Combined gas law and ideal gas equation PV = nRT

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WHY do we even need this?

We already know three separate experimental laws:

Law	Holds what constant	Relation	Meaning
Boyle	$n, T$	$PV = \text{const}$	squeeze → pressure up
Charles	$n, P$	$V/T = \text{const}$	heat → expands
Avogadro	$P, T$	$V/n = \text{const}$	more gas → more volume

Carrying three laws is clumsy. The WHAT we want is one master equation from which all three fall out as special cases. That master equation is $PV = nRT$.

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HOW to derive it from first principles

Why can we multiply proportionalities? If a quantity depends on several variables and we know how it responds to each one separately (others held fixed), the joint dependence is the product. So combine:

$V \propto \frac{nT}{P}$

Introduce a proportionality constant $R$ (the universal gas constant — same for every ideal gas, that's the whole point):

$V = R\,\frac{nT}{P} \quad\Longrightarrow\quad \boxed{PV = nRT}$

Why two values of $R$? Same constant, different units. If $P$ is in atm and $V$ in litres, use $0.0821$; if SI (Pa, m³, J), use $8.314$. Always keep $T$ in kelvin — never Celsius.

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Getting the Combined Gas Law (WHY it's just $PV=nRT$ in disguise)

For a fixed amount of gas, $n$ and $R$ never change. So rearrange:

$\frac{PV}{T} = nR = \text{constant}$

Therefore between any two states of the same sealed sample:

Why is this so useful? You never need to know $n$ or $R$ — they cancel. Perfect for "gas goes from condition 1 to condition 2" problems.

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Useful spin-offs (derive, don't memorise)

Density form. Moles $n = \dfrac{m}{M}$ where $m$ = mass, $M$ = molar mass. Substitute:

$PV = \frac{m}{M}RT \;\Longrightarrow\; PM = \frac{m}{V}RT = \rho RT$

$\boxed{\rho = \frac{PM}{RT}}$

Why this matters: density of a gas rises with pressure and molar mass, falls with temperature — exactly what your intuition expects.

Molar volume at STP. At STP ($T=273.15$ K, $P=1$ atm), for $n=1$: $V = \frac{nRT}{P} = \frac{1 \times 0.0821 \times 273.15}{1} \approx 22.4\ \text{L}$

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Worked Examples

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Common Mistakes

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Active Recall

Recall Quick self-test (hide answers!)

• Derive $PV=nRT$ from Boyle, Charles, Avogadro. → multiply proportionalities $V\propto nT/P$, add constant $R$.
• Why does $n$ vanish in the combined gas law? → it's constant for a sealed sample.
• Value & units of $R$? → $0.0821$ L·atm·K⁻¹·mol⁻¹ or $8.314$ J·K⁻¹·mol⁻¹.
• Derive $\rho = PM/RT$. → sub $n=m/M$ then $\rho=m/V$.

Recall Feynman: explain to a 12-year-old

Imagine a balloon full of bouncing balls. Pressure is how hard the balls hit the walls, volume is how big the balloon is, temperature is how fast the balls move, and moles is how many balls there are. The equation $PV=nRT$ just says: if you add more balls or make them faster, they push harder or need more room. Squish the balloon smaller and each hit lands more often, so pressure climbs. One tidy rule connects all four.

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Connections

• Boyle's Law — the $P$–$V$ special case
• Charles's Law — the $V$–$T$ special case
• Avogadro's Law — the $V$–$n$ special case
• Kinetic Theory of Gases — WHERE $PV=nRT$ comes from microscopically
• Real Gases and van der Waals Equation — what breaks at high P / low T
• Dalton's Law of Partial Pressures — uses $PV=nRT$ per component

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Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, gas ke andar chhote-chhote particles idhar-udhar bounce kar rahe hain. Unke behaviour ko samajhne ke liye chaar cheezein important hain: Pressure ($P$), Volume ($V$), amount yaani moles ($n$), aur Temperature ($T$, hamesha Kelvin mein). Ye chaaron ek single formula se jude hain — $PV = nRT$. Yahi ideal gas equation hai, aur isse hi Boyle, Charles aur Avogadro teenon laws nikal aate hain. Basically ek hi "master rule" hai jismein sab kuch aa jaata hai.

Derivation simple hai: Boyle kehta hai $V \propto 1/P$, Charles kehta hai $V \propto T$, aur Avogadro kehta hai $V \propto n$. Teenon ko multiply karo to $V \propto nT/P$, aur proportionality constant $R$ daal do — bas ban gaya $PV = nRT$. $R$ ek universal constant hai, matlab har ideal gas ke liye same. Units ka dhyan rakho: atm-litre mein $R = 0.0821$, aur SI (Pa, m³) mein $R = 8.314$.

Jab gas ek sealed container mein ho aur uski condition change ho (pehle state 1, phir state 2), tab $n$ aur $R$ constant rehte hain, to woh cancel ho jaate hain aur milta hai combined gas law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. Exam mein "gas ko compress kiya aur heat kiya, naya pressure batao" type questions mein yahi lagana hai — bina $n$ jaane bhi answer aa jaata hai.

Do sabse badi galtiyaan avoid karo: (1) Temperature ko Celsius mein mat daalo, hamesha Kelvin ($°C + 273$). (2) $R$ ki value aur units match karo — agar pressure atm mein hai to $0.0821$, warna galat answer. Bonus: density form $\rho = PM/RT$ yaad rakho, isse molar mass nikalna easy ho jaata hai.