Kinetic theory — pressure derivation, temperature as mean KE: Foundations

1.7.9 · D1 · Physics › Thermodynamics › Kinetic theory — pressure derivation, temperature as mean KE

Yeh page assume karta hai ki tum kuch bhi nahi jaante. Pehle hum parent note (the topic itself) padhne se pehle har ek symbol build karenge, ek ek brick karke, har brick pichle brick pe tikki hui.

1. Balls ko count karna: $N$ , $m$ , aur "point particle"

Topic ko $N$ aur $m$ kyun chahiye: pressure build hoti hai sab $N$ molecules ki pushes ko jodkar, aur har push us mass $m$ pe depend karti hai jo wall mein jaati hai.

2. Position, direction, aur box: $L$ , $V$

Topic ko $L$ kyun chahiye: ek molecule jo box ke across round-trip karta hai woh $2 L$ distance cover karta hai (jaana aur aana), jo set karta hai kitni baar woh wall se takrata hai. Aur force ko wall area $L^{2}$ se divide karne pe force pressure ban jaati hai.

Figure — Kinetic theory — pressure derivation, temperature as mean KE

3. Speed vs velocity: $v$ aur uske components $v_{x}, v_{y}, v_{z}$

Yeh poore topic ki sabse important picture hai.

Ek arrow ko teen mein kyun split karein? Kyunki ek wall sirf woh motion toward it care karti hai. Right wall $x$ -direction face karti hai, toh sirf $v_{x}$ decide karta hai ki molecule kitna hard aur kitni baar usse hit karta hai. Baaki do ( $v_{y}, v_{z}$ ) molecule ko wall ke saath sideways slide karte hain — woh impact cause nahi karte.

4. Squares ka average: $\overset{v}{ˉ}$ , $\overline{v^{2}}$ aur $v_{r m s}$

Symbol ke upar bar ka matlab hai "ka average."

Topic ko $\overline{v^{2}}$ kyun chahiye: kyunki pressure aur kinetic energy dono speed ke square ke saath badhte hain, toh plug in karne ke liye sahi average squares ka average hai — plain average speed $\overset{v}{ˉ}$ nahi.

5. Momentum $p$ aur uska change $Δ p$

Topic ko $Δ p_{x}$ kyun chahiye: jab ek molecule right wall se bounce karta hai, uski $x$ -motion reverse hoti hai. Uske $x$ -momentum ka change exactly wahi hai jo wall ne usse push kiya (aur, Newton ke law of pairs se, jo usne wall ko push kiya).

6. Force as a rate: $F = \frac{Δ p}{Δ t}$

7. Force ko Pressure mein: $P = \frac{F}{A}$

Topic ko yeh kyun chahiye: molecules wall pe total force deliver karte hain, lekin jo hum gauge se measure karte hain woh pressure hai. Wall area $L^{2}$ se divide karna bridge banata hai.

8. Kinetic energy $E_{k}$ , aur constants $R$ , $k_{B}$ , $N_{A}$ , $T$

Topic ko yeh kyun chahiye: poori punchline — "temperature is average kinetic energy" — equation hai $\frac{1}{2} m \overline{v^{2}} = \frac{3}{2} k_{B} T$ . $k_{B}$ ke bina woh sentence symbols mein likhne ka koi tarika nahi hai.

Foundations topic ko kaise feed karte hain

Neeche har box ek foundation hai jo tumne abhi build ki; arrows dikhate hain yeh kya possible banata hai. Upar se neeche padho — counting aur velocity-splitting momentum-per-bounce build karte hain, jo "force is a rate" aur "sum over all molecules" aur "pressure is force over area" ke through pressure result build karta hai — aur use kinetic energy aur $k_{B}$ ke saath pair karke temperature punchline milti hai.

Har foundation downstream kahan reappear karta hai

Do short formulas jo upar build ki gayi hain woh parent ki derivation ke pehle do steps hain — yeh lo, reuse ke liye ready:

$∣ Δ p_{x} ∣ = 2 m ∣ v_{x} ∣ Δ t = \frac{2 L}{∣ v _{x} ∣}$

Foundation	Topic mein pehli baar kab use hota hai
$∣ Δ p_{x} ∣ = 2 m ∣ v_{x} ∣$	Pressure derivation ka Step 1
$Δ t = 2 L / ∣ v_{x} ∣$	Step 2 (hit frequency)
$F = Δ p /Δ t$	Step 3 (ek molecule se force)
$\overline{v^{2}}, N$	Step 4 (saare molecules ko sum karna)
$v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$	Step 5 (factor $\frac{1}{3}$ )
$P = F / A$ , $A = L^{2}$	Step 6 (final pressure)
$E_{k} = \frac{1}{2} m v^{2}$ , $k_{B}$ , $T$	Temperature as mean KE

Yahi ideas aage Ideal Gas Law PV=nRT, Maxwell-Boltzmann Speed Distribution, Degrees of Freedom and Molar Heat Capacity, aur Internal Energy of Ideal Gas mein branch karte hain.

Equipment checklist

Right side cover karo aur khud ko test karo — tum parent note ke liye tab hi ready ho jab har line instant ho.

N

kya count karta hai?

Box mein molecules ki sankhya (ek finite, bahut bada pura number).

"Point particle" kya hai?

Ek molecule jise zero size (ek dot) maana jaata hai.

L

side wale cube ka volume?

V = L^{3}

; ek wall ki area

L^{2}

hai.

Speed aur velocity mein fark?

Speed ek non-negative number hai; velocity speed with direction hai (ek arrow).

Kya

v_{x}

jaisa component negative ho sakta hai?

Haan — uska sign dikhata hai woh axis ke saath kis taraf move kar raha hai; plain speed

v

nahi ho sakti.

Velocity ko

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

mein kyun split karein?

Ek wall sirf woh component feel karti hai jo uski taraf point karta hai (jaise right wall ke liye

v_{x}

Speed ke liye 3D Pythagoras?

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

\overset{v}{ˉ}

kya hai?

Speeds ka plain arithmetic mean.

\overline{v^{2}}

mein bar ka kya matlab hai?

Average — yahan, squared speeds ka average.

v_{r m s}

define karo.

v_{r m s} = \overline{v^{2}}

— square, phir mean, phir root.

Kya

v_{r m s}

plain average speed

\overset{v}{ˉ}

ke barabar hai?

Nahi;

v_{r m s}

hamesha thoda bada hota hai (barabar tab hi jab saari speeds match karein).

Momentum

p

kya hai?

Mass times velocity,

p = m v

(ek axis ke saath sign carry karta hai).

Δ

ka kya matlab hai?

"Change in" — final minus initial.

Ek elastic bounce mein momentum change (size)?

∣ Δ p_{x} ∣ = 2 m ∣ v_{x} ∣

Ek wall pe hits ke beech round-trip time?

Δ t = 2 L / ∣ v_{x} ∣

Un do formulas mein absolute values kyun?

Ek push-size aur ek time positive hone chahiye chahe travel direction koi bhi ho.

Newton's second law ka momentum form?

F = Δ p /Δ t

(force momentum change ki rate hai).

F = ma

ki jagah

F = Δ p /Δ t

kyun use karein?

Molecules momentum sudden bounces mein change karte hain, smooth acceleration mein nahi.

Pressure ki definition?

Force per unit area,

P = F / A

Ek molecule ki kinetic energy?

E_{k} = \frac{1}{2} m v^{2}

N_{A}

kya hai?

Avogadro's number,

\approx 6.02 \times 1 0^{23}

— ek mole mein molecules.

k_{B}

ka role?

Woh conversion factor jo temperature ko energy per molecule mein turn karta hai.

k_{B}

aur

R

mein link?

k_{B} = R / N_{A}