Decay kinetics — first-order; half-life, mean life, activity

5.2.3 · Chemistry › Nuclear & Radiochemistry

1. First-order kyun? (Derivation scratch se)

SHURU mein physical statement kya hai? Har nucleus ki same probability $λ d t$ hoti hai ek choti si time $d t$ mein decay karne ki. Yeh probability age par depend nahi karti (1-second-old nucleus aur 1000-year-old nucleus identical hote hain). Yeh "memoryless" property hi key hai.

YEH rate law kaise deta hai? Agar $N$ nuclei hain, toh $d t$ mein decay hone wale nuclei ki expected number hai $N \times (λ d t)$ . Toh $N$ mein decrease:

$- d N = λ N d t \Rightarrow \frac{d N}{d t} = - λ N$

Yeh first-order hai kyunki rate $\propto N^{1}$ hai. Dhyan do: rate law mein koi concentration units, collisions, ya temperature dependence nahi hai — yeh ek pure nuclear property hai, isliye $λ$ ek true constant hai.

2. Equation solve karna → exponential law

Variables separate kyun karte hain? Kyunki $N$ aur $t$ cleanly opposite sides par aa jaate hain.

$\frac{d N}{N} = - λ d t$

Dono sides ko $t = 0$ (jahan $N = N_{0}$ ) se $t$ tak integrate karo:

$\int_{N_{0}}^{N} \frac{d N}{N} = - λ \int_{0}^{t} d t$ $ln N - ln N_{0} = - λ t$ $N = N_{0} e^{- λ t}$

Figure — Decay kinetics — first-order; half-life, mean life, activity

3. Half-life $t_{1/2}$ — derived, memorized nahi

YEH kya hai? Woh time jisme sample half apni current amount par aa jaaye: $N = N_{0} /2$ .

KAISE? Master law mein plug karo:

$\frac{N _{0}}{2} = N_{0} e^{- λ t_{1/2}} \Rightarrow \frac{1}{2} = e^{- λ t_{1/2}} \Rightarrow ln 2 = λ t_{1/2}$

$t_{1/2} = \frac{ln 2}{λ} = \frac{0.693}{λ}$

4. Mean life (average lifetime) $τ$ — derived

KYA hai? Ek nucleus ke decay hone se pehle average time kitna hota hai.

KAISE (derivation): $t$ aur $t + d t$ ke beech decay hone wala fraction hai $\frac{- d N}{N _{0}} = λ e^{- λ t} d t$ . Mean life time-weighted average hai:

$τ = \int_{0}^{\infty} t λ e^{- λ t} d t$

Integration by parts use karo (ya yaad rakho $\int_{0}^{\infty} t e^{- a t} d t = 1/ a^{2}$ ):

$τ = λ \cdot \frac{1}{λ ^{2}} = \frac{1}{λ}$

$τ = \frac{1}{λ}, t_{1/2} = τ ln 2 = 0.693 τ$

5. Activity $A$ — jo detectors actually measure karte hain

Kyunki $- d N / d t = λ N$ :

$A = λ N = λ N_{0} e^{- λ t} = A_{0} e^{- λ t}$

Activity SAME exponential se kyun decay karti hai? Kyunki $A = λ N$ sirf $N$ ko ek constant $λ$ se scale karna hai — isliye yeh identical curve aur identical half-life follow karta hai.

6. Worked examples

7. Common mistakes (Steel-man + fix)

8. Forecast-then-Verify

Recall Check karne se pehle Forecast karo

Q: Agar ek nuclide ki activity 40 days mein apni initial value ke 6.25% tak gir jaaye, toh $t_{1/2}$ kya hai? Forecast: $6.25% = 1/16 = (1/2)^{4}$ , toh 4 half-lives = 40 days → $t_{1/2} = 10$ days. Verify: $A / A_{0} = e^{- λ t}$ , $λ = ln (16) /40 = 2.773/40 = 0.0693 day^{- 1}$ , $t_{1/2} = 0.693/0.0693 = 10$ days ✓

9. Flashcards

Radioactive decay first-order kyun hai?

Har nucleus ki ek fixed decay probability per unit time hoti hai, isliye rate

= λ N \propto N^{1}

hai.

Master decay equation kya hai?

N = N_{0} e^{- λ t}

d N / d t = - λ N

ko integrate karke derive kiya gaya.

λ

se half-life ka formula?

t_{1/2} = ln 2/ λ = 0.693/ λ

(

N = N_{0} /2

set karo).

Mean life ka formula?

τ = 1/ λ

; aur

τ = t_{1/2} /0.693 = 1.44 t_{1/2}

Mean life, half-life se badi kyun hoti hai?

Mean lambe time tak zinda bachne wale nuclei ki wajah se upar khinch jaata hai; half-life median hai,

τ

mean hai.

Activity ki definition?

A = - d N / d t = λ N

; yeh

A = A_{0} e^{- λ t}

ki tarah decay karti hai.

Kis

N / N_{0}

par mean life reach hoti hai?

N / N_{0} = 1/ e \approx 0.37

n

half-lives ke baad kitna fraction bachta hai?

(1/2)^{n}

1 curie kitne Bq ke barabar hai?

3.7 \times 1 0^{10}

disintegrations per second.

Kya

t_{1/2}

starting amount ya temperature par depend karta hai?

Nahi — yeh sirf

λ

par depend karta hai, jo ek nuclear constant hai.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek giant bag popcorn kernels ka jahan har kernel ki same tiny chance hai har second pop karne ki. Aap predict nahi kar sakte kaunsa kernel next pop karega, lekin lakho kernels ke saath aap keh sakte ho "approximately itne har second pop honge." Jitne zyada unpopped kernels bache honge, utni zyada pops sunoge — isliye jaise bag khaali hota hai, popping slow hoti jaati hai. Half-life woh time hai jisme half kernels pop ho jaayein. Agla half ussi time mein hota hai, phir agla... isliye bag kabhi bilkul khaali nahi hota. "Popping sound rate" activity hai, aur yeh exactly usi tarah fade hoti hai.