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

1.7.8 · D5 · Physics › Thermodynamics › Ideal gas law PV = nRT — derivation from kinetic theory

Sach ya jhooth — justify karo

Gas law collision picture ke bina galat hai; har item tumse kyun karwata hai bolna.

Molecules ki sankhya

N

ko double karna fixed

V

aur

T

par pressure ko double kar deta hai.

Sach —

P = N k_{B} T / V

mein

N

linear hai; double molecules matlab double wall-kicks per second, toh double push. Ye $P=Nk_BT/V$ seedha padha hua hai.

Same temperature par, bhaare molecules aur halke molecules same pressure daalte hain (same

N

V

Sach — pressure sirf

N

V

T

par depend karta hai, mass par nahi. Bhaare molecules slower chalte hain (

v_{r m s} \propto 1/ m

) lekin har collision mein zyada jor se marte hain; dono effects exactly cancel hote hain kyunki

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

dono ke liye same hai.

⟨ v_{x} ⟩ = 0

ek still container mein equilibrium mein gas ke liye hota hai.

Sach — equal molecules

+ x

aur

- x

dono mein chalti hain, toh average velocity component cancel hoti hai. Agar ye zero nahi hoti toh gas wind ki tarah drift karti, still nahi rehti.

Kyunki

⟨ v_{x} ⟩ = 0

hai, molecules wall par zero average force daalti hain.

Jhooth — force

⟨ v_{x}^{2} ⟩

ke saath scale karta hai,

⟨ v_{x} ⟩

ke saath nahi. Squares hamesha positive hote hain, toh cancel nahi hote; yahi reason hai ki pressure survive karta hai.

Agar collisions thodi inelastic hoti, toh pressure phir bhi hamesha constant rehta.

Jhooth — inelastic collisions kinetic energy ko wall mein bleed karte, gas ko cool karte aur pressure drop karte time ke saath. Elasticity hi hai jo

⟨ v^{2} ⟩

(aur is tarah

T

) ko steady rakhti hai.

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

mein

\frac{1}{3}

factor experiments se fit kiya gaya fudge factor hai.

Jhooth — ye pure geometry se aata hai:

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

aur isotropy teeno ko equal banata hai, toh har ek

\frac{1}{3} ⟨ v^{2} ⟩

hai. Ye exact hai, empirical nahi.

Temperature badhana pressure ko sirf isliye increase karta hai kyunki molecules tezi se chalti hain.

Jhooth — heating pressure ko ek saath do tarike se badhata hai: har molecule zyada jor se marti hai (

2 m v_{x}

bada) aur zyada baar marti hai (rate

v_{x} /2 L

bada). Dono factors mein

v_{x}

hai, isliye result

v_{x}^{2} \propto T

par depend karta hai.

Derivation assume karti hai ki molecules ek dusre se constantly collide karti hain.

Jhooth — ideal-gas derivation assume karti hai ki molecules sirf walls se collide karti hain; inter-molecular collisions sirf speeds redistribute karti hain aur isotropy preserve karti hain, toh pressure result unchanged rehta hai. (Dekho Kinetic Theory of Gases.)

P V = n R T

real gas ke liye bahut zyada pressure par bhi hold karta hai.

Jhooth — high pressure par molecules ki apni volume aur unke mutual attractions matter karte hain, toh ideal assumptions toot jaati hain. Tab tumhe Real Gases and Van der Waals Equation chahiye.

Constant

R

k_{B}

se zyada fundamental hai.

Jhooth —

k_{B}

per-molecule fundamental constant hai;

R = N_{A} k_{B}

sirf

k_{B}

ko ek mole tak scale kiya gaya hai. Nature molecules count karti hai, moles nahi.

Error dhundo

Har statement mein ek flaw hai. Use naam do.

"Jab ek molecule

x

-wall se takraati hai, uska momentum change

- m v_{x}

hai."

Error: ye

- 2 m v_{x}

hona chahiye. Velocity reverse hoti hai

+ v_{x}

- v_{x}

tak, toh change hai

- m v_{x} - (+ m v_{x}) = - 2 m v_{x}

, sirf

m v_{x}

lose nahi.

"Wall par force poori speed

v

use karta hai kyunki molecule itni tezi se travel karti hai."

Error: sirf perpendicular component

v_{x}

reverse hoti hai aur momentum deliver karti hai; parallel components (

v_{y}, v_{z}

) wall ke saath slide karte hain aur us wall ko koi push nahi dete.

"Same wall par hits ke beech time

L / v_{x}

hai."

Error: molecule ko door wali wall aur wapis jaana padta hai, distance

2 L

, toh

Δ t = 2 L / v_{x}

hai,

L / v_{x}

nahi.

"Hum

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩

set kar sakte hain pipe mein flow kar rahe gas ke liye."

Error: jab bulk flow ho toh isotropy fail hoti hai — ek preferred direction exist karti hai, toh teeno mean squares differ karte hain. Equality sirf tab hold karti hai jab gas mein koi net drift na ho.

⟨ E_{k} ⟩ = \frac{3}{2} k_{B} T

substitute karna ek mathematical identity hai jo humne pehle derive ki thi."

Error: ye ek physical definition/input hai, Equipartition Theorem se aata hai, koi cheez nahi jo mechanics ne diya. Ye woh bridge hai jo Newton's laws ko thermodynamics mein convert karta hai.

"Room temperature ke liye

T = 25

use karna

P V = n R T

mein theek hai jab tak tum consistent ho."

Error:

T

absolute (Kelvin) hona chahiye kyunki ye kinetic energy ke proportional hai;

25^{\circ} C = 298.15

0^{\circ} C

par molecules phir bhi chalti hain, toh Celsius galti se zero energy predict karta.

⟨ v^{2} ⟩ = ⟨ v ⟩^{2}

, toh hum sirf average speed ko square kar sakte hain."

Error: squares ka mean nahi hota mean ka square. Speeds mein spread ki wajah se,

⟨ v^{2} ⟩ > ⟨ v ⟩^{2}

; sirf

⟨ v^{2} ⟩

(mean-square) pressure mein enter karta hai. Dekho Root Mean Square Speed.

Kyun wale sawaal

Apne aap ko reason explain karne par force karo, fact restate mat karo.

Wall ko steady pressure kyun feel hota hai aur jittery taps ki series nahi?

Kyunki ~

1 0^{23}

molecules har second mein strike karti hain; individual taps ek smooth average push mein blur ho jaate hain, jaise roof par raindrops continuous sunai deti hain. Pressure wahi time-average hai.

Pressure

⟨ v^{2} ⟩

par depend kyun karta hai na ki average speed

⟨ v ⟩

par?

Kyunki momentum per hit (

\propto v_{x}

) aur hit-frequency (

\propto v_{x}

) dono velocity ke saath scale karte hain, toh unka product

v_{x}^{2}

carry karta hai. Tezi molecules do baar count hoti hain.

Hum sum

\sum v_{x, i}^{2}

N ⟨ v_{x}^{2} ⟩

se kyun replace kar sakte hain?

Kyunki

⟨ v_{x}^{2} ⟩

defined hai

\frac{1}{N} \sum v_{x, i}^{2}

ke roop mein; average ko

N

se multiply karna sirf average ko undo karta hai. Wahan koi physics assume nahi ki — ye bookkeeping hai.

Derivation mein constant pressure dene ke liye collisions elastic kyun honi chahiye?

Elastic ka matlab hai kinetic energy conserve hoti hai, toh

⟨ v^{2} ⟩

(aur isliye

T

) fixed rehta hai. Agar energy walls mein leak karti, gas cool hoti aur

P

neeche drift karta.

P V = n R T

mein molecule ke mass ka koi reference kyun nahi hai, jabki derivation

m

se bhari hui hai?

Mass cancel ho jaata hai jab hum temperature bridge insert karte hain:

N m ⟨ v^{2} ⟩ = N \cdot 3 k_{B} T

, toh

m

T

mein disappear ho jaata hai. Temperature pehle se hi mass-speed combination encode karta hai.

Isotropy (

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩

) ek physical assumption kyun hai aur automatic nahi?

Ye state karta hai gas ka koi preferred direction nahi hai — ye sirf equilibrium mein sach hai jab koi flow, gravity gradient, ya field na ho. Ye state ke baare mein ek assumption hai, isliye moving ya stratified gas ise violate kar sakti hai.

Box ko squeeze karna (smaller

V

) pressure kyun badhata hai agar

T

unchanged hai?

Walls ke beech shorter path

2 L

ke saath, har molecule per second zyada baar marti hai, toh same area par per second zyada momentum dump hota hai. Ye hai

1/ V

P = N k_{B} T / V

mein.

Edge cases

Woh scenarios jo tidy final formula quietly assume away kar deta hai.

Kinetic model pressure ke liye kya predict karta hai jab

T \to 0

⟨ v^{2} ⟩ \to 0

, toh

P \to 0

— molecules chalna band kar deti hain aur walls ko kick karna band kar deti hain. Ye absolute zero ko translational motion khatam hone ke point ke roop mein define karta hai.

Kya

P V = n R T

ek single molecule (

N = 1

) ke liye valid hai?

Sirf time-averaged, statistical sense mein — ek molecule wildly fluctuating, spiky "pressure" deti hai. Law meaningful hai kyunki

N

astronomically large hai, fluctuations ko smooth karta hai.

Jab density

N m / V

bahut large ho jaaye toh ideal-gas prediction ka kya hoga?

Ye fail hota hai, kyunki molecular volume aur attractions ko ab ignore nahi kiya ja sakta; real molecules points nahi hain. Ye exactly Real Gases and Van der Waals Equation ka regime hai.

Kya derivation container ki shape par kuch khayal karti hai (humne cube use kiya)?

Nahi — cube sirf convenient hai. Saari walls aur orientations par sum karne ke baad, sirf total volume

V

survive karta hai, toh koi bhi shape same

P V = n R T

deta hai.

Do alag molecular masses ke gas ke mixture ke liye, temperature

T

par har species ka partial pressure kya determine karta hai?

Har species independently

P_{i} V = N_{i} k_{B} T

satisfy karti hai; total sum hai (Dalton's law). Mass drop out hota hai kyunki dono same

T

share karte hain, isliye same average kinetic energy.

Agar container wall move kar rahi hai (jaise piston andar push ho raha hai), kya collision abhi bhi hamari sense mein "elastic" hai?

Nahi — ek moving wall rebound speed change karti hai, gas par kaam karti hai aur use heat karti hai. Static derivation rigid, stationary walls assume karti hai; moving walls kaam aur internal energy ke study mein aati hain.

Kya same

T

N

V

par ek monatomic aur ek diatomic gas ka same pressure hota hai?

Haan — pressure sirf translational kinetic energy

\frac{3}{2} k_{B} T

par depend karta hai, jo dono ke liye identical hai. Diatomic ke extra rotational modes energy store karte hain lekin walls ko push nahi karte (dekho Equipartition Theorem).

Recall Lock in karne ke liye ek-line summary

Pressure = momentum bookkeeping; $\frac{1}{3}$ geometry hai (isotropy); $T$ ek definition hai ( $⟨ E_{k} ⟩ = \frac{3}{2} k_{B} T$ ); aur law exactly tab break hota hai jab "point particles, no forces, elastic, isotropic" assumptions break hoti hain.