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

Why do gases obey any law at all? (First principles)

WHAT is pressure microscopically? Molecules hit the walls and bounce. Each hit gives the wall a tiny push. Pressure = total push per unit area per unit time.

Pressure depends on:

How often molecules hit the wall → more molecules per volume (density) or faster molecules = more hits.
How hard each hit is → faster molecules hit harder.

Temperature $T$ (absolute, in Kelvin) is a measure of average kinetic energy of molecules. Hotter = faster.

From kinetic theory (derived elsewhere), for $n$ moles: $PV = nRT$ Every gas law below is just this equation with one variable frozen. That's the whole secret — memorise the mechanism, not three separate boxes.

Boyle's Law

HOW to derive it from $PV = nRT$ : freeze $T$ and $n$ . Then the right side $nRT$ is a constant. So $PV = \underbrace{nRT}_{\text{const}} \implies P_1V_1 = P_2V_2.$

Charles's Law

HOW to derive: from $PV = nRT$ , freeze $P$ and $n$ . Rearranging, $V = \dfrac{nR}{P}\,T$ , and $\dfrac{nR}{P}$ is constant, so $V/T =$ 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)

HOW to derive: from $PV = nRT$ , freeze $V$ and $n$ : $P = \dfrac{nR}{V}\,T$ , coefficient constant, so $P/T =$ const.

Forecast-then-Verify

Recall Predict before reading the answer

A gas at $1\ \text{atm}$ , $273\ \text{K}$ is heated to $546\ \text{K}$ at constant $P$ . What happens to $V$ ? Forecast: doubling absolute $T$ at constant $P$ (Charles) → $V$ doubles. Verify: $V_2 = V_1(546/273) = 2V_1$ . ✔ Note: doubling $^\circ$ C ( $0\to$ ? ) is meaningless; only Kelvin doubling counts.

Common mistakes (Steel-man + fix)

Combined gas law (the unifier)

Since $\dfrac{PV}{T} = nR =$ const for fixed $n$ : $\boxed{\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}}$ Set any one variable equal on both sides and it collapses back to Boyle ( $T$ same), Charles ( $P$ same), or Gay-Lussac ( $V$ same). 80/20: learn this ONE box + which quantity cancels.

Recall Feynman: explain to a 12-year-old

Imagine a box full of tiny bouncy balls banging on the walls. The banging is the pressure.

Boyle: make the box smaller — the balls have less room, so they bang the walls more often → more pressure. Small box = big push.
Charles: heat the balls so they zoom faster; to keep the pushing-strength the same, the box has to grow bigger. Hot = big.
Gay-Lussac: lock the box so it can't grow, then heat the balls — they zoom faster and bang harder → the walls feel much more push. Hot locked can = big pressure (can go boom!).

Connections

Ideal Gas Equation PV=nRT — the parent law all three come from.
Kinetic Theory of Gases — microscopic origin of pressure & temperature.
Absolute Zero and Kelvin Scale — why Charles/Gay-Lussac need Kelvin.
Combined Gas Law — merges all three.
Dalton's Law of Partial Pressures — mixtures.
Avogadro's Law — the $n$ (amount) knob.

Flashcards

Boyle's law statement (constant T, n)

P \propto 1/V

, i.e.

PV =

constant;

P_1V_1 = P_2V_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

Which quantity is held constant in Boyle's law?

Temperature (and amount n).

Which quantity is held constant in Charles's law?

Pressure (and amount n).

Which quantity is held constant in Gay-Lussac's law?

Volume (and amount n).

Why must Charles & Gay-Lussac use Kelvin, not Celsius?

They are direct proportionalities to absolute thermal energy; volume/pressure hit zero at 0 K = −273.15 °C, so only Kelvin gives valid ratios.

Convert 27 °C to Kelvin

27 + 273 = 300 K.

Derive Boyle from PV=nRT

Fix T, n → nRT constant → PV = constant.

Shape of P vs 1/V graph (Boyle)

Straight line through origin, slope = nRT.

Shape of P vs V graph (Boyle)

Hyperbola (isotherm).

If absolute T doubles at constant P, what happens to V?

V doubles (Charles).

If T doubles at constant V, what happens to P?

P doubles (Gay-Lussac).

Keyword "rigid tank" signals which law?

Constant volume → Gay-Lussac.

The combined gas law

P_1V_1/T_1 = P_2V_2/T_2

(n constant).

What is absolute zero in °C?

−273.15 °C, the temperature where ideal-gas volume extrapolates to zero.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, teeno gas laws basically ek hi cheez hain — $PV = nRT$ ka ek-ek version. Gas ke paas teen "knobs" hain: pressure $P$ , volume $V$ , aur temperature $T$ . Har law bolta hai: ek knob ko fix kar do, aur baaki do kaise trade-off karte hain woh dekho. Boyle mein temperature constant — gas ko dabao (V kam), toh molecules zyada baar wall pe takraate hain, pressure badh jaata hai. Isliye $PV$ = constant, inverse relation.

Charles mein pressure constant rehta hai (jaise balloon ya free piston). Agar tum gas ko garam karoge, molecules tez ho jaate hain, aur same push maintain karne ke liye volume badhna padta hai. Isliye $V \propto T$ . Gay-Lussac mein volume constant (rigid ya sealed can) — garam karo toh molecules zyada zor se takraate hain lekin phail nahi sakte, toh pressure badh jaata hai. Isiliye deo-spray can aag mein phat jaata hai!

Sabse important baat, aur yahin students marks kho dete hain: Charles aur Gay-Lussac ke liye Kelvin use karo, Celsius nahi. Kyunki yeh laws absolute energy ke proportional hain, aur $0^\circ$ C pe molecules ruk nahi jaate — asli zero (absolute zero) $-273.15^\circ$ C pe hai. Formula: $T(K) = t(^\circ C) + 273$ . Boyle mein $T$ cancel ho jaata hai isliye conversion ki zaroorat nahi, bas $T$ constant rakhna hai.

Ek jugaad: agar question mein "rigid/sealed" likha ho toh Gay-Lussac, "balloon/constant pressure" ho toh Charles, "isothermal/same temperature" ho toh Boyle. Aur agar teeno mein se koi bhi constant nahi, toh combined law $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ laga do — yeh sabka baap hai.

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

Why do gases obey any law at all? (First principles)

Boyle's Law

Charles's Law

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

Forecast-then-Verify

Common mistakes (Steel-man + fix)

Combined gas law (the unifier)

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — States of Matter (Quantitative)

Connections