Phase rule — Gibbs phase rule — Hinglish notes

2.4.7 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

Phase rule HAI kya?

Key words ka matlab:

Phase: ek physically distinct, mechanically alag kiya ja sakne wala, homogeneous region (ice, liquid water, vapour — ye ek substance ke 3 phases hain).
Component: minimum number of independent chemical species jo har phase ki composition specify karne ke liye chahiye.
Degree of freedom: ek intensive variable (amount nahi) jise aap freely change kar sako.

YEH SACH kyun hai? — Derivation scratch se

Hum bas variables count karte hain, phir equations subtract karte hain. Yahi poora khel hai.

Step 1 — Intensive variables count karo. Har phase ki intensive state $T$ , $P$ , aur uski composition se fix hoti hai. $C$ components ke saath, ek phase mein mole fractions $x_{1}, x_{2}, \dots, x_{C}$ hain, lekin unka sum 1 hota hai: $\sum_{i = 1}^{C} x_{i} = 1$ Yeh step kyun? Mole fractions saare independent nahi hote — ek baaki se determine ho jaata hai. Isliye har phase mein $C - 1$ independent composition variables hote hain.

$P$ phases mein: composition variables $= P (C - 1)$ . Do shared variables $T$ aur $P$ add karo: $Total variables = P (C - 1) + 2$

Step 2 — Equilibrium constraints count karo. Equilibrium mein, har component ka chemical potential har phase mein equal hona chahiye (warna matter tab tak flow karta rahega jab tak ho na jaaye): $μ_{i}^{(α)} = μ_{i}^{(β)} = \dots = μ_{i}^{(P)} for each component i$ Yeh step kyun? Mechanical equilibrium pehle se $P$ ko sab jagah equal karta hai, thermal equilibrium $T$ ko — lekin chemical equilibrium woh nayi constraints add karta hai jinhe humein count karna hai.

EK component ke liye, $P$ phases ek chain of equalities dete hain — yeh $P - 1$ independent equations hain. Saare $C$ components ke liye: $Total constraints = C (P - 1)$

Step 3 — Subtract karo. $F = variables [P (C - 1) + 2] - constraints C (P - 1)$

Expand karo: $F = P C - P + 2 - C P + C = C - P + 2$

KAISE use karein — components count karna aur $+ 2$

$C$ count karna: $C = (chemical species ki sankhya) - (independent chemical reactions) - (additional constraints)$ .

Worked examples

Steel-manned mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek control panel hai jisme temperature, pressure, aur "har ingredient kitni hai" ke dials hain. Jab ice, water, aur steam sab saath khushi se rehte hain, universe bahut zyada choosy ho jaata hai — woh unhe sirf ek special temperature aur pressure par coexist karne deta hai. Toh har dial lock ho jaata hai: zero free dials. Agar sirf water aur steam coexist karein, universe thoda relax karta hai aur temperature choose karne deta hai — lekin phir pressure khud choose kar leta hai. Phase rule bas woh bookkeeping hai jo kehti hai: har extra phase jo saath forced ho = ek aur locked dial.

Active-recall flashcards

F = C - P + 2

mein har symbol ka kya matlab hai?

F

=degrees of freedom (independent intensive variables),

C

=independent components,

P

=coexisting phases,

2

=global variables

T

aur

P

Phase rule mein constant

2

kyun hai?

Yeh do intensive variables count karta hai jo saare phases mein shared hain: temperature aur pressure.

Constraints subtract karne se pehle variables ka count derive karo.

P

phases ×

(C - 1)

composition vars each

+ 2

(

T, P

)

= P (C - 1) + 2

Kitne equilibrium constraints hain, aur kyun?

C (P - 1)

, kyunki har component ka chemical potential saare phases mein equal hona chahiye, jo har component ke liye

P - 1

equations deta hai.

Water ke triple point par

F

kya hai?

F = 1 - 3 + 2 = 0

(ek invariant point).

Ek one-component system mein maximum kitne phases coexist kar sakte hain?

3 (

F = 0

C = 1

set karo ⇒

P = 3

Condensed (reduced) phase rule kya hai aur kab use hota hai?

F = C - P + 1

, jab pressure fixed rakha jaaye (ek global variable hata diya).

CaC O_{3} ⇌ CaO + C O_{2}

ke liye

C

kya hai aur kyun?

C = 2

: 3 species minus 1 independent reaction.

P

–

T

diagram par

F = 2, 1, 0

kis dimension se correspond karta hai?

F = 2

→area,

F = 1

→line,

F = 0

→point.

Kya do immiscible liquids ek phase hain ya do?

Do phases — phases physically distinct homogeneous regions hote hain, sirf states of matter nahi.

Connections

Chemical potential — equality $μ_{i}^{(α)} = μ_{i}^{(β)}$ jo har constraint supply karti hai.
Phase diagrams of pure substances — triple point, sublimation/fusion/vaporisation curves.
Clausius–Clapeyron equation — $F = 1$ coexistence lines ka slope.
Components and independent reactions — $C$ sahi se count karna.
Condensed phase rule — fixed-pressure version $F = C - P + 1$ .
Gibbs free energy and equilibrium — kyun $μ$ equality = minimum $G$ .