Mean free path, mean speed, RMS speed — derivations: Foundations

1.7.11 · D1 · Physics › Thermodynamics › Mean free path, mean speed, RMS speed — derivations

Hum dhire dhire aur order mein chalte hain. Har naya symbol plain words mein define hoga, picture ke roop mein draw hoga, aur us question se justify hoga jiska wo jawab deta hai. Koi bhi cheez use nahi hogi pehle usse build kiye bina.

1. "Molecule" kya hai aur hum use kaise picture karte hain?

Ball kyun? Kyunki collisions ke liye jo cheez matter karti hai wo sirf yeh hai ki do centres touch karne se pehle kitne paas aate hain. Ek ball woh ek single number $d$ se capture karta hai. Hum molecule ki real fuzzy shape ki parwah nahi karte.

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

$d$ (metres) — ek molecule ka diameter. Figure mein do touching balls dekho: contact ke moment mein centres exactly $d$ apart hain. Yeh single fact poori mean-free-path derivation ko drive karta hai.

2. Molecules count karna: $N$ , $V$ aur number density $n$

Hum $n$ kyun chahiye aur sirf $N$ kyun nahi? Kyunki "crowdedness" hi collisions ko control karti hai, raw count nahi. Ek stadium mein ek million molecules empty hai; ek thimble mein ek million ek traffic jam hai. Number density $n$ crowding ko directly measure karta hai.

3. Speed, velocity, aur uske teen components $v_{x}, v_{y}, v_{z}$

3-D mein koi bhi arrow teen sideways pieces mein split ho sakta hai: $x$ ke along kitna point karta hai, $y$ ke along, $z$ ke along. Yeh pieces components $v_{x}, v_{y}, v_{z}$ hain. Speed $v$ ke unlike, har component negative ho sakta hai — $v_{x} < 0$ ka matlab sirf " $- x$ direction mein move kar raha hai" hai.

Arrow ki length uske pieces se 3-D Pythagoras rule se relate hoti hai (note: ismein plain speed $v$ involve hai, bold vector nahi):

$v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$

Definition Isotropy se do facts — aur woh alag alag hain

Isotropy ka matlab hai gas har direction mein same dikhti hai — molecules equally likely hain left jaane ke liye jitna right, upar jitna neeche. Do alag consequences follow karte hain, aur pressure derivation ko dono chahiye:

Fact 1 — average component zero hai: $\overline{v_{x}} = \overline{v_{y}} = \overline{v_{z}} = 0$ . Har molecule jo right drift kar raha hai ( $v_{x} > 0$ ) ke liye ek left drift kar raha hai ( $v_{x} < 0$ ); signed values cancel ho jaate hain. Isliye gas poori tarah drift nahi karti — aur isliye sirf squared components (jo hamesha $\geq 0$ hain aur kabhi cancel nahi hote) pressure formula mein survive kar sakte hain.

Fact 2 — squared components equally share karte hain: $\overline{v_{x}^{2}} = \overline{v_{y}^{2}} = \overline{v_{z}^{2}} = \frac{1}{3} \overline{v^{2}}$ . Koi direction special nahi, toh teen equal squared-averages ko $\overline{v^{2}}$ mein add hona chahiye.

Saath mein yeh exactly woh hai jo wall-pressure result ko parent derivation ke step 5 mein RMS speed mein turn karta hai.

4. Bar aur "mean": $\overset{v}{ˉ}$ aur $\overline{v^{2}}$ ka matlab

Order kyun matter karta hai? Averaging se pehle squaring, averaging karke phir squaring se same nahi hai, jab speeds spread out hoti hain. Fast molecules squaring se zyada blow up ho jaate hain, toh woh $v^{2}$ ke average ko upar khench lete hain. Yahi gap exactly wajah hai ki RMS speed mean speed se zyada hoti hai — agla section dekho.

5. Jo tools sneak in karte hain: integrals, $e^{- x}$ , aur $\propto$ sign

Yahan integral kyun? Molecules teen speeds mein nahi aate; woh speeds ki ek continuous smear mein aate hain. Smear ke upar average karne ke liye tum finitely add nahi kar sakte — integrate karna padta hai.

Common mistake Yeh sochna ki speed distribution sirf

$e^{- m v^{2} /2 k_{B} T}$ hai Kyun sahi lagta hai: exponential eye-catching part hai, aur woh sach mein kehta hai "fast rare hai". Fix: Saccha Maxwell–Boltzmann speed distribution hai $f (v) d v = 4 π N (\frac{m}{2 π k _{B} T})^{3/2} v^{2} e^{- \frac{m v ^{2}}{2 k _{B} T}} d v .$ Do pieces jo bare exponential miss karta hai:

ek $v^{2}$ prefactor — jaise $v$ badhta hai ek length $v$ ka arrow utni zyada directions mein point kar sakta hai, toh distribution $v = 0$ par $0$ hai, rise karti hai, phir exponential ise wapas pull karta hai. Yahi ise non-zero speed $v_{p}$ par peak karta hai, $v = 0$ par nahi.
ek normalisation constant $4 π N (m /2 π k_{B} T)^{3/2}$ jo isliye choose kiya gaya hai ki fractions add up hokar sab $N$ molecules ban sakein. Full shape aur uski origin Maxwell-Boltzmann distribution mein hai. Yahan hum sirf ratio form $v^{2} e^{- a v^{2}}$ borrow karte hain, $a = m / (2 k_{B} T)$ ke saath, jahan constants cancel ho jaate hain.

6. Mean speed $\overset{v}{ˉ}$ actually distribution se kaise build hoti hai

Parent note state karta hai $\overset{v}{ˉ} = 8 k_{B} T / π m$ ; yahan hum zero se dikhate hain ki integral kyun appear hota hai aur number kaise fall out hota hai — taaki symbol $\overset{v}{ˉ}$ fully earned ho.

Kyunki $f (v)$ mein har constant upar aur neeche same hai, woh cancel ho jaata hai. Sirf shape $v^{2} e^{- a v^{2}}$ survive karta hai, $a = m / (2 k_{B} T)$ ke saath:

= \frac{\displaystyle\int_0^\infty v^3 e^{-av^2}\,dv}{\displaystyle\int_0^\infty v^2 e^{-av^2}\,dv}.$$ Do standard results hain $\int_0^\infty v^3 e^{-av^2}dv=\dfrac{1}{2a^2}$ aur $\int_0^\infty v^2 e^{-av^2}dv=\dfrac{\sqrt\pi}{4a^{3/2}}$. Divide karne par: $$\bar v = \frac{1/(2a^2)}{\sqrt\pi/(4a^{3/2})} = \frac{2}{\sqrt{\pi a}}.$$ Ab $a=m/(2k_BT)$ wapas daalo: > [!formula] Mean speed (result parent se match karta hai) > $$\boxed{\;\bar v = \frac{2}{\sqrt{\pi a}} = \sqrt{\frac{8k_BT}{\pi m}}=\sqrt{\frac{8RT}{\pi M}}\;}$$ Toh phrase "weight by $v^2e^{-av^2}$ aur mean speed milti hai" exactly yeh integral hai — kuch bhi hand-waved nahi. --- ## 7. Physics constants: $m$, $M$, $T$, $k_B$, $R$, $N_A$, $P$ > [!definition] Core physical symbols > - $m$ — **ek** molecule ki mass (kg). Tiny, e.g. $\sim 10^{-26}$ kg. > - $M = m\,N_A$ — **molar mass** (kg/mol): ek mole molecules ki mass. > - $N_A$ — **Avogadro's number** $\approx 6.022\times10^{23}\ \text{mol}^{-1}$ (units: *per mole*): ek mole banane ke liye kitne molecules chahiye. > - $T$ — **absolute temperature** (kelvin, K). Average molecular energy ka ek measure. Yahan kabhi °C mein nahi. > - $k_B$ — **Boltzmann constant** $\approx 1.38\times10^{-23}$ J/K: temperature se energy *per molecule* ka bridge. > - $R = k_B N_A \approx 8.314$ J/(mol·K): wahi bridge, lekin *per mole*. > - $P$ — **pressure** (pascal, Pa): molecular bombardment se wall area par force per unit. > [!intuition] Do constants $k_B$ aur $R$ kyun? > Woh *same cheez* do scales par kehte hain. $k_B T$ **ek molecule** ki energy scale hai; $R T$ **ek mole** ki energy scale hai. Tum konsa use karte ho yeh sirf depend karta hai ki tum single molecules count kar rahe ho ($m$, $n$) ya moles ($M$). Isliye $v_{rms}=\sqrt{3k_BT/m}=\sqrt{3RT/M}$ — upar aur neeche dono $N_A$ (units $\text{mol}^{-1}$) se multiply hote hain, toh ratio unchanged rehta hai. > [!mistake] Molar mass grams mein > **Kyun sahi lagta hai:** hum nitrogen ke liye $M$ ko "28" quote karte hain. > **Fix:** SI ko $M$ **kg/mol** mein chahiye ($0.028$). Grams ek speed dete hain jo $\sqrt{1000}\approx32\times$ zyada badi hoti hai. --- ## 8. Collision geometry: cross-section $\sigma$, moving-target $\sqrt2$, aur mean free path $\lambda$ > [!definition] Collision cross-section $\sigma$ > $\sigma = \pi d^2$ — "danger disc" ka area jo ek moving molecule sweep karta hai. Koi bhi molecule jिसका centre is disc ke andar land kare, wo hit ho jaata hai. > **$d$ kyun, $d/2$ kyun nahi?** Collision tab hoti hai jab do *centres* $d$ distance par hote hain (har ball ki radius $d/2$ hai). Toh effective target radius $d$ hai, aur disc area $\pi d^2$ hai. > [!intuition] $\sqrt2$ kyun appear hota hai — targets bhi move kar rahe hain > Sabse simple picture pretend karti hai ki *baaki sab molecules khade hain* jabki ek unke through fly karta hai. Tab time $t$ mein woh $\sigma\,\bar v\,t$ volume ka ek cylinder sweep karta hai, $n\sigma\bar v\,t$ targets ko hit karta hai, aur hits ke beech average distance hai > $$\lambda_{\text{naive}} = \frac{\bar v\,t}{n\sigma\bar v\,t} = \frac{1}{n\sigma}.$$ > Lekin targets **khade nahi hain** — har molecule zoom kar raha hai. Jo collision rate set karta hai woh ground ke against hamara molecule ki speed nahi, balki uski speed **target ke relative** hai. Equal mean speed ke independent molecules ke sab pairs par average karne par, woh *relative* speed $\sqrt2\,\bar v$ nikalta hai (do equal-length random arrows ka vector difference average par $\sqrt2$ times lamba hota hai). Faster approach → zyada collisions per second → shorter flights, toh hum rate mein $\bar v$ ko $\sqrt2\,\bar v$ se replace karte hain: > [!formula] Mean free path (moving-target correction ke saath) > $$\boxed{\;\lambda = \frac{1}{\sqrt2\,n\sigma} = \frac{1}{\sqrt2\,n\pi d^2}\;}$$ > Denominator mein $\sqrt2$ *sirf* isliye hai kyunki baaki molecules move karte hain. Ise drop karo aur tum $\lambda$ ko $\sqrt2\approx1.41$ factor se overestimate karoge. > [!definition] Mean free path $\lambda$ > ==Mean free path== $\lambda$ woh average straight-line distance hai jo ek molecule **do collisions ke beech** fly karta hai. Bada $\lambda$ = khaali gas = lambi free flights. Yeh ($\sigma$, $\sqrt2$, aur $\lambda$) parent ke Section 4 ka poora content hain; upar ke har doosre symbol in mein feed hote hain. --- ## 9. Sab kuch kaise fit hota hai Ise ek wiring diagram ki tarah nahi balki ek short story ki tarah padho. Teen streams topic mein flow karti hain: > [!intuition] Teen streams > **Stream 1 — kitna fast (ek ball).** Velocity **arrow** $\mathbf{v}$ $v_x,v_y,v_z$ mein split hota hai; isotropy $\overline{v_x}=0$ aur $\overline{v_x^2}=\tfrac13\overline{v^2}$ deta hai. Yeh wall-se-bounce argument mein feed karo aur pressure milta hai, aur pressure se **RMS speed** $v_{rms}$. Speeds ko Maxwell–Boltzmann shape $v^2e^{-av^2}$ (Section 6) se weight karo aur **mean speed** $\bar v$ milti hai. > > **Stream 2 — kitna crowded.** Volume $V$ mein count $N$ density $n=N/V$ deta hai — "kitne targets hain" wala number. > > **Stream 3 — target kitna bada.** Diameter $d$ collision disc $\sigma=\pi d^2$ deta hai. > > **Confluence.** Crowding ($n$), target size ($\sigma$), aur ek molecule room mein kitna fast tour karta hai ($\bar v$) — moving-target $\sqrt2$ se correct kiya gaya — saath mein **mean free path** $\lambda$ fix karte hain. RMS speed, mean speed, aur mean free path — teen headline results — sab in teen streams se trace back karte hain. ```mermaid graph TD V["velocity arrow v"] --> S1["components vx vy vz"] S1 -->|isotropy| WALL["wall bounce pressure"] WALL --> VRMS["v_rms"] V -->|weight by MB shape| VBAR["v_bar mean speed"] N["count N"] --> DEN["density n"] VOL["volume V"] --> DEN DIAM["diameter d"] --> SIG["cross section pi d^2"] DEN --> LAM["mean free path lambda"] SIG --> LAM VBAR --> LAM SQRT2["moving target root 2"] --> LAM ``` Agar tum bell-and-prefactor weighting ki deeper origin chahte ho, woh [[Maxwell-Boltzmann distribution]] mein hai; pressure argument [[Kinetic theory of pressure]] mein expand kiya gaya hai; $PV=Nk_BT$ link [[Ideal gas law]] hai; "energy per component" idea [[Equipartition theorem]] hai. Yahi numbers [[Diffusion and effusion]], [[Viscosity and thermal conductivity of gases]], aur [[Speed of sound in gases]] mein wapas appear karte hain. --- ## Equipment checklist Self-test: kya tum reveal karne se **pehle** har ek ko ek sentence mein state kar sakte ho? Overbar $\overline{(\cdot)}$ ka kya matlab hai? ::: Har molecule ke liye quantity add karo aur $N$ se divide karo — ek average. $N$ aur $n$ mein difference? ::: $N$ ek plain count hai; $n=N/V$ molecules per cubic metre hai (crowding). Velocity vs speed kaise likhte hain? ::: Velocity bold/arrow vector $\mathbf{v}$ hai (direction hai); speed $v=|\mathbf{v}|$ uski length hai, hamesha $\ge 0$. RMS ke liye averaging se pehle square kyun karein? ::: Kinetic energy $\propto v^2$ hai; $v^2$ average karna typical energy capture karta hai, aur squaring fast molecules ko zyada weight deta hai. $\overline{v_x}$ kya hai aur kyun? ::: Zero — har molecule jo $+x$ ja raha hai ke liye ek aur $-x$ ja raha hai, toh signed components cancel ho jaate hain; sirf squared components survive karte hain. $\overline{v_x^2}=\tfrac13\overline{v^2}$ kyun? ::: Isotropy — koi direction special nahi, toh teen squared components $\overline{v^2}$ equally share karte hain. $e^{-av^2}$ mein constant $a$ kya stand karta hai? ::: $a=m/(2k_BT)$, toh $av^2$ kinetic energy over thermal energy hai. $\bar v$ ek integral ke roop mein kaise define hoti hai? ::: $\bar v=\int_0^\infty v\,f(v)\,dv \big/ \int_0^\infty f(v)\,dv$ — (speed × kitne) ka sum count par. Woh integral kya value deta hai? ::: $\bar v=2/\sqrt{\pi a}=\sqrt{8k_BT/\pi m}$. Kya speed distribution sirf $e^{-mv^2/2k_BT}$ hai? ::: Nahi — iska ek $v^2$ prefactor aur ek normalisation constant hai, isliye yeh non-zero speed par peak karta hai. Collision picture mein $d$ kya hai? ::: Molecule ka diameter; do molecules tab collide karte hain jab unke centres $d$ apart hote hain. $\sigma=\pi d^2$ kyun hai aur $\pi (d/2)^2$ kyun nahi? ::: Effective target radius poora $d$ hai (do radii ka sum), toh disc area $d$ use karta hai. $\lambda$ mein $\sqrt2$ kahan se aata hai? ::: Targets bhi move karte hain; average relative speed $\sqrt2\,\bar v$ hai, collision rate badhata hai aur $\lambda$ chota karta hai. $k_B$ aur $R$ mein difference? ::: $k_B$ energy-per-molecule per kelvin hai; $R=k_B N_A$ energy-per-mole per kelvin hai. $\sqrt{3RT/M}$ mein $M$ ke units? ::: Kilograms per mole (SI), e.g. $\text{N}_2$ ke liye $0.028$. $N_A$ ke units? ::: Per mole, $\text{mol}^{-1}$ ($\approx 6.022\times10^{23}\,\text{mol}^{-1}$). $\lambda$ physically kya measure karta hai? ::: Woh average distance jo ek molecule collisions ke beech fly karta hai. $A\propto B$ tumhe kya batata hai? ::: $A$, $B$ ka ek fixed multiple hai; $B$ double karo aur $A$ double ho jaata hai.

1. "Molecule" kya hai aur hum use kaise picture karte hain?

2. Molecules count karna: N, V aur number density n

3. Speed, velocity, aur uske teen components vx​,vy​,vz​

4. Bar aur "mean": vˉ aur v2 ka matlab

5. Jo tools sneak in karte hain: integrals, e−x, aur ∝ sign

6. Mean speed vˉ actually distribution se kaise build hoti hai

2. Molecules count karna: $N$ , $V$ aur number density $n$

3. Speed, velocity, aur uske teen components $v_{x}, v_{y}, v_{z}$

4. Bar aur "mean": $\overset{v}{ˉ}$ aur $\overline{v^{2}}$ ka matlab

5. Jo tools sneak in karte hain: integrals, $e^{- x}$ , aur $\propto$ sign

6. Mean speed $\overset{v}{ˉ}$ actually distribution se kaise build hoti hai