{B}T/m$ ek speed-squared hi hai — yeh set karta hai ki bheed kitni tezi se daurti hai.

7. Molecular mass $m$

Picture: bhaari molecules sluggish bowling balls hain; halke molecules ping-pong balls hain. Ek hi energy budget $k_{B}T$ dene par, halke wale zyada tezi se chalenge ($v\propto 1/\sqrt{m}$).

Topic ko yeh kyun chahiye: $m$ propulsion punchline hai. Kyunki har speed $\propto 1/\sqrt{m}$ hoti hai, halki gases (hydrogen) sabse tezi se nikalti hain — yahi wajah hai ki woh Rocket Propulsion & Specific Impulse mein jeetti hain.

8. Exponential $e^{-\beta v^2}$ — "falling" factor

Topic ko yeh kyun chahiye: $e^{-\beta v^2}$ woh factor hai jo super-fast molecules ko suppress karta hai. Yeh poore curve ko shape karne wale two tug-of-war factors mein se ek hai.

9. Shell factor $4\pi v^2$ — directions count karna

Topic ko yeh kyun chahiye: $4\pi v^2$ ke bina aapke paas velocity-vector density hoti, speed density nahi. Ise bhool jaana sabse common derivation error hai (parent ki mistakes section dekho).

10. Saare pieces ek saath

Final law sirf in do bricks ka multiplication hai, times bookkeeping constants:

$f(v)=\underset{\text{§9: zyada directions}}{\underbrace{4\pi v^2}}\times \underset{\text{§6,7: normalization}}{\underbrace{{\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}}}\times \underset{\text{§8: rare fast ones}}{\underbrace{e^{-\frac{m{v}^2}{2k_{B}T}}}}.$

Badhta $v^2$ aur girta $e^{-\beta v^2}$ ladte hain; inका product ek ek taraf jhuktī huii pahadi hai — Maxwell–Boltzmann curve.

11. Prerequisite map

Ise upar se neeche padhein: components speed banate hain, speed spheres aur shell factor banati hai; exponential plus energy scale width $\beta$ banata hai; fractions integral aur averages banate hain — ye saare final $f(v)$ aur teen speeds ko feed karte hain.

Equipment checklist

Khud test karo — sirf apna jawab bolne ke baad reveal karo.

Connections

• Kinetic Theory of Gases — molecules-ki-bheed model jo in symbols describe karta hai
• Equipartition Theorem — $\frac{1}{2}{k}_{B}T$ energy scale supply karta hai (§6)
• Boltzmann Distribution — energy version $e^{-E/k_{B}T}$ hamare exponential ke peeche
• Mean Free Path — §5 ke averages $\bar{v}$ pe build hota hai
• Rocket Propulsion & Specific Impulse — kyun $1/\sqrt{m}$ mass dependence matter karti hai
• Effusion and Graham's Law — $\bar{v}\propto 1/\sqrt{m}$ ka ek aur consequence