Heat and internal energy — microscopic vs macroscopic

1.7.3 · Physics › Thermodynamics

YEH QUANTITIES HAIN KYA?

DO DESCRIPTIONS KYUN? (Microscopic vs Macroscopic)

Intuition Usi reality ke do zoom levels

	Microscopic	Macroscopic
Variables	har molecule ka $v_{i}$ , $r_{i}$	$T$ , $P$ , $V$ , $U$
Kitne track kiye	$\sim 1 0^{23}$	mutthi bhar
Tool	statistics / averages	thermodynamic laws
"Temperature" hai...	average molecular KE	ek thermometer reading

Macroscopic quantities microscopic chaos ki averages hain. Aap detail chhod ke tractability lete ho.

Figure — Heat and internal energy — microscopic vs macroscopic

KAISE: First principles se ideal gas ka $U$ derive karna

Hum microscopic motion ko macroscopic $U$ se connect karna chahte hain.

Step 1 — Molecular collisions se pressure. $L$ side wale ek cube (volume $V = L^{3}$ ) mein $m$ mass ke $N$ molecules lo. Ek molecule lo jiska $x$ -velocity $v_{x}$ hai. Jab woh right wall se elastically bounce karta hai, uska momentum $Δ p = 2 m v_{x}$ se change hota hai.

Yeh step kyun? Pressure = force/area hai, aur force = wall par momentum delivery ki rate hai — isliye hum har bounce mein momentum transfer se shuru karte hain.

Step 2 — Usi wall par dobara hit karne ka time. Woh usi wall par successive hits ke beech $2 L$ travel karta hai, isliye hits ke beech ka time $= \frac{2 L}{v _{x}}$ hai.

Step 3 — Ek molecule se force. $F_{1} = \frac{Δ p}{Δ t} = \frac{2 m v _{x}}{2 L / v _{x}} = \frac{m v _{x}^{2}}{L}$

Yeh step kyun? Average force = deliver kiya gaya total momentum ÷ time.

Step 4 — Saare molecules par sum karo, averages use karo. $P = \frac{F _{total}}{L ^{2}} = \frac{m}{L ^{3}} \sum_{i} v_{x, i}^{2} = \frac{N m}{V} ⟨ v_{x}^{2} ⟩$ Symmetry se $⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩$ , isliye $P V = \frac{1}{3} N m ⟨ v^{2} ⟩$

Step 5 — Ideal gas law $P V = N k_{B} T$ se compare karo. $\frac{1}{3} N m ⟨ v^{2} ⟩ = N k_{B} T \Rightarrow \frac{1}{2} m ⟨ v^{2} ⟩ = \frac{3}{2} k_{B} T$

Yeh step kyun? Yahi bridge hai: yeh temperature ko average translational KE ki tarah define karta hai.

Step 6 — Total internal energy. Monatomic ideal gas ke liye (sirf translation, koi intermolecular PE nahin): $U = N (\frac{1}{2} m ⟨ v^{2} ⟩) = \frac{3}{2} N k_{B} T = \frac{3}{2} n R T$

Link: First Law of Thermodynamics

Flashcards

Internal energy

U

(process / state) function hai?

State function — sirf current state par depend karta hai, path par nahin.

Heat woh energy hai jo transfer hoti hai ek

boundary ke paar temperature difference ki wajah se.

"Body mein heat stored hai" galat kyun hai?

Bodies internal energy store karti hain; heat sirf

Δ T

ki wajah se transfer ke dauran exist karta hai.

Molecule ki average translational KE barabar hai

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

ke.

Har degree of freedom mein energy (equipartition)

\frac{1}{2} k_{B} T

Monatomic ideal gas ka

U

U = \frac{3}{2} n R T

(sirf

T

par depend karta hai).

Ideal-gas

U

sirf

T

par kyun depend karta hai?

Koi intermolecular forces nahin ⇒ koi potential-energy term nahin ⇒ sirf KE, jo

T

se set hoti hai.

First law of thermodynamics

Δ U = Q - W

(

W

= gas ke dwara kiya gaya work).

Isothermal ideal-gas expansion mein,

Δ U = ?

aur

Q = ?

Δ U = 0

;

Q = W

Kinetic theory se pressure

P V = \frac{1}{3} N m ⟨ v^{2} ⟩

Micro KE ko macro

T

se kya jodta hai?

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

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek box mein choti-choti uchhalti huyi balls (molecules) bhari hain. Internal energy hai total bounciness — woh sab kitni tezi se aur kitna move kar rahe hain. Temperature sirf ek ball ki average speed hai. Heat woh hota hai jab aap apna garam box ek thande box ke paas rakhte ho: tezi se chalti balls dheemi walon se takraati hain jab tak dono boxes equally bouncy na ho jaayein — energy flow karti hai. Woh flow karne wali energy heat hai; andar ki bounciness internal energy hai. Aap kabhi nahin kahoge "box flowing se bhara hua hai" — flowing kuch aisa hai jo hota hai, koi cheez nahin jo store ho. Heat ke saath bilkul waise hi!

Connections

First Law of Thermodynamics
Kinetic Theory of Gases
Equipartition Theorem
Specific Heats Cv and Cp
Isothermal and Adiabatic Processes
Temperature and the Zeroth Law
Degrees of Freedom and Molecular Structure