Real gases — deviations from ideality, compressibility factor Z

2.4.7 · Chemistry › States of Matter (Quantitative)

Compressibility factor HAI kya?

Aise kyun define karte hain? Kyunki $Z$ ek ratio hai real duniya ka ideal duniya se. Jo measure kiya ( $P V_{m}$ ) usse divide karo ideal law ke prediction ( $R T$ ) se. Agar match ho, $Z = 1$ . 1 se koi bhi departure non-ideal behaviour ki seedhi pehchaan hai.

Real gases deviate kyun karte hain? (Do competing effects)

Van der Waals equation scratch se HOW derive karein

Hum ideal law ke do jhooton ko ek ek karke theek karte hain.

Step 1 — Volume ke jhoote ko theek karo. Ideal: $V$ free space hai. Real: molecules ka har mole ek excluded volume $b$ leta hai. Toh available volume $(V_{m} - b)$ ho jaati hai. Kyun? Do hard spheres overlap nahi kar sakti, isliye kuch space permanently unavailable hai.

Step 2 — Pressure ke jhoote ko theek karo. Wall se takrane wala ek molecule apne neighbours se peeche kheencha jaata hai, isliye real pressure ideal se kam hoti hai. Correction proportional hai (pullers ki density) × (pulled ki density) $\propto (n / V)^{2}$ . Per mole: correction $= a / V_{m}^{2}$ . Squared kyun? Ek factor density ka wall se takrane wale molecule ke liye, ek factor un molecules ke liye jo pull kar rahe hain — ek two-body interaction hai.

Toh bina attraction ke jo "true" internal pressure exist karti, wo hai $P + \frac{a}{V _{m}^{2}}$ .

Figure — Real gases — deviations from ideality, compressibility factor Z

$Z$ van der Waals se HOW aata hai

vdW ko $P$ ke liye solve karo aur $Z = P V_{m} / R T$ mein daalo: $P = \frac{R T}{V _{m} - b} - \frac{a}{V _{m}^{2}}$ $Z = \frac{P V _{m}}{R T} = \frac{V _{m}}{V _{m} - b} - \frac{a}{R T V _{m}}$

Low-pressure limit ( $V_{m}$ bada, isliye $b / V_{m} ≪ 1$ ): $\frac{V _{m}}{V _{m} - b} = \frac{1}{1 - b / V _{m}} \approx 1 + \frac{b}{V _{m}}$ expand karo: $Z \approx 1 + \frac{1}{V _{m}} (b - \frac{a}{R T})$

Ye tumhe $Z$ vs $P$ ka initial slope batata hai (kyunki low P par $1/ V_{m} \propto P$ ):

Agar $\frac{a}{R T} > b$ (attraction jeetta hai) → $Z$ 1 se neeche shuru hota hai.
Agar $b > \frac{a}{R T}$ (size jeetti hai) → $Z$ 1 se upar shuru hota hai (jaise $H_{2}$ , $He$ room T par).

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Compressibility factor

Z

define karo.

Z = P V_{m} / R T = P V / n R T

; ideal gas ke liye 1 ke barabar hota hai.

Z > 1

ka physically kya matlab hai?

Gas ideal se less compressible hai; molecular size/repulsion dominate karta hai (high P).

Z < 1

ka physically kya matlab hai?

Gas ideal se more compressible hai; intermolecular attraction dominate karti hai (low P).

n

moles ke liye van der Waals equation likho.

(P + a n^{2} / V^{2}) (V - nb) = n R T

Constant

a

kya represent karta hai?

Intermolecular attractive forces ki strength.

Constant

b

kya represent karta hai?

Molecules ka excluded (finite) volume per mole.

vdW se

Z

aur uske do competing terms do.

Z = \frac{V _{m}}{V _{m} - b} - \frac{a}{R T V _{m}}

; pehla term (size, Z badhata hai), doosra (attraction, Z ghatata hai).

Boyle temperature define karo aur uska formula do.

Wo temperature jahan Z ka low-P slope zero hai;

T_{B} = a / (R b)

H_{2}

aur

He

ka room temperature par

Z > 1

kyun hota hai?

Bahut chhota

a

, isliye

b > a / R T

; size effect shuroo se dominate karta hai.

Bahut high pressure par har real gas ke

Z

ka kya hota hai?

Ye 1 se upar uth jaata hai (finite molecular volume dominate karti hai).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek kamre ki kalpana karo logo se bhara hua. "Ideal gas" maan leta hai ki log invisible dots hain jo kabhi nahi touch karte aur kabhi haath nahi pakadte. Lekin real log space lete hain (tum utne log fit nahi kar sakte jitne dots), aur kabhi kabhi ye haath pakad lete hain (attract karte hain). $Z$ ek score hai: agar ye 1 hai, pretend ke rules kaam karte hain. Agar $Z$ 1 se kam hai, haath pakadna jeet raha hai — sab log ek saath ghoom jaate hain aur expected se kam jagah lete hain. Agar $Z$ 1 se zyada hai, log itne packed hain ki unke bodies jagah le leti hain aur ye aur press nahi ho sakta. Itna press karo (high pressure) aur "bodies space leti hain" effect hamesha jeet jaati hai.

Connections

Ideal Gas Law — $Z = 1$ baseline jise ye note correct karta hai.
Intermolecular Forces — constant $a$ aur $Z < 1$ dip ki origin.
Critical Constants and Liquefaction — $a, b$ ka $T_{c}, P_{c}, V_{c}$ se relation.
Boyle Temperature — jahan low-P behaviour ideal ho jaati hai.
Kinetic Theory of Gases — point-particle assumptions jo yahan relax ho rahi hain.