Maxwell-Boltzmann speed distribution — derivation (key for propulsion): Question bank

1.7.12 · D5 · Physics › Thermodynamics › Maxwell-Boltzmann speed distribution — derivation (key for p

Shuru karne se pehle, ek reminder — taaki neeche har symbol samjha hua lage:

$v$ = speed (length per time, hamesha $\geq 0$ — koi direction nahi).
$v_{x}, v_{y}, v_{z}$ = teen velocity components: kisi molecule ki velocity space mein ek arrow hoti hai, aur $v_{x}$ batata hai ki woh $x$ -direction mein kitni tezi se move kar raha hai (yeh negative ho sakta hai — matlab leftward — speed ke unlike). Speed inse $v = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$ se rebuild hoti hai.
$T$ = temperature, $m$ = ek molecule ka mass, $k_{B}$ = Boltzmann's constant (woh number jo temperature ko energy per molecule mein convert karta hai).
$β$ = shorthand constant $\frac{m}{2 k _{B} T}$ jo exponent $e^{- β v^{2}}$ mein baitha hai; yeh measure karta hai ki distribution high speeds ko kitni sharply penalise karti hai (bada $β$ = thanda/heavy = steep drop-off).
$v_{p} = 2 k_{B} T / m$ (peak), $\overset{v}{ˉ} = 8 k_{B} T / π m$ (mean), $v_{r m s} = 3 k_{B} T / m$ (root-mean-square).

Woh picture jis par neeche har question point karta hai — iska shape yaad kar lo:

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

True or false — justify karo

True or false: Agar temperature fixed hai, to gas ka har molecule

v_{r m s}

par move karta hai.

False —

v_{r m s}

sirf ek poore spread ka summary number hai; molecules near

0

se arbitrarily large speeds tak occupy karte hain, sirf

f (v)

se weighted hote hain.

True or false:

f (v)

ek symmetric bell curve hai jaise Gaussian.

False —

v^{2}

factor curve ko

v = 0

par zero push karta hai aur right side mein ek lamba tail stretch karta hai, isliye yeh skewed hai, symmetric nahi.

True or false: Jab gas ko heat karo to

f (v)

ke neeche ka area change ho jaata hai.

False — area hamesha

1

hota hai (yeh hona hi chahiye, kyunki koi na koi speed toh zaroor hogi); heating curve ko wider aur lower karta hai lekin total area conserve hoti hai.

True or false:

T

double karne se

v_{p}

double ho jaata hai.

False —

v_{p} \propto T

hai, isliye

T

double karne se

v_{p}

sirf

2 \approx 1.41

se multiply hota hai.

True or false: Ek halki gas same temperature par ek broader, faster distribution rakhti hai.

True — saari speeds

1/ m

ke saath scale karti hain, isliye chota

m

poori curve ko higher speed ki taraf shift aur spread karta hai (isliye H₂ ek prized propellant hai, dekho Rocket Propulsion & Specific Impulse).

True or false: Most probable speed

v_{p}

average speed

\overset{v}{ˉ}

ke barabar hoti hai.

False — skew mean ko peak ke upar khenchti hai, jo fixed order

v_{p} < \overset{v}{ˉ} < v_{r m s}

deta hai.

True or false: Same temperature par, do gases jinka molecular mass equal hai, unki speed distributions identical hoti hain.

True —

f (v)

sirf

m

aur

T

par depend karta hai (

k_{B} T / m

ke through), chemistry par nahi, isliye equal

m

at equal

T

ek identical curve deta hai.

True or false: Velocity component

v_{x}

ki distribution shape speed

v

ki distribution shape jaisi hoti hai.

False — har component

v_{x}

ek symmetric Gaussian hai (

0

par peaked, negative ho sakta hai), jabki speed one-sided hai aur

0

se door peak karti hai kyunki

4 π v^{2}

shell factor hota hai.

True or false:

⟨ v^{2} ⟩ = ⟨ v ⟩^{2}

False — squares ka mean, mean ke square se variance se zyada hota hai, jo speeds ke kisi bhi real spread ke liye strictly positive hoti hai.

Why questions

Kisi gas mein ek shared speed ki jagah speeds ki distribution kyun hoti hai?

Continuous collisions randomly molecules ke beech energy trade karti hain, isliye koi bhi single-speed state turant destroy ho jaati hai; sirf ek statistical spread stable hoti hai (Kinetic Theory of Gases).

Har velocity component exponential of a quadratic,

e^{- β v_{x}^{2}}

, kyun nikalta hai?

Teen facts chain hote hain — (1)

v_{x}, v_{y}, v_{z}

ki independence joint distribution ko ek product

g (v_{x}) g (v_{y}) g (v_{z})

banati hai; (2) isotropy demand karti hai ki woh product sirf

v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

par depend kare; (3) woh ek hi function jo product ko sum-of-squares ka function banata hai woh exponential hai, kyunki

e^{a} e^{b} e^{c} = e^{a + b + c}

, jo har exponent ko

\propto v_{x}^{2}

aur negative hone par majboor karta hai taaki finite rahe.

Constant

β = \frac{m}{2 k _{B} T}

waise kyun nikalta hai?

Equipartition Theorem mean energy per component ko

\frac{1}{2} k_{B} T

fix karta hai, jo

⟨ v_{x}^{2} ⟩ = 1/2 β = k_{B} T / m

force karta hai aur isliye

β = m /2 k_{B} T

hota hai.

Speed distribution mein

4 π v^{2}

factor kyun hota hai?

Ek given speed ke saare velocity vectors velocity space mein radius

v

ke ek sphere par lie karte hain, aur aisi states ki number us sphere ki surface area

4 π v^{2}

ke saath scale karti hai.

High-speed tail propulsion aur reactions ke liye sabse zyada important kyun hai?

Fastest molecules escape, reaction rates, aur effusion dominate karte hain; thodi si temperature rise is tail ko dramatically mota kaar deti hai chahe peak barely move kare (Effusion and Graham's Law).

Same temperature par hydrogen CO₂ se zyada exhaust speed kyun deta hai?

Har characteristic speed

1/ m

ke saath scale karti hai, isliye H₂ ka tiny mass uski poori distribution ko kaafi higher speeds tak push karta hai, specific impulse badhata hai.

Ratio

v_{p} : \overset{v}{ˉ} : v_{r m s}

har gas ke liye har temperature par same kyun hai?

Har speed ek pure number times

k_{B} T / m

hai, aur woh common factor ratio mein cancel ho jaata hai, universal

2 : 8/ π : 3

chhod ke.

Boltzmann Distribution

e^{- E / k_{B} T}

Maxwell–Boltzmann speed law ke "peeche" kyun hai?

General Boltzmann weight

e^{- E / k_{B} T}

mein kinetic energy

E = \frac{1}{2} m v^{2}

substitute karne par exactly

f (v)

mein exponential factor milta hai.

Edge cases

v = 0

par

f (v)

exactly kya hai?

Exactly

0

, kyunki

v^{2}

shell factor wahan vanish ho jaata hai — koi bhi molecule literally zero speed par finite probability density ke saath nahi hoti.

v \to \infty

par

f (v)

ka kya hota hai?

Yeh

0

tak decay ho jaata hai, kyunki exponential

e^{- m v^{2} /2 k_{B} T}

large speeds ke liye polynomial

v^{2}

growth ko crush kar deta hai.

T \to 0

ki limit mein, distribution kaisi dikhti hai?

Yeh

v = 0

par ek spike ki taraf collapse ho jaati hai — exponential penalty infinitely steep ho jaati hai, isliye almost saare molecules near-zero speed par freeze ho jaate hain.

T \to \infty

ki limit mein, teen characteristic speeds ka kya hota hai?

Unme se saari (

v_{p}, \overset{v}{ˉ}, v_{r m s}

)

T

ke saath unbounded grow karti hain, aur curve indefinitely right ki taraf flatten aur stretch hoti hai.

Agar fixed

T

par molecular mass

m \to \infty

ho, to peak kahan jaata hai?

v = 0

ki taraf, kyunki

v_{p} \propto 1/ m

— bahut heavy molecules sluggish hote hain aur low speed par cluster karte hain.

Kya ek single molecule ek instant mein Maxwell–Boltzmann distribution "rakhta" hai?

Nahi —

f (v)

kaafi saare molecules ke ensemble ko describe karta hai (ya ek molecule ko lambe time mein sample kiya gaya ho); ek instant mein ek single molecule ka sirf ek definite speed hoti hai.

Kya kisi speed ke liye

f (v)

kabhi negative hota hai?

Kabhi nahi — yeh

v^{2} \geq 0

, ek positive exponential, aur ek positive constant ka product hai, isliye yeh har jagah non-negative hai, jaise kisi bhi probability density ko hona chahiye.

Connections

Kinetic Theory of Gases — kyun collisions ek statistical spread force karti hain
Equipartition Theorem — $β = m /2 k_{B} T$ fix karta hai
Boltzmann Distribution — $f (v)$ ka $e^{- E / k_{B} T}$ ancestor
Mean Free Path — $\overset{v}{ˉ}$ use karta hai
Rocket Propulsion & Specific Impulse — low- $m$ , high-speed payoff
Effusion and Graham's Law — tail- aur $\overset{v}{ˉ}$ -driven escape rates

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

True or false — justify karo

Error pakdo

Why questions

Edge cases

Connections