Gas laws — Boyle (PV const at T), Charles (V - T const at P), Gay-Lussac (P - T const at V)

2.4.1 · Chemistry › States of Matter (Quantitative)

Gas koi bhi law kyun follow karta hai? (First principles)

KYA hota hai pressure microscopically? Molecules walls se takrate hain aur bounce karte hain. Har takkar wall ko ek choti si push deti hai. Pressure = total push per unit area per unit time.

Pressure depend karta hai:

Kitni baar molecules wall se takrate hain → zyada molecules per volume (density) ya faster molecules = zyada hits.
Kitni zyada takkar → faster molecules zyada zor se takrate hain.

Temperature $T$ (absolute, Kelvin mein) molecules ki average kinetic energy ka measure hai. Zyada garam = zyada fast.

Kinetic theory se (baaki jagah derive kiya gaya hai), $n$ moles ke liye: $P V = n R T$ Neeche har gas law bas is equation ka ek variable frozen karna hai. Yahi poora raaz hai — mechanism yaad karo, teen alag-alag boxes nahi.

Boyle's Law

Kaise derive karein $P V = n R T$ se: $T$ aur $n$ freeze karo. Tab right side $n R T$ ek constant hai. To $P V = const n R T ⟹ P_{1} V_{1} = P_{2} V_{2} .$

Charles's Law

Kaise derive karein: $P V = n R T$ se, $P$ aur $n$ freeze karo. Rearrange karo, $V = \frac{n R}{P} T$ , aur $\frac{n R}{P}$ constant hai, to $V / T =$ const.

Figure — Gas laws — Boyle (PV const at T), Charles (V - T const at P), Gay-Lussac (P - T const at V)

Gay-Lussac's Law (Amontons's Law)

Kaise derive karein: $P V = n R T$ se, $V$ aur $n$ freeze karo: $P = \frac{n R}{V} T$ , coefficient constant hai, to $P / T =$ const.

Forecast-then-Verify

Recall Answer padhne se pehle predict karo

Ek gas $1 atm$ , $273 K$ par hai aur constant $P$ par $546 K$ tak garam ki jaati hai. $V$ ka kya hoga? Forecast: absolute $T$ double karna at constant $P$ (Charles) → $V$ double hoga. Verify: $V_{2} = V_{1} (546/273) = 2 V_{1}$ . ✔ Note: $^{\circ}$ C double karna ( $0 \to$ ?) meaningless hai; sirf Kelvin doubling count karti hai.

Common mistakes (Steel-man + fix)

Combined gas law (the unifier)

Kyunki $\frac{P V}{T} = n R =$ const fixed $n$ ke liye: $\frac{P _{1} V _{1}}{T _{1}} = \frac{P _{2} V _{2}}{T _{2}}$ Koi bhi ek variable dono sides par equal set karo aur yeh Boyle ( $T$ same), Charles ( $P$ same), ya Gay-Lussac ( $V$ same) mein collapse ho jaata hai. 80/20: yeh EK box yaad karo + kaun sa quantity cancel hota hai.

Recall Feynman: ek 12-saal ke bachhe ko explain karo

Socho ek box full of tiny bouncy balls jo walls se takra rahe hain. Yeh takrana hi pressure hai.

Boyle: box chota karo — balls ke paas kam jagah hoti hai, to wo walls se zyada baar takrate hain → zyada pressure. Chota box = badi push.
Charles: balls ko garam karo taaki wo zyada fast zoom karein; pushing-strength same rakhne ke liye, box ko bada hona padega. Garam = bada.
Gay-Lussac: box ko lock karo taaki grow na kar sake, phir balls ko garam karo — wo zyada fast zoom karte hain aur zyada baar takrate hain → walls bahut zyada push feel karti hain. Garam locked can = bada pressure (can phoot sakta hai!).

Connections

Ideal Gas Equation PV=nRT — parent law jisse teeno aate hain.
Kinetic Theory of Gases — pressure & temperature ka microscopic origin.
Absolute Zero and Kelvin Scale — kyun Charles/Gay-Lussac ko Kelvin chahiye.
Combined Gas Law — teeno ko merge karta hai.
Dalton's Law of Partial Pressures — mixtures.
Avogadro's Law — $n$ (amount) knob.

Flashcards

Boyle's law statement (constant T, n)

P \propto 1/ V

, yaani

P V =

constant;

P_{1} V_{1} = P_{2} V_{2}

Charles's law statement (constant P, n)

V \propto T

(Kelvin);

V / T

= const;

V_{1} / T_{1} = V_{2} / T_{2}

Gay-Lussac's law statement (constant V, n)

P \propto T

(Kelvin);

P / T

= const;

P_{1} / T_{1} = P_{2} / T_{2}

Boyle's law mein kaun sa quantity constant hota hai?

Temperature (aur amount n).

Charles's law mein kaun sa quantity constant hota hai?

Pressure (aur amount n).

Gay-Lussac's law mein kaun sa quantity constant hota hai?

Volume (aur amount n).

Charles & Gay-Lussac Kelvin kyun use karte hain, Celsius kyun nahi?

Yeh absolute thermal energy ke direct proportionalities hain; volume/pressure 0 K = −273.15 °C par zero ho jaate hain, to sirf Kelvin valid ratios deta hai.

27 °C ko Kelvin mein convert karo

27 + 273 = 300 K.

PV=nRT se Boyle derive karo

T, n fix karo → nRT constant → PV = constant.

Shape of P vs 1/V graph (Boyle)

Origin se seedhi line, slope = nRT.

Shape of P vs V graph (Boyle)

Hyperbola (isotherm).

Agar absolute T constant P par double ho, V ka kya hoga?

V double hoga (Charles).

Agar T constant V par double ho, P ka kya hoga?

P double hoga (Gay-Lussac).

Keyword "rigid tank" kaun sa law indicate karta hai?

Constant volume → Gay-Lussac.

Combined gas law

P_{1} V_{1} / T_{1} = P_{2} V_{2} / T_{2}

(n constant).

Absolute zero °C mein kya hota hai?

−273.15 °C, woh temperature jahan ideal-gas volume extrapolate karke zero ho jaata hai.