Ideal gas law PV = nRT — derivation from kinetic theory: Foundations

1.7.8 · D1 · Physics › Thermodynamics › Ideal gas law PV = nRT — derivation from kinetic theory

Yeh page kuch bhi assume nahi karta. derivation ko chhoone se pehle, tumhe symbols ka ek chhota toolbox chahiye. Hum har ek ko earn karenge — plain meaning, uska picture, aur kyun topic aage nahi badh sakta uske bina.

1 · Box: $L$ , $V$ , aur "ek wall"

Ek cube-shaped gas container ka picture banao. Har side ki length $L$ hai (ek length, metres mein measure ki gayi).

Figure — Ideal gas law PV = nRT — derivation from kinetic theory

Topic ko iske zarorat kyun hai. Do walls ek dusre ke saamne $L$ doori par hain. Ek ball jo ek wall se bounce karti hai, usे box ke paar aur wapas jaana padta hai — ek round trip of $2 L$ — pehle ki woh usi wall ko dobara hit kar sake. Woh distance $2 L$ baad mein set karta hai ki collisions kitni baar hoti hain. Aur $V = L^{3}$ exactly wahi $V$ hai jo $P V = n R T$ mein aata hai, isliye box ki size literally final law ke chaar letters mein se ek hai.

2 · Speed, velocity, aur uske components $v_{x}, v_{y}, v_{z}$

Ek ball ke paas sirf speed nahi hoti (kitni tez); uski ek direction bhi hoti hai. Speed + direction milake velocity kehlata hai.

Teeno components milke puri speed $v$ dete hain, 3-D Pythagoras rule se: $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$

Topic ko iske zarorat kyun hai. Jab ek ball right-hand wall se takraati hai, sirf uski $x$ -motion matter karti hai — wall $x$ -direction face karti hai. Sideways parts $v_{y}, v_{z}$ wall ke saath-saath phisalte hain aur kuch nahi badlate. Isliye hume full speed se $v_{x}$ ko alag karna aana chahiye. Yahi splitting derivation ka core hai.

3 · Mass $m$ aur momentum $p = m v$

Topic ko iske zarorat kyun hai. Wall kabhi "speed" feel nahi karti — woh ek kick feel karti hai, aur kick ek change in momentum hai. Jab ball hit karke wapas bounce karti hai, uska $x$ -momentum $+ m v_{x}$ se $- m v_{x}$ flip ho jaata hai, $2 m v_{x}$ ka change. Woh $2 m v_{x}$ hi wall ko milne wale dhakke ka size hai. Momentum woh currency hai jisme poori derivation ki payment hoti hai.

4 · Force $F$ aur Newton's second law $F = Δ p /Δ t$

Topic ko iske zarorat kyun hai. Har bounce $2 m v_{x}$ momentum deliver karta hai. Agar ye bounces har $Δ t$ seconds mein aate hain, toh momentum ko time se divide karne par wall par average force milta hai. Yeh "ek ball bounce hui" se "wall ko ek steady push feel hoti hai" tak ka bridge hai. Deeper version ke liye Pressure and Newton's Second Law dekho.

5 · Pressure $P = F / A$

Topic ko iske zarorat kyun hai. $P$ $P V = n R T$ ka pehla hi letter hai. Upar sab kuch — components, momentum, force — sirf yahi ek number compute karne ke liye hai.

6 · Averages: $⟨ ⟩$ aur kyun $⟨ v_{x} ⟩ = 0$ lekin $⟨ v_{x}^{2} ⟩ \neq = 0$

Hum billions of balls ko ek ek track nahi kar sakte. Hum averages track karte hain, jo angle brackets $⟨ \cdot ⟩$ se likhe jaate hain.

Topic ko iske zarorat kyun hai. Push depend karta hai ki balls kitni hard aur kitni baar hit karti hain — dono $v_{x}$ ke saath badhte hain, isliye per ball force $v_{x}^{2}$ ke saath scale karta hai. Saare balls par average karne se messy sum ek clean $N ⟨ v_{x}^{2} ⟩$ ban jaata hai.

7 · Isotropy: $⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩$

Isko $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$ ke saath combine karo, dono sides ka average lo: $⟨ v^{2} ⟩ = 3 ⟨ v_{x}^{2} ⟩ \Rightarrow ⟨ v_{x}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩$

Topic ko iske zarorat kyun hai. Yahan se $P = \frac{1}{3} \frac{N m ⟨ v ^{2} ⟩}{V}$ mein famous factor $\frac{1}{3}$ janam leta hai. Full swarm picture ke liye Kinetic Theory of Gases dekho aur $⟨ v^{2} ⟩$ ka matlab jaanne ke liye Root Mean Square Speed dekho.

8 · Gas ko count karna: $N$ , $n$ , $N_{A}$ , aur constants $k_{B}$ , $R$ , $T$

Topic ko iske zarorat kyun hai. Ye mechanics result $P V = \frac{1}{3} N m ⟨ v^{2} ⟩$ ko thermodynamic $P V = n R T$ mein convert karte hain. "Temperature = average kinetic energy" wala link Equipartition Theorem aur Temperature and Internal Energy se aata hai; constants khud Boltzmann Constant and Gas Constant mein unpack kiye gaye hain.

Prerequisite map

Equipment checklist

Right side hide karo aur khud test karo — tum derivation ke liye tabhi ready ho jab har line automatic ho.

V = L^{3}

ka kya matlab hai aur

2 L

special kyun hai?

V

cube ka volume hai;

2 L

woh round-trip distance hai jo ek ball same wall ko dobara hit karne se pehle travel karti hai.

Jab ball

x

-wall se takraati hai toh sirf

v_{x}

kyun matter karta hai?

Wall

x

-direction face karti hai;

v_{y}, v_{z}

uske saath-saath slide karte hain aur kuch nahi badlate.

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

milke speed kaise dete hain?

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

(Pythagoras do baar).

Momentum kya hai aur elastic bounce par kitna change hota hai?

p = m v

;

x

-momentum flip hota hai

+ m v_{x} \to - m v_{x}

, size

2 m v_{x}

ka change.

Newton's second law ko momentum form mein batao.

F = Δ p /Δ t

— force woh rate hai jis par momentum change hota hai.

Pressure define karo aur cube ke liye wall area batao.

P = F / A

; wall area

A = L^{2}

hai.

⟨ v_{x} ⟩ = 0

kyun hai lekin

⟨ v_{x}^{2} ⟩ \neq = 0

kyun?

Signed velocities cancel ho jaati hain; squares sab positive hote hain isliye cancel nahi hote.

Isotropy se kya milta hai?

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩

N

n

N_{A}

ko relate karo.

N = n N_{A}

, jahaan

N_{A} \approx 6.022 \times 1 0^{23}

per mole.

R

aur

k_{B}

kaise related hain, aur

T

kelvin mein kyun hona chahiye?

R = N_{A} k_{B}

;

T

energy measure karta hai isliye absolute zero se start karna zaroori hai.