Foundations — Decay law — N = N₀ e^(−λt), half-life, activity

2.3.22 · D1 · Physics › Modern Physics › Decay law — N = N₀ e^(−λt), half-life, activity

Yeh page har letter, symbol, aur idea ko build karta hai jis par parent note tika hua hai — ek smart 12-saal ke bache se shuru karte hain jisne pehle kabhi kuch nahi dekha. Upar se neeche padho; har block agले ke liye ek brick hai.

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1. Atoms ko count karna: $N$ aur $N_0$

Topic ko dono ki zaroorat kyun hai? Kyunki poori story ek change ke baare mein hai: hum "abhi" ($N$) ko "shuruat" ($N_0$) se compare karte hain. Bina ek fixed starting mark ke tum yeh nahi keh sakte ki kitna shrink hua.

Figure 1. Do jars. Left mein, $t=0$ par, har dot present hai — poora count $N_0$ hai. Right mein, time mein baad mein, orange crosses pehle hi decay ho chuke hain aur chale gaye hain; sirf blue dots bache hain, aur unka count $N$ hai. Dekho ki right mein blue count chhota hai — yahi shrinkage poora topic hai.

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2. Time $t$ aur tiny slice $dt$

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3. Decay constant $\lambda$

Greek letter $\lambda$ (kaho "lambda") ek ulta y jaisa dikhta hai. Yeh ek number ko represent karta hai.

Kya picture? Socho har atom har second ek weighted die roll karta hai. $\lambda$ woh hai jo die ko "abhi toot jao" ki taraf load karta hai. Important baat, die pichle rolls yaad nahi rakhti — chance pehle roll aur ek million roll par same hi hota hai. Atoms ki umar nahi hoti ya ghiste nahi hain.

Topic ko $\lambda$ ki zaroorat kyun hai: yeh ek akela knob hai jo sab kuch set karta hai — decay ki speed, half-life, aur activity sab seedhe $\lambda$ se aate hain.

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4. Proportionality aur master equation $dN=-\lambda Ndt$

Step A — slice mein decays count karo. Ek proportion ko equation mein badalne ke liye sirf ek multiplier chahiye — aur woh multiplier exactly $\lambda$ hai:

$\text{decays in }dt=\lambda Ndt$

Step B — har decay ek atom hata deta hai. Count mein change, $dN$, minus us number ke barabar hai jo decay huye (minus, kyunki woh chale jaate hain):

$dN=-(\text{decays in }dt)=-\lambda Ndt$

Yeh poore chapter ki master equation hai. Dono sides ko $dt$ se divide karo toh ise rate ke roop mein padh sako:

Figure 2. Do jars ko same tiny slice $dt$ mein dekha gaya. Fuller jar (bada $N$) half-full jar se zyada atoms (red X's) khota hai, aur woh unhe same proportion mein khota hai. Yeh $dN=-\lambda Ndt$ ko visible banata hai: same $\lambda$, same $dt$, lekin jahan $N$ zyada hai wahan zyada decays.

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5. Fractions aur ratios: $N/N_0$

Topic ko iske zaroorat kyun hai: decay curve absolute atoms ki sankhya ki parwah nahi karta, sirf fraction ki.