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

2.3.22 · Physics › Modern Physics

Decay law KYUN exist karta hai?

$N$ ke proportional KYUN? Agar har nucleus independently probability $λ d t$ se ek chote time $d t$ mein decay karta hai, toh $N$ nuclei ki population mein, $d t$ mein decay karne wale nuclei ki expected sankhya sirf $N \times λ d t$ hoti hai. Zyada nuclei → zyada decays. Bas itna hi hai.

$N = N_{0} e^{- λ t}$ KAISE derive karen (scratch se)

Step 1 — Rate of change likhein. $d t$ mein jo nuclei decay karte hain unki sankhya $λ N d t$ hai, aur har decay ek nucleus hatati hai, isliye: $d N = - λ N d t$ Minus KYUN? Kyunki $N$ kam hota hai — $d N$ negative hai.

Step 2 — Variables alag karein. $\frac{d N}{N} = - λ d t$ Yeh step KYUN? Hum saare $N$ ek taraf aur saare $t$ doosri taraf collect karte hain, taaki dono sides ko alag-alag integrate kiya ja sake.

Step 3 — Dono sides integrate karein start se ( $t = 0$ , $N = N_{0}$ ) time $t$ tak: $\int_{N_{0}}^{N} \frac{d N}{N} = - λ \int_{0}^{t} d t$ $ln N - ln N_{0} = - λ t \Rightarrow ln \frac{N}{N _{0}} = - λ t$

Step 4 — Exponentiate karein. $N = N_{0} e^{- λ t}$ Exponential KYUN? Kyunki rate, amount ke proportional hai — yahi exponential decay ki defining property hai. Equal time intervals mein haari gayi fraction hamesha same hoti hai.

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

Half-life $T_{1/2}$ — ise derive karein

$N = N_{0} /2$ set karein: $\frac{N _{0}}{2} = N_{0} e^{- λ T_{1/2}} \Rightarrow \frac{1}{2} = e^{- λ T_{1/2}}$ $ln$ lein: $- ln 2 = - λ T_{1/2}$ $T_{1/2} = \frac{ln 2}{λ} = \frac{0.693}{λ}$

Mean life $τ$

Activity $A$ — woh cheez jo aap actually measure karte hain

$d N / d t = - λ N$ se: $A = \frac{d N}{d t} = λ N$ $N = N_{0} e^{- λ t}$ substitute karein: $A = λ N_{0} e^{- λ t} = A_{0} e^{- λ t}$ Activity SAME law aur SAME half-life se decay karti hai jaise $N$ . Isliye ek measurable quantity aapko invisible nuclei ke baare mein batati hai.

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Sochiye ek bade bag mein popcorn kernels hain jo ek garam pan mein hain. Har kernel random time par pops karta hai — aap nahi keh sakte kaunsa aage popega. Lekin aap keh sakte hain: "un-popped walon mein se lagbhag har 10 mein se 1 har second mein pop karta hai." Toh shuruaat mein bahut zyada popping hoti hai (shor!), aur jaise-jaise kam un-popped kernels bachte hain, popping slow ho jaati hai. Radioactive atoms bilkul aise hi hain. Half-life = woh time jisme aadhe kernels pop ho jaayein. "Per second popping ki awaaz" = activity — pehle tez, baad mein dheemi, lekin bahut lambe time tak bilkul band nahi hoti.

Forecast-then-Verify

Connections

Radioactivity — alpha, beta, gamma decay
Binding energy and mass defect
Nuclear reactions and Q-value
Exponential functions and natural log
First-order chemical kinetics (identical math)
Carbon dating (decay law ka direct application)

Decay law equation kya hai?

N = N_{0} e^{- λ t}

N_{0}

= nuclei ki initial sankhya,

λ

= decay constant.

Decay rate N ke proportional KYUN hai?

Har nucleus ki constant probability

λ d t

hoti hai decay karne ki, isliye total expected decays

= N λ d t

Radioactive decay ka differential equation derive karo.

d N = - λ N d t

(har decay ek nucleus hatati hai; minus sign kyunki N girta hai).

λ

ke terms mein half-life ka formula?

T_{1/2} = ln 2/ λ = 0.693/ λ

Half-life derive kaise hoti hai?

N = N_{0} e^{- λ t}

mein

N = N_{0} /2

set karo;

1/2 = e^{- λ T_{1/2}}

milta hai, toh

T_{1/2} = ln 2/ λ

n

half-lives ke baad remaining fraction?

(1/2)^{n}

Activity kya hai aur uska formula?

Decays per second;

A = λ N = A_{0} e^{- λ t}

Kya activity ka half-life N jaisa hi hota hai?

Haan — dono

e^{- λ t}

follow karte hain same

λ

ke saath.

Mean life

τ

kya hai aur half-life se uska kya relation hai?

τ = 1/ λ

;

T_{1/2} = τ ln 2 \approx 0.693 τ

, isliye

τ > T_{1/2}

Ek mean life ke baad remaining fraction?

e^{- 1} \approx 0.37

(37%).

Activity ke units?

Becquerel (1 decay/s); 1 Ci

= 3.7 \times 1 0^{10}

Bq.

N

exactly zero KYUN nahi pahunchta?

Exponential decay asymptotic hai; baar-baar aadha karne par hamesha nonzero amount bachti hai.

Agar

λ

double ho, toh half-life ka kya hoga?

Half ho jaayegi (inversely proportional).