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

2.3.22 · D2 · Physics › Modern Physics › Decay law — N = N₀ e^(−λt), half-life, activity

Har symbol pehle banaya gaya hai, phir use kiya gaya hai. Aao un sab "characters" ki list dekhe jo hum ek-ek karke milenge:

• $N$ — wo nuclei ki count jo abhi tak decay nahi hue, bilkul abhi.
• $t$ — bita hua time, us waqt se measure kiya jab humne dekhna shuru kiya.
• $\lambda$ (Greek letter "lambda") — decay constant: kisi bhi ek nucleus ke decay hone ki chance, per second.
• $dN$, $dt$ — $N$ mein ek choti si change aur time ka ek chota sa slice. Chhota "$d$" matlab "ek sliver of."

Related maths Exponential functions and natural log mein hai; parent the decay-law topic note hai.

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Step 1 — Pile ko Picture karo

KYA HAI. Undecayed nuclei ka ek dher se shuru karo. Count ko $N$ kaho. Bilkul shuruaat mein ($t=0$) is starting count ko hum $N_0$ naam dete hain — chhota "0" bas itna kehta hai "time zero pe value."

KYUN. Kisi bhi equation se pehle, hum agree karte hain ki hum kya count kar rahe hain. Aage jo bhi hoga wo is pile ke simatne ke baare mein statement hoga.

PICTURE. Figure dekho: har dot ek undecayed nucleus hai. Bahut saare hain — yahi "bahut saare" randomness ko smooth banata hai.

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Step 2 — Ek Nucleus, Ek Coin-flip

KYA HAI. Ek single nucleus pe zoom in karo. Ek chote time slice $dt$ mein, ya to wo decay hota hai ya nahi. Uske decay hone ki chance $\lambda dt$ hai.

YE FORM KYUN. Probability $\lambda \times dt$ kyun likhi, sirf $\lambda$ kyun nahi? Kyunki chance ek pure number hona chahiye 0 aur 1 ke beech, lekin $\lambda$ ki units "per second" hain. Time slice $dt$ (seconds mein) se multiply karne par "per second" cancel ho jaata hai aur ek bara probability milta hai. Lamba slice → decay ki zyada chance. $\lambda$ ka poora matlab yahi hai: chance per second.

PICTURE. Figure mein ek nucleus dikha hai aur ek choti ghadi $dt$ se guzar rahi hai. Usse do arrows nikalte hain: ek plum arrow "stays" (probability $1-\lambda dt$) aur ek burnt-orange arrow "decays" (probability $\lambda dt$).

$\underset{\text{a pure number}}{\underbrace{P_{\text{decay}}}}=\underset{\text{chance per second}}{\underbrace{\lambda }}\cdot \underset{\text{tiny seconds}}{\underbrace{dt}}$

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Step 3 — Ek Nucleus se Poori Pile tak

KYA HAI. Ab slice $dt$ mein poori pile ke decays count karo. Agar $N$ nuclei mein se har ek independently $\lambda dt$ chance rakhta hai jaane ki, to jo expected number decay karte hain wo hai $N\times \lambda dt$.

KYUN. "Expected number" bas itna kehta hai: chance-per-item times number-of-items. Das dice, har ek mein $1/6$ chance ek six ki → expect karo $10\times 1/6$ sixes. Same logic, koi nayi idea nahi.

PICTURE. Figure mein burnt orange mein wo kuch nuclei highlight hain jo is ek slice mein pop hue — unki count is baat ke proportional hai ki pile abhi kitni bhari hai.

$\underset{\text{orange dots}}{\underbrace{(\text{decays in }dt)}}=\underset{\text{how many are left}}{\underbrace{N}}\cdot \underset{\text{each one’s chance}}{\underbrace{\lambda dt}}$

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Step 4 — Rate of Change Likho

KYA HAI. Har decay pile se ek nucleus hatata hai, isliye $dt$ ke dauran $N$ mein change decay hone walon ki count ka negative hai: $dN=-\lambda Ndt.$

MINUS SIGN KYUN. $dN$ hai "$N$ mein change." Pile sirf members khoti hai, kabhi gain nahi karti, isliye $dN$ negative aana chahiye. Minus sign decoration nahi hai — yeh physics ka kehna hai "simat raha hai."

CALCULUS YAHAN KYUN CHAHIYE. Hum pile ko fixed jumps mein track karne ki koshish kar sakte the, lekin decay continuously hoti hai — nuclei har instant mein jaate hain, kisi metronome pe nahi. Tools $dN$ aur $dt$ ("slivers") hamare liye instantaneous rate ke baare mein baat karna possible banate hain. Yeh woh waqt hai jab calculus apni jagah banata hai: yeh "rate jo chalte chalte change hoti hai" ke liye single language hai.

PICTURE. Figure mein $N$ ek slice $dt$ mein thoda girta dikha hai; woh chota step-down $dN$ hai, aur iska steepness is baat se set hota hai ki pile abhi kitni oonchi hai.

$\underset{\text{tiny change in }N}{\underbrace{dN}}=\underset{\text{it falls}}{\underbrace{-}}\underset{\text{current decay rate}}{\underbrace{\lambda N}}\underset{\text{tiny time}}{\underbrace{dt}}$

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Step 5 — Do Variables Alag Karo

KYA HAI. Hum rearrange karte hain taaki $N$ ke baare mein sab kuch left side pe ho aur $t$ ke baare mein sab kuch right side pe: $\frac{dN}{N}=-\lambda dt.$

KYUN. Hamare paas do changing quantities hain, $N$ aur $t$, ek saath uljhi hui. Unke slivers add karne (integrate karne) ke liye hume unhe suljhana hoga — ek variable per side. Is trick ko separation of variables kehte hain, aur yeh theek isliye kaam karti hai kyunki hamari equation ek clean product $N\cdot dt$ hai.

PICTURE. Figure mein ek simple balance hai: left pan mein, fractional shrink $dN/N$; right pan mein, fixed drip $-\lambda dt$. Equal pans ka matlab: har equal fractional loss equal time leta hai. Yeh ek sentence poore exponential ka seed hai.

$\underset{\text{fractional loss}}{\underbrace{\frac{dN}{N}}}=\underset{\text{a fixed drip}}{\underbrace{-\lambda dt}}$

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Step 6 — Saare Slivers Add Karo (Integrate)

KYA HAI. Hum shuruaat ($t=0$, $N=N_0$) se lekar abhi tak ($t$, $N$) ke har ek tiny sliver ko sum karte hain: ${\int }_{N_0}^N\frac{dN}{N}=-\lambda {\int }_0^{t}dt.$ Left sum natural logarithm hai; right sum sirf $\lambda t$ hai: $\ln N-\ln N_0=-\lambda t⟹\ln \frac{N}{N_0}=-\lambda t.$

LOGARITHM KYUN AATA HAI. $\frac{1}{N}$ ke slivers add karna ek special sum hai jiska answer $\ln N$ hai. Yahi natural log ki definition hai — yeh $1/N$ ka running total hai. Isliye log hat se nahi nikla; yeh exactly woh quantity hai jo "fractional loss" accumulate karti hai. Dekho Exponential functions and natural log.

PICTURE. Figure mein $1/N$ curve ke neeche $N_0$ aur $N$ ke beech ka area shade kiya gaya hai — woh shaded area hi $\ln (N/N_0)$ hai, aur right side pe height $\lambda$ times width $t$ ke barabar hai.

$\underset{\text{accumulated fractional loss}}{\underbrace{\ln \frac{N}{N_0}}}=\underset{\text{steady drip}\times \text{time}}{\underbrace{-\lambda t}}$

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Step 7 — Logarithm ko Undo Karo

KYA HAI. $N$ ko log ke andar se nikalne ke liye, log ka opposite operation lagao: exponential $e^{(\cdot )}$. Kyunki $e^{\ln x}=x$: $\frac{N}{N_0}=e^{-\lambda t}⟹\boxed{N=N_0{e}^{-\lambda t}}.$

$e$ KYUN, KOI AUR BASE KYUN NAHI. Step 6 mein jo log aaya woh natural log tha (base $e$), kyunki yahi woh hai jiska running total exactly $1/N$ hai. Iska perfect inverse $e^{(\cdot )}$ hai. Koi aur base use karne se ek extra clutter factor aa jaata. Natural base equation ko clean rakhta hai.

PICTURE. Figure mein finished curve dikhi hai: $N_0$ se shuru hoti hai, neeche curve hoti hai, zero ki taraf flatten hoti hai — lekin kabhi chhuti nahi.

$\underset{\text{left now}}{\underbrace{N}}=\underset{\text{started with}}{\underbrace{N_0}}\underset{\text{the shrinking factor}}{\underbrace{e^{-\lambda t}}}$

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Step 8 — Curve pe Half-life Kahan Rehti Hai

KYA HAI. Pucho: pile apne start ki half hone mein kitna time lagta hai? $N=N_0/2$ set karo: $\frac{1}{2}=e^{-\lambda T_{1/2}}\Rightarrow -\ln 2=-\lambda T_{1/2}\Rightarrow \boxed{T_{1/2}=\frac{\ln 2}{\lambda }=\frac{0.693}{\lambda }}.$

$\ln 2$ KYUN AATA HAI. "Halving" ka matlab hai shrinking factor $e^{-\lambda t}$ $\frac{1}{2}$ tak pahunche. Exponential ko log se undo karne par "one half" $\ln 2\approx 0.693$ ban jaata hai. Toh $0.693$ koi mysterious cheez nahi — yeh bas $\ln 2$ hai, "one half" ka log.

PICTURE. Figure mein equal ledges mark hain: curve $N_0$ se $N_0/2$ tak same horizontal width $T_{1/2}$ mein girta hai jo baad mein $N_0/2$ se $N_0/4$ tak girne mein lagta hai, aur phir $N_0/8$ tak. Equal widths, halving heights — decay ki staircase.

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Step 9 — Edge Cases (Inhe Kabhi Skip Mat Karo)

KYA HAI & PICTURE. Figure teen regimes overlay karta hai taaki tumhe koi unseen scenario kabhi na mile:

• $t=0$ (shuruaat). $e^{-\lambda \cdot 0}=e^0=1$, isliye $N=N_0$. Curve bilkul top se shuru hoti hai. Sanity check: abhi kuch decay nahi hua.
• $t\to \infty$ (door ka future). $e^{-\lambda t}$ 0 ki taraf shrink karta hai lekin hamesha positive rehta hai, isliye $N$ approach karta hai — lekin kabhi zero nahi pohuncha. Curve asymptotic hai: mathematically pile kabhi poori tarah empty nahi hoti. (Reality mein aakhri kuch nuclei jaate hain, kyunki $N$ ek whole number hai, lekin smooth law kehta hai "vanishingly small.")
• $\lambda =0$ (ek stable nucleus). Zero chance per second → $e^0=1$ sabhi $t$ ke liye → $N=N_0$ hamesha. Ek flat line: kuch kabhi decay nahi hota. Yeh degenerate case hai, aur formula isko automatically handle karta hai.
• Large vs small $\lambda$. Bada $\lambda$ = steeper drop = choti half-life; chota $\lambda$ = gentle slope = lambi half-life. Same shape, alag stretch.

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Ek Picture mein Poora Summary

Yeh final figure saare nau steps ko ek single storyboard mein compress karta hai: pile (Step 1) → ek nucleus apna coin flip karta hua (Step 2) → rate law $dN=-\lambda Ndt$ (Steps 3–4) → separate karo aur $\ln$ mein integrate karo (Steps 5–6) → $N=N_0{e}^{-\lambda t}$ paane ke liye exponentiate karo (Step 7) → curve se half-life ledges padho (Step 8), edge cases apni jagah pin kiye hue (Step 9).

Recall Feynman: poori walk ko seedhe shabdon mein batao

Socho jumping beans ka ek bada jar, aur har bean ko har second jar se bahar koodne ki same choti si chance hai — woh chance hai $\lambda$. Kyunki unki chance same hai, abhi right now jo bahar koodte hain unki count bas is baat pe depend karti hai ki andar kitne abhi bhi hain: bhari hui jar tez beans khoti hai. Ise likho "tiny change equals minus lambda times kitne bache hain, times ek tiny time." Itne lambe stretch mein jar kaise khaali hoti hai ye jaanne ke liye, hum saari tiny changes add karte hain — woh add-karna (calculus) naturally natural logarithm produce karta hai, aur log undo karne se $e^{-\lambda t}$ milta hai. Jar ki count bhari shuru hoti hai ($N_0$), neeche curve hoti hai, aur empty hone ki taraf slide karti hai kabhi quite bottom touch kiye bina. Half beans khoye bina time hamesha same length ka hota hai, kyunki half khona half khona hai chahe tum kahan se shuru karo — aur woh repeated equal step hi half-life hai, $T_{1/2}=0.693/\lambda$, jahan $0.693$ simply one-half ka log hai.

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Connections

• Parent topic
• Exponential functions and natural log (kyun $\ln$ aur $e$ aate hain)
• Radioactivity — alpha, beta, gamma decay
• First-order chemical kinetics (same mathematics)
• Carbon dating (yahi derivation, apply ki hui)