1.7.3 · D1Heat and internal energy — microscopic vs macroscopic
Gas ka ek box billions of tiny bouncing balls ka collection hai, lekin hum kabhi bhi sirf kuch bulk numbers hi measure karte hain. Yeh poora topic do languages seekhna hai — har ball ki microscopic speed-and-position, aur macroscopic dials T P V U andar stored energy (U wall cross karne wali energy (Q W
Yeh page ek toolbox hai. Parent note v x ⟨ v 2 ⟩ k B Δ U = Q − W
Neechey sab kuch ek image mein rehta hai: ek cube jiska side L
Gas jinse bana hai unme se ek tiny particle . Picture mein yeh ek single ball hai. Hum unme se ∼ 1 0 23 ek ko dekhte hain, phir crowd par average karte hain.
Is cube ko apne dimaag mein rakho. Har symbol jo hum define karte hain woh ya toh ek ball ki property hai, ya crowd ki property hai, ya box ki property hai.
L A V
L side length (ek distance, metres mein measure hoti hai).A = L 2 ek wall ka area (ek flat surface jise ball hit karti hai).V = L 3 volume , andar ki space ki matra.
Picture: L A V
Topic ko yeh kyun chahiye: pressure ek force hai jo wall ke area A volume V
Molecules count karne ke do tarike hain, aur do matching constants hain. Yeh sabko trip karta hai, toh hum dhire chalte hain.
N N kitne balls actually box mein hain woh hai. Real gas ke liye yeh ek bahut bada number hota hai jaise 6 × 1 0 23 individual molecules count karta hai.
n mole sirf ek giant fixed bundle hai — jaise ek "dozen" lekin bundle ka size Avogadro's number N A = 6.022 × 1 0 23 n kitne bundles tumhare paas hain:
N = n N A
k B R temperature ko ek energy mein convert karte hain. Yeh sirf is baat mein differ karte hain ki tum konsi counting use kar rahe ho:
k B 1.38 × 1 0 − 23 J/K per molecule kaam karta hai.R 8.314 J/(mol⋅K) per mole kaam karta hai.
Yeh bundle size se linked hain: R = N A k B
N R n k B Kyun hota hai: yeh interchangeable lagte hain.
Fix: pair match karo. Per-molecule energy mein N aur k B n aur R U = 2 3 N k B T aur U = 2 3 n R T same number hain, kyunki N k B = ( n N A ) ( R / N A ) = n R
Micro-fact N k B
Macro-fact n R
Ab hum ek single molecule ki motion describe karte hain.
r Ek chosen origin se ball jahan hai wahan tak ka arrow . Chhota bar
v Ek arrow jo dikhata hai ball kahan ja rahi hai, jiski length uski speed hai . Lamba arrow = tezi se chal rahi hai.
v x v y v z x y z
Topic ko yeh kyun chahiye: jab ek ball right wall se hit karti hai, sirf uski x v x v
i v x , i i x i 1 , 2 , 3 , … , N
∑ i ∑ har ball ke liye yeh quantity add karo :
∑ i v x , i 2 = v x , 1 2 + v x , 2 2 + ⋯ + v x , N 2
Topic ko yeh kyun chahiye: wall ko sabhi molecules ki combined pounding feel hoti hai, toh pressure i
1 0 23 average .
⟨ ⟩ ⟨ Q ⟩ Q ⟨ v 2 ⟩ = N 1 ∑ i v i 2 ek typical ball se replace karo jo unhe sabko represent kare.
⟨ v x 2 ⟩ = 3 1 ⟨ v 2 ⟩ x y z teen equal shares mein split hoti hai, ek per direction. Yahi woh poora reason hai ki parent ka factor 3 1
Topic ko yeh kyun chahiye: averaging do languages ke beech bridge hai. Microscopic ⟨ v 2 ⟩ temperature ban jaata hai. Averaging literally wahi hai jo "macroscopic" ka matlab hai.
p = m v Mass times velocity — ek measure "is ball ko rokna kitna mushkil hai." Ek heavy ya fast ball ke paas bahut momentum hota hai.
Δ Δ final value minus starting value . Jab ek ball seedha wall se bounce karti hai, uski x + v x − v x Δ p = m ( + v x ) − m ( − v x ) = 2 m v x 2 m v x
Topic ko yeh kyun chahiye: force momentum delivery ki rate hai — yeh parent ke pressure-from-collisions argument ka poora seed hai Step 1 mein.
F push ya pull . Precisely, F = Δ t Δ p
P Force area par spread hoti hai : P = A F
T deep meaning , jo topic derive karta hai, hai ek molecule ki average translational kinetic energy . Zyada hot = faster average bouncing.
Topic ko yeh kyun chahiye: P V T P V = N k B T T Temperature and the Zeroth Law jab T matlab samjhna ho ek molecule count karne se pehle.
Definition Kinetic energy2 1 m v 2 motion ki energy . Speed double karo → energy chaar guna ho jaati hai (note the square).
Definition Internal energyU Saari balls ki kinetic energies ka grand total (plus koi stored potential energy) . Yeh box ke andar rehti hai. Yeh ek state function hai: yeh sirf is par depend karta hai ki abhi cheezein kaise hain , woh journey par nahi jo wahan pohonchi.
Q W Kyun sahi lagta hai: ek hot object "heat se bhara" lagta hai.
Fix: sirf U Q W sirf tab exist karte hain jab energy move kar rahi ho . Yeh woh single distinction hai jo poora topic guard karta hai.
Stored energy symbol U
Transfer-by-temperature symbol Q
Transfer-by-piston symbol W
One ball velocity v and vx
Momentum p and change delta p
Force F equals rate of momentum
Pressure P equals F over A
Ideal gas law P V equals N kB T
First law delta U equals Q minus W
Arrows follow karo: geometry + counting + ek ball ki motion → averages → pressure → ideal gas law → temperature → internal energy → first law. Har node ek symbol hai jo ab tumhara apna hai.
Counting & averaging poori Kinetic Theory of Gases ko power karta hai.
"Energy per direction" Equipartition Theorem ban jaata hai — 2 1 k B T
Kitni directions mein ek molecule energy store kar sakta hai woh uski Degrees of Freedom and Molecular Structure se set hoti hai.
Stored-vs-transferred split Specific Heats Cv and Cp aur Isothermal and Adiabatic Processes ko power karta hai.
V L V = L 3 A = L 2
N n N n N = n N A
Konsa constant N n k B N R n R = N A k B
v x ek molecule ki velocity ka woh part jo x
Speed components se kaise relate karti hai v 2 = v x 2 + v y 2 + v z 2
∑ i har molecule i = 1 … N
⟨ ⟩ average — saare molecules par sum N
⟨ v x 2 ⟩ = 3 1 ⟨ v 2 ⟩ teen directions speed-squared equally share karte hain.
Jab ball wall se bounce karti hai toh momentum change Δ p = 2 m v x
Force in terms of momentum F = Δ p /Δ t
Pressure in terms of force P = F / A
Temperature T ek molecule ki average translational kinetic energy.
Kya stored hai vs kya boundary cross karta hai U Q W
First-law bookkeeping Δ U = Q − W
