1.7.10 · D1Internal energy of ideal gas U = (f - 2)nRT
Ek ideal gas tiny particles ki ek bheed hai, aur uski internal energy kuch nahi balki unki saari random motion ki total energy hai — isliye gas ko heat karna (unhe faster move karana) hi wo ek tarika hai us energy ko badhane ka. U = 2 f n R T kitni motion , kitne particles , aur kitne independent tarike unke move karne ke .
Parent formula par trust karne se pehle, aapko usmein har ek letter bina rukke padhna aana chahiye. Yeh page har symbol ko absolute zero se build karta hai, ek aisi order mein jahan har ek cheez sirf pehle wali cheez par depend karti hai. Kuch bhi assume nahi kiya gaya.
Ek sealed box full of tiny hard balls imagine karo, jo har direction mein zoom kar rahi hain, walls aur ek doosre se bounce kar rahi hain. Yahi hamara gas hai. Neeche jo kuch bhi hai woh is swarm ke baare mein kuch count karne ya measure karne ka tarika hai. Yeh picture apne dimaag mein rakho — har symbol isi par point karta hai.
Definition Molecules ki sankhyaN N kitne individual molecules box ke andar hain. Ek plain count, jaise marbles ginna. Koi units nahi.
Woh number enormous hota hai (trillions of trillions), isliye ek-ek karke ginna hopeless hai. Iske bajaye hum bundles mein count karte hain jise moles kehte hain.
Definition Amount of substancen n 6.022 × 1 0 23 6.022 × 1 0 23
Definition Avogadro's numberN A N A = 6.022 × 1 0 23 kitne molecules ek mole banate hain yeh batata hai. Yeh honest count N n N = n N A
Intuition Moles ki zaroorat kyun hai?Count N n R
Yeh teen cheezein woh hain jo aap box ke bahar se measure kar sakte ho, individual molecules dekhe bina.
V V box kitna space enclose karta hai yeh hai. Container ki size imagine karo. Units: cubic metres m 3
P P molecules walls par kitna hard push karte hain, per unit area hai. Ek hail of tiny balls ko walls par drumming karte imagine karo; unke saare tiny pushes ko ek square metre par add karo. Units: pascals Pa
T T kelvin , K molecules kitne vigorously jiggle karte hain uska measure hai — random motion ki average kinetic energy. Imagine karo: hotter = faster balls. Kelvin absolute zero par shuru hota hai (sari motion ruk jaaye).
T U T hi depend karta hai. Yeh tabhi believable hai jab aap accept kar lo ki T P V arrangement describe karte hain, lekin T speed describe karta hai, aur motion ki energy speed ke baare mein hai.
Common mistake °C use karna K ki jagahKyun sahi lagta hai: rozana ki temperature Celsius mein hoti hai.
The fix: In sab formulas ko kelvin chahiye, kyunki sirf kelvin actual molecular energy ke proportional hai. T ( K ) = T ( ° C ) + 273.15
Ab bahar ke measurements (P , V , T N , n Ideal gas law .
Do constants aate hain. Yeh same idea hain do counting scales par.
R R = 8.314 J mol − 1 K − 1 ek mole ko per degree kitni energy milti hai . Yeh n
Definition Boltzmann constantk B k B = 1.381 × 1 0 − 23 J K − 1 ek single molecule ko per degree kitni energy milti hai . Yeh N
Intuition Is bridge ki zaroorat kyun haiKinetic theory naturally molecules mein bolti hai (woh ek ball ke baare mein baat karti hai), isliye woh k B moles mein bolte hain, isliye woh R N k B = n R
Definition Kinetic energy (ek molecule ki)Woh energy jo ek moving ball carry karta hai kyunki woh move karta hai: 2 1 m v 2 m v
Alag-alag molecules alag-alag speeds par move karte hain, isliye hume ek average chahiye.
Definition Mean-square speed⟨ v 2 ⟩ ⟨ ⟩ saare molecules ka average "; ⟨ v 2 ⟩ squared ka average hai. Hum pehle square karte hain kyunki kinetic energy v 2
Intuition Speed khud ki jagahv 2 v 2 v
Isliye kinetic-theory result jo parent quote karta hai,
P V = 3 1 N m ⟨ v 2 ⟩ , ⟨ v 2 ⟩ ⟨ v ⟩ total kinetic energy ke baare mein ek statement hai jo pressure ki form mein dressed hai. Dekho Kinetic theory of gases .
Definition Degree of freedomEk degree of freedom ek independent tarika hai jisme ek molecule motion ki energy store kar sake . Imagine karo ek mixing desk par separate sliders: har ek ek channel hai jisme energy flow ho sakti hai.
Ek single point-like atom teen independent directions mein move kar sakta hai — left/right, forward/back, up/down. Yeh 3 translational degrees of freedom hain. Ek dumbbell-shaped diatomic molecule do axes ke around end-over-end tumble bhi kar sakta hai, jo 2 rotational degrees of freedom add karta hai, aur f = 5 Degrees of freedom .
f f us gas ke ek molecule ke liye active degrees of freedom ki total sankhya hai. Monatomic: f = 3 T f = 5
U f Equipartition theorem ). Toh zyada channels wala molecule same temperature par zyada energy store karta hai. f f f U = 2 f n R T
Formula mein 2 f f × 2 1 2 1 har ek channel mein daalti hai: 2 1 k B T f 2 f k B T N N k B → n R U = 2 f n R T
Definition Internal energyU U poore swarm ki total microscopic kinetic energy . Units: joules J
Δ Δ X X Δ U = U final − U initial Δ T = T final − T initial
Q W Q heat ke roop mein pour ki gayi energy ; W gas jo energy apne surroundings ko push karne mein spend karta hai . Yeh U First law of thermodynamics ke through connect hote hain: Δ U = Q − W Δ U
Kinetic theory and mean square speed
Translational KE per molecule
Equipartition half kB T per DOF
Energy per molecule = f over 2 kB T
First law delta U = Q minus W
Right side cover karo aur derivation page par jaane se pehle khud ko test karo.
N box mein individual molecules ka plain count.
n moles ki sankhya = bundles of 6.022 × 1 0 23
N A Avogadro's number 6.022 × 1 0 23 N = n N A
T random molecular motion ki average kinetic energy.
T sirf kelvin actual molecular energy ke proportional hai aur absolute zero par shuru hota hai.
Molecule form mein ideal gas law P V = N k B T
Dono constants ke beech bridge relation N k B = n R R = N A k B
k B molecule count N R n
v 2 v energy v 2
Ek degree of freedom hai ek independent tarika jisme molecule motion ki energy store kar sake.
f 3 / 5.
U f energy har channel mein equally share hoti hai, isliye zyada channels zyada energy store karte hain.
Δ U final internal energy minus initial internal energy.
U Q W Δ U = Q − W
