Mean free path, mean speed, RMS speed — derivations

1.7.11 · Physics › Thermodynamics

1. Setup karna: Maxwell–Boltzmann speed distribution

Sabhi molecules ek hi speed se nahi chalte. $[v, v + d v]$ mein speed wale fraction ka formula hai:

$f (v) d v = 4 π N (\frac{m}{2 π k _{B} T})^{3/2} v^{2} e^{- \frac{m v ^{2}}{2 k _{B} T}} d v$

WHY teen alag numbers? Kyunki $⟨ v ⟩ \neq = ⟨ v^{2} ⟩$ kisi bhi spread-out distribution ke liye. Average karne se pehle square karna fast molecules ko zyada weight deta hai, isliye $v_{r m s} > \overset{v}{ˉ} > v_{p}$ hamesha.

2. RMS speed derive karna — pehle pressure first principles se

WHAT: $v_{r m s}$ nikaalo bina poori distribution ki zaroorat ke, kinetic theory of pressure use karke.

HOW (derivation scratch se):

Ek molecule of mass $m$ consider karo ek cube of side $L$ mein, jo $x$ -velocity $v_{x}$ se move kar raha hai. Woh ek wall se bounce karta hai.

Ek wall se collision per momentum change: $Δ p = m v_{x} - (- m v_{x}) = 2 m v_{x}$ Ye step kyun? Wall $v_{x}$ ko reverse karta hai; momentum ek vector hai isliye change magnitude ka double hota hai.
Us wall pe hits ke beech ka time: molecule ko $2 L$ travel karna hoga (jaana aur aana): $Δ t = \frac{2 L}{v _{x}}$
Ek molecule se force = rate of momentum transfer: $F = \frac{Δ p}{Δ t} = \frac{2 m v _{x}}{2 L / v _{x}} = \frac{m v _{x}^{2}}{L}$
$N$ molecules se pressure on area $L^{2}$ : $P = \frac{\sum F}{L ^{2}} = \frac{m}{L ^{3}} \sum v_{x}^{2} = \frac{N m}{V} \overline{v_{x}^{2}}$ Ye step kyun? $L^{3} = V$ , aur hum $v_{x}^{2}$ ko sabhi molecules pe average karte hain.
Isotropy: koi direction special nahi hai, isliye $\overline{v_{x}^{2}} = \overline{v_{y}^{2}} = \overline{v_{z}^{2}}$ , aur $\overline{v^{2}} = \overline{v_{x}^{2}} + \overline{v_{y}^{2}} + \overline{v_{z}^{2}} = 3 \overline{v_{x}^{2}}$ Thus $\overline{v_{x}^{2}} = \frac{1}{3} \overline{v^{2}}$ aur:

Ideal gas $P V = N k_{B} T$ se compare karo: $\frac{1}{3} m \overline{v^{2}} = k_{B} T \Rightarrow \overline{v^{2}} = \frac{3 k _{B} T}{m}$

3. Distribution se mean speed $\overset{v}{ˉ}$ derive karna

$\overset{v}{ˉ} = \frac{\int _{0}^{\infty} v f ( v ) d v}{\int _{0}^{\infty} f ( v ) d v}$

Maano $a = \frac{m}{2 k _{B} T}$ . Shape hai $v^{2} e^{- a v^{2}}$ (constants ratio mein cancel ho jaate hain).

Numerator $\int_{0}^{\infty} v \cdot v^{2} e^{- a v^{2}} d v = \int_{0}^{\infty} v^{3} e^{- a v^{2}} d v = \frac{1}{2 a ^{2}}$ .
Denominator $\int_{0}^{\infty} v^{2} e^{- a v^{2}} d v = \frac{π}{4 a ^{3/2}}$ .

Ye integrals kyun? Ye standard Gaussian moments hain; odd wala ( $v^{3}$ ) elementary hai (sub $u = v^{2}$ ), even wala $\int_{0}^{\infty} e^{- a v^{2}} d v = \frac{1}{2} π / a$ ko $a$ ke w.r.t. differentiate karke milta hai.

$\overset{v}{ˉ} = \frac{1/ ( 2 a ^{2} )}{π / ( 4 a ^{3/2} )} = \frac{4 a ^{3/2}}{2 a ^{2} π} = \frac{2}{π a} = \frac{4}{π a}$

$a = \frac{m}{2 k _{B} T}$ substitute karo:

Similarly peak se $v_{p} = \frac{2 k _{B} T}{m}$ milta hai.

Figure — Mean free path, mean speed, RMS speed — derivations

4. Mean free path $λ$ derive karna

WHAT: Ek molecule collisions ke beech average kitni distance travel karta hai.

HOW:

Molecules ko spheres of diameter $d$ maano. Do molecules collide karte hain jab unke centres $d$ ke andar aa jaate hain. Toh moving molecule ek collision cylinder of radius $d$ sweep karta hai, yaani cross-section: $σ = π d^{2}$ $d /2$ nahi balki $d$ kyun? Collision tab hoti hai jab centre-to-centre distance $= d$ ho (dono ka radius $d /2$ hai). Effective target radius $d$ hai.
Time $t$ mein ek molecule volume $σ \overset{v}{ˉ} t$ sweep karta hai. Agar $n = N / V$ number density hai, toh collisions ki sankhya $= nσ \overset{v}{ˉ} t$ hai.
Mean free path = total distance / collisions ki sankhya: $λ = \frac{v ˉ t}{nσ v ˉ t} = \frac{1}{nσ} = \frac{1}{nπ d ^{2}}$
Correction: baaki molecules bhi move kar rahe hain. Do molecules ke beech relative speed average $2 \overset{v}{ˉ}$ hota hai (equal mean speed ki do independent velocities ka vector difference). Toh collision rate mein $\overset{v}{ˉ}$ ki jagah $2 \overset{v}{ˉ}$ lagao:

Formula padhna: $λ \propto 1/ P$ (gas ko squeeze karo → zyada bheed → chhote flights) aur $λ \propto T$ fixed $P$ par.

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active recall

Recall Self-test (answers chhupao)

$v_{r m s} > \overset{v}{ˉ}$ kyun hai?
$λ$ mein $2$ kahaan se aata hai?
$\overline{v_{x}^{2}} = \frac{1}{3} \overline{v^{2}}$ kyun hai?
Fixed $P$ par $λ$ ki temperature dependence kya hai? $v_{r m s}$ ki?

Recall Feynman (12-saal ke bachche ko explain karo)

Ek room imagine karo jismein bouncing super-balls bhari hain. Agar tum poochho "ek typical ball kitni tez hai?" toh tum teen tareekon se jawab de sakte ho. Sabse sahi energy-answer har speed ko square karta hai, average karta hai, phir square-root leta hai (RMS) — kyunki fast balls bahut zyada punch carry karti hain. Plain average ( $\overset{v}{ˉ}$ ) thoda slow hota hai. Aur "ek ball kisi doosre se takraane se pehle kitni door jaati hai?" ye depend karta hai ki room kitna packed hai: do baar zyada balls pack karo, toh har ek ball crash se pehle half utni door jaayegi. Yahi crash distance mean free path hai.

Flashcards

T ke terms mein RMS speed ka formula

v_{r m s} = 3 k_{B} T / m = 3 R T / M

Mean speed ka formula

\overset{v}{ˉ} = 8 k_{B} T / π m = 8 R T / π M

Most probable speed

v_{p} = 2 k_{B} T / m

Ratio

v_{p} : \overset{v}{ˉ} : v_{r m s}

2 : 8/ π : 3 \approx 1.41 : 1.60 : 1.73

Mean free path ka formula

λ = 1/ (2 nπ d^{2}) = k_{B} T / (2 π d^{2} P)

λ

mein

2

kyun hai

Targets bhi move karte hain; mean relative speed

2 \overset{v}{ˉ}

hai

Collision cross-section

π d^{2}

kyun hai

π (d /2)^{2}

nahi

Collision centre separation

d

par hoti hai, toh effective radius

d

hai

\overline{v_{x}^{2}} = \frac{1}{3} \overline{v^{2}}

kyun hai

Isotropy:

\overline{v^{2}} = \overline{v_{x}^{2}} + \overline{v_{y}^{2}} + \overline{v_{z}^{2}}

aur teeno equal hain

P V

ka mean square speed se relation

P V = \frac{1}{3} N m \overline{v^{2}}

Pressure-derived link jo

v_{r m s}

deta hai

\frac{1}{3} m \overline{v^{2}} = k_{B} T

Pressure par

λ

ki dependence (T fixed)

λ \propto 1/ P

(P double karne par

λ

half ho jaata hai)

Temperature par

v_{r m s}

ki dependence

v_{r m s} \propto T

Kinetic energy kaunsi speed govern karti hai

v_{r m s}

, kyunki

\overline{K E} = \frac{1}{2} m \overline{v^{2}} = \frac{3}{2} k_{B} T

Connections

Maxwell-Boltzmann distribution — teeno speeds ka parent
Kinetic theory of pressure — $P V = \frac{1}{3} N m \overline{v^{2}}$ ka source
Ideal gas law — $T$ nikaalne ke liye loop close karta hai
Equipartition theorem — $\frac{3}{2} k_{B} T$ per molecule explain karta hai
Diffusion and effusion — $\overset{v}{ˉ}$ aur $λ$ use karta hai
Viscosity and thermal conductivity of gases — dono $\propto λ \overset{v}{ˉ}$
Speed of sound in gases — $v_{r m s}$ se comparable