Combined gas law and ideal gas equation PV = nRT

2.4.2 · Chemistry › States of Matter (Quantitative)

Hume iski zaroorat kyun hai?

Humein pehle se teen alag experimental laws pata hain:

Law	Kya constant rakhta hai	Relation	Matlab
Boyle	$n, T$	$P V = const$	dabao → pressure upar
Charles	$n, P$	$V / T = const$	garam karo → failta hai
Avogadro	$P, T$	$V / n = const$	zyada gas → zyada volume

Teen laws yaad rakhna awkward hai. WHAT hum chahte hain woh hai ek master equation jisse teeno special cases ki tarah nikalein. Woh master equation hai $P V = n R T$ .

Ise first principles se derive kaise karein

Hum proportionalities ko multiply kyun kar sakte hain? Agar ek quantity kaafi saari variables par depend karti hai aur hum jaante hain ki woh har ek par alag-alag kaise respond karti hai (baaki fixed hain), toh joint dependence unka product hota hai. Toh combine karo:

$V \propto \frac{n T}{P}$

Ek proportionality constant $R$ introduce karo (universal gas constant — har ideal gas ke liye same, yahi toh poora point hai):

$V = R \frac{n T}{P} ⟹ P V = n R T$

$R$ ki do values kyun? Same constant hai, bas alag units hain. Agar $P$ atm mein hai aur $V$ litres mein, toh $0.0821$ use karo; SI (Pa, m³, J) ke liye $8.314$ use karo. $T$ hamesha kelvin mein rakho — kabhi Celsius mein nahi.

Combined Gas Law nikalna (WHY yeh sirf $P V = n R T$ ka disguise hai)

Gas ki fixed amount ke liye, $n$ aur $R$ kabhi nahi badlate. Toh rearrange karo:

$\frac{P V}{T} = n R = constant$

Isliye usi sealed sample ke kisi bhi do states ke beech:

Yeh itna useful kyun hai? Tumhe $n$ ya $R$ jaanna zaruri nahi — woh cancel ho jaate hain. "Gas condition 1 se condition 2 mein jaati hai" wale problems ke liye perfect hai.

Figure — Combined gas law and ideal gas equation PV = nRT

Useful spin-offs (derive karo, memorise mat karo)

Density form. Moles $n = \frac{m}{M}$ jahan $m$ = mass, $M$ = molar mass. Substitute karo:

$P V = \frac{m}{M} R T ⟹ P M = \frac{m}{V} R T = ρR T$

$ρ = \frac{P M}{R T}$

Yeh kyun matter karta hai: gas ki density pressure aur molar mass ke saath badhti hai, temperature ke saath kam hoti hai — bilkul waisa hi jaisa tumhari intuition expect karti hai.

STP par Molar volume. STP par ( $T = 273.15$ K, $P = 1$ atm), $n = 1$ ke liye: $V = \frac{n R T}{P} = \frac{1 \times 0.0821 \times 273.15}{1} \approx 22.4 L$

Worked Examples

Common Mistakes

Active Recall

Recall Quick self-test (answers chhupao!)

Boyle, Charles, Avogadro se $P V = n R T$ derive karo. → proportionalities $V \propto n T / P$ multiply karo, constant $R$ add karo.
Combined gas law mein $n$ kyun gayab ho jaata hai? → yeh sealed sample ke liye constant hota hai.
$R$ ki value aur units? → $0.0821$ L·atm·K⁻¹·mol⁻¹ ya $8.314$ J·K⁻¹·mol⁻¹.
$ρ = P M / R T$ derive karo. → $n = m / M$ substitute karo phir $ρ = m / V$ .

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek balloon bouncing balls se bhara hai. Pressure woh hai kitni zor se balls walls se takraati hain, volume woh hai balloon kitna bada hai, temperature woh hai balls kitni tez chal rahi hain, aur moles woh hai kitni balls hain. Equation $P V = n R T$ bas yeh kehti hai: agar aur balls daalo ya unhe tez karo, woh zyada zor se push karengi ya unhe zyada jagah chahiye hogi. Balloon ko chhota dabao aur har takkar zyada baar padegi, toh pressure badhta hai. Ek saaf rule charo ko connect karta hai.

Connections

Boyle's Law — $P$ – $V$ special case
Charles's Law — $V$ – $T$ special case
Avogadro's Law — $V$ – $n$ special case
Kinetic Theory of Gases — microscopically $P V = n R T$ KAHAN se aata hai
Real Gases and van der Waals Equation — high P / low T par kya toot jaata hai
Dalton's Law of Partial Pressures — har component ke liye $P V = n R T$ use karta hai

Ideal gas equation

P V = n R T

P V = n R T

mein chaar quantities

Pressure, Volume, moles (n), Temperature (Kelvin mein)

R ki value L·atm units mein

0.0821 L atm K^{- 1} mol^{- 1}

R ki value SI units mein

8.314 J K^{- 1} mol^{- 1}

Combined gas law

\frac{P _{1} V _{1}}{T _{1}} = \frac{P _{2} V _{2}}{T _{2}}

Combined gas law mein n kyun cancel hota hai

n aur R fixed (sealed) sample ke liye constant hain

Ideal gas law ki density form

ρ = \frac{P M}{R T}

STP par ideal gas ka molar volume

\approx 22.4 L

Boyle's law from PV=nRT (fix n,T)

P V =

constant

Charles's law from PV=nRT (fix n,P)

V / T =

constant

Temperature Celsius mein honi chahiye ya Kelvin mein

Kelvin (absolute), hamesha

Celsius ko Kelvin mein convert karo

T (K) = T (° C) + 273.15