Avogadro's law and Avogadro's number N_A = 6.022 × 10²³

1.1.11 · Chemistry › Matter, Measurement & the Mole

1. Avogadro's Law (the gas law)

YE KYA KEHTA HAI: agar tum gas molecules ki number double karo jabki $T$ aur $P$ same rakho, toh volume bhi double ho jaata hai. Gas volume ko fark nahi padta kaun se molecules hain (H₂, O₂, CO₂) — sirf kitne hain yeh matter karta hai.

YE KYU SACH HAI (ideal gas law se derivation):

Ideal gas equation se start karo (jo khud Boyle + Charles + Gay-Lussac ka summary hai): $P V = n R T$

Ab $V / n$ ke liye solve karo: $\frac{V}{n} = \frac{R T}{P}$

Yeh step kyun? Hum us quantity ko isolate kar rahe hain jiske baare mein Avogadro's law baat karta hai ( $V$ per mole).

Fixed $T$ aur $P$ par, right-hand side $\frac{R T}{P}$ ek constant hai. Isliye $V / n$ constant hai — jo exactly Avogadro's Law hi hai.

2. Avogadro's Number $N_{A}$

YEH EXACT NUMBER KYUN? Historically, $N_{A}$ ko define kiya gaya tha taaki carbon-12 ke 1 mole ka weight exactly 12 grams ho. Isse molar mass g/mol mein numerically equal ho jaati hai atomic/molecular mass u (atomic mass units) ke — yeh ek bahut convenient coincidence hai by design.

$N_{A}$ grams ko atomic mass units se kaise link karta hai: Definition ke according $1 u = \frac{1}{12} \times$ (ek C-12 atom ki mass), aur C-12 ka 1 mol = 12 g. Toh:

\quad\Rightarrow\quad 1\ \text{u} = \frac{1\ \text{g}}{N_A} = \frac{1}{6.022\times10^{23}}\ \text{g} = 1.66\times10^{-24}\ \text{g}$$ - *Yeh step kyun?* $N_A$ atoms mein 12 g divide karne se ek atom ki mass milti hai; uska 1/12 lene se 1 u milta hai. ![[1.1.11-Avogadro's-law-and-Avogadro's-number-N_A-=-6.022-×-1023.png]] --- ## 3. Worked examples > [!example] (a) 18 g water mein kitne molecules hain? > **Given:** $m=18$ g, $M(\text{H}_2\text{O}) = 2(1)+16 = 18$ g/mol. > $$n = \frac{m}{M} = \frac{18}{18} = 1\ \text{mol} > \quad\Rightarrow\quad N = n N_A = 6.022\times10^{23}\ \text{molecules}$$ > *Kyun:* pehle grams → moles ($m/M$), phir moles → count ($\times N_A$). > **Bonus:** hydrogen ke atoms $= 2N$ (har molecule mein 2 H hain) $=1.204\times10^{24}$. > [!example] (b) $\text{CO}_2$ ke 0.5 mol mein atoms count karo > $$N_{\text{molecules}} = 0.5 \times 6.022\times10^{23} = 3.011\times10^{23}$$ > Har CO₂ mein 3 atoms hain (1 C + 2 O), toh total atoms $= 3 \times 3.011\times10^{23} = 9.03\times10^{23}$. > *Kyun:* $N_A$ us *entity* ko count karta hai jo tumne *specify* ki — yahan molecules — phir atoms per molecule se multiply karo. > [!example] (c) Ek single sodium atom ki mass > $M(\text{Na}) = 23$ g/mol. > $$m_{\text{atom}} = \frac{M}{N_A} = \frac{23}{6.022\times10^{23}} = 3.82\times10^{-23}\ \text{g}$$ > *Kyun:* ek mole (23 g) ko $N_A$ atoms mein share karo. > [!example] (d) Gas-law style: volume comparison > 8 g O₂ kisi $T,P$ par 5.0 L occupy karta hai. Same $T,P$ par 8 g He kitna volume occupy karega? > $n(\text{O}_2)=8/32=0.25$ mol; $n(\text{He})=8/4=2.0$ mol. > Avogadro's law ke anusaar $V\propto n$: > $$V_{\text{He}} = 5.0 \times \frac{2.0}{0.25} = 40\ \text{L}$$ > *Kyun:* same $T,P$ ⇒ volume purely mole count ke saath scale karta hai, mass ya identity se nahi. --- ## 4. Common mistakes (steel-manned) > [!mistake] "Gases ke equal *masses* ke equal volumes hote hain." > **Kyun sahi lagta hai:** Avogadro's law kehta hai volume sirf amount par depend karta hai, aur mass 'amount' jaisi lagti hai. > **Fix:** Avogadro's law **molecules (moles) ki number** ke baare mein hai, mass ke baare mein *nahi*. H₂ ke 8 g (4 mol) mein O₂ ke 8 g (0.25 mol) se bahut zyada molecules hain, isliye yeh 16 guna volume occupy karta hai. Pehle *moles* mein convert karo. > [!mistake] "$N_A$ molecules = $N_A$ atoms." > **Kyun sahi lagta hai:** hum laparwaahi se "$N_A$ particles" kehte hain. > **Fix:** $N_A$ us entity ko count karta hai jo tum *specify* karte ho. O₂ ka 1 mol = $N_A$ *molecules* = $2N_A$ *atoms*. Hamesha poochho: "KIS CHEEZ ke moles?" > [!mistake] "Molar volume hamesha 22.4 L hota hai." > **Kyun sahi lagta hai:** yeh ek memorized number hai. > **Fix:** 22.4 L *purane* STP (1 atm, 273 K) ke liye hai. Modern IUPAC STP (1 bar, 273 K) **22.7 L** deta hai. Aur *dono mein se koi bhi* non-STP conditions ya non-ideal gases par apply nahi hota — hamesha $PV=nRT$ par wapas jao. > [!mistake] $M$ ko kg/mol mein use karna $R=8.314$ ke saath aur grams expect karna. > **Kyun sahi lagta hai:** SI ko kg chahiye. > **Fix:** Units consistent rakho. Agar $M$ g/mol mein hai, toh $m$ bhi g mein hona chahiye. $PV=nRT$ ke liye $R=8.314$ J/(mol·K) use karo $P$ Pa mein, $V$ m³ mein. --- ## 5. Flashcards #flashcards/chemistry Avogadro's Law (statement) ::: Gases ke equal volumes mein same T aur P par molecules ki equal numbers hoti hain; yani $V\propto n$ fixed $T,P$ par. Avogadro's number ki value ::: $6.022\times10^{23}$ (per mole). Ideal gas law se $V/n=$ const derive karo ::: $PV=nRT \Rightarrow V/n = RT/P$, fixed $T,P$ par constant. Moles se particles ki number ka formula ::: $N = n\,N_A$. Mass aur count ko link karne wala formula ::: $N = \dfrac{m}{M}\,N_A$. STP par ideal gas ka molar volume (1 bar, 273 K) ::: $22.7$ L (purana STP 1 atm par 22.4 L deta hai). 18 g water mein molecules ::: $6.022\times10^{23}$ (18 g = 1 mol H₂O). 1 atomic mass unit ki mass grams mein ::: $1/N_A = 1.66\times10^{-24}$ g. 1 mol CO₂ mein atoms ::: $3N_A = 1.807\times10^{24}$ (3 atoms per molecule). Molar mass g/mol mein atomic mass u ke equal kyun hoti hai ::: Kyunki $N_A$ ko define kiya gaya tha taaki C-12 ka 1 mol exactly 12 g weighs kare. --- > [!recall]- Feynman: ek 12-saal ke bachche ko explain karo > Socho marbles itni tiny hain ki tum unhe haath se kabhi count nahi kar sakte. Toh hum ek "counting box" invent karte hain jise **mole** kehte hain — har box mein exactly ek hi huge number ke marbles hote hain: lagbhag 602,200,000,000,000,000,000,000 ($6.022\times10^{23}$). Kyunki chemists ne yeh number cleverly choose kiya, carbon marbles ka ek box 12 grams weighs karta hai, water molecules ka ek box 18 grams weighs karta hai, aur aise hi — isliye kuch bhi weighing karna tumhe batata hai kitne boxes (moles) tumhare paas hain. Aur yahan ek cool gas trick hai: agar *kisi bhi* gas marbles ki same number same temperature par ho aur squeeze karo, toh woh same room lete hain — helium ka ek box aur oxygen ka ek box same size mein phufte hain. Yahi Avogadro's law hai. > [!mnemonic] Yaad rakho > **"Same T, same P, same *count*, same size."** — yahi gas law hai. > Number ke liye: **6-0-2-2** ek phone number ki tarah — "$6.022$, times $10$ to the twenty-three." Ya: *"Six oh two two, particles per mole for me and you."* ## Connections - [[The mole concept]] — $N_A$ *hi* mole ki size ki definition hai. - [[Ideal gas law PV=nRT]] — Avogadro's law uska $T,P$-constant slice hai. - [[Molar mass and atomic mass unit]] — kyun g/mol = u numerically. - [[Empirical and molecular formulas]] — mole ratios $N_A$ ke through counting se aate hain. - [[Stoichiometry]] — moles har balanced-equation calculation ki currency hain. - [[Boyle's law]] aur [[Charles's law]] — $PV=nRT$ ke baaki components. ## 🖼️ Concept Map ```mermaid flowchart TD IGL[Ideal gas law PV=nRT] -->|solve for V/n| AL[Avogadro's Law V/n constant] AL -->|same T,P equal molecules| EQ[Equal volumes equal N] AL -->|set n=1 at STP| MV[Molar volume 22.7 L] MOLE[Mole] -->|counts| NA[Avogadro's number 6.022e23] NA -->|defined via C-12 = 12 g| MM[Molar mass M g/mol] NA -->|N = n × N_A| PART[Particle count N] MM -->|n = m/M| NMOL[Moles n] NMOL -->|multiply by N_A| PART MASS[Mass m in grams] -->|divide by M| NMOL AL -.->|both named Avogadro| NA ```

1. Avogadro's Law (the gas law)

2. Avogadro's Number NA​

2. Avogadro's Number $N_{A}$