5.2.4
Intuition The big pictureHeavy nuclei jaise 238 U chain of unstable daughters ke through decay karte hain, har step mein ek α β stable isotope of lead par nahi pahunch jaate. Decay ka yeh poora staircase radioactive (disintegration) series kehlata hai.
Series kyun hoti hai, ek decay kyun nahi? Ek α A β A = 238 A ≈ 206 α
Definition Exactly four series kyun hote hainEk α A β − A ek chain ke har member ka mass number parent se 4 ke multiple se differ karta hai . Matlab A mod 4 conserved rehta hai. Kyunki A mod 4 ∈ { 0 , 1 , 2 , 3 } four possible families hote hain.
Series
Type
Parent
Parent ka Half-life
Final stable nuclide
Thorium 4 n 90 232 Th 1.4 × 1 0 10 82 208 Pb
Neptunium 4 n + 1 93 237 Np 2.1 × 1 0 6 83 209 Bi
Uranium (radium) 4 n + 2 92 238 U 4.5 × 1 0 9 82 206 Pb
Actinium 4 n + 3 92 235 U 7.0 × 1 0 8 82 207 Pb
4 n + 1 237 Np ∼ 2.1 × 1 0 6 4.5 × 1 0 9 bahut chota . Isliye koi bhi natural neptunium kab ka decay ho chuka hai. Baaki teen parents ke half-lives Earth ki age ke comparable (ya zyada) hain, isliye wo survive karte hain aur aaj bhi milte hain.
Derivation Parent → product seα β ( A 1 , Z 1 ) ( A 2 , Z 2 ) x y − Mass-number balance (sirf α A − 4 A 1 = A 2 + 4 x ⇒ x = 4 A 1 − A 2 Atomic-number balance (α − 2 β − + 1 Z 1 = Z 2 + 2 x − y ⇒ y = Z 2 − Z 1 + 2 x + 2 x − y x Z 2 x y Z y Z 2 − Z 1
Worked example Example 1 — Uranium series:92 238 U → 82 206 Pb x x = 4 238 − 206 = 4 32 = 8 Yeh step kyun? Sirf α y y = Z 2 − Z 1 + 2 x = 82 − 92 + 16 = 6 Yeh step kyun? 8 alphas akele Z 92 − 16 = 76 raise karna hoga, jiske liye 82 − 76 = 6 Answer: 8 α 6 β − 238 = 4 n + 2
Worked example Example 2 — Thorium series:90 232 Th → 82 208 Pb x = 4 232 − 208 = 6 Kyun? Mass 24 se drop hota hai, yani 24/4 = 6 y = 82 − 90 + 2 ( 6 ) = 82 − 90 + 12 = 4 Kyun? 6 alphas Z 90 − 12 = 78 4 Answer: 6 α 4 β − 232 = 4 ( 58 ) ⇒ 4 n
Worked example Example 3 — Actinium series:92 235 U → 82 207 Pb x = 4 235 − 207 = 4 28 = 7 y = 82 − 92 + 14 = 4 Answer: 7 α 4 β − 235 = 4 ( 58 ) + 3 ⇒ 4 n + 3
Worked example Example 4 — Pehle forecast, phir verify (Neptunium series)Forecast: 93 237 Np → 83 209 Bi α , β = 237 − 209 = 28 ⇒ 7 α Z 7 α 93 − 14 = 79 83 4 β Verify: x = 28/4 = 7 y = 83 − 93 + 14 = 4 7 α 4 β − 4 n + 1
Common mistake "Main sirf atomic numbers subtract karke beta count nikaalta hoon."Yeh sahi kyun lagta hai: β Z y = Z 2 − Z 1 Yeh galat kyun hai: α Z 2 Z Fix: pehle hamesha x y = Z 2 − Z 1 + 2 x + 2 x
Common mistake "End product hamesha206 Pb Yeh sahi kyun lagta hai: lead famous stable endpoint hai aur U-238 famous series hai. Yeh galat kyun hai: har series ek alag stable nuclide par end hoti hai — 206 Pb 207 Pb 208 Pb 209 Bi Fix: yaad rakho ki 4 n + k k A mod 4
β Yeh sahi kyun lagta hai: ek electron nucleus se ja raha hai, zaroor kuch mass change hoga. Yeh galat kyun hai: electron ek neutron→proton conversion se aata hai; total nucleon count unchanged rehta hai, isliye A Fix: sirf α A A mod 4
Recall Answers padhne se pehle yeh try karo
Exactly four radioactive series kyun hote hain?
Kaunsi natural quantity ek chain mein conserved rehti hai?
Neptunium series nature mein absent kyun hai?
Har natural series ka parent aur stable end-product batao.
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek giant, wobbly Jenga tower (ek heavy atom). Yeh itna tall hai ki khada nahi reh sakta, isliye yeh ek move mein chhota steady tower nahi ban sakta. Isliye yeh ek ek block drop karta hai: kabhi ek bada chunky block of 4 (yeh alpha hai), kabhi ek tiny invisible flick jo sirf ek block ka color badal deti hai lekin height nahi (yeh beta hai). Tab tak drop karta rehta hai jab tak ek chhota, rock-solid tower nahi ban jaata — woh lead hai, jo kabhi nahi girta. Kyunki bade blocks hamesha exactly 4 hote hain, ek family mein har tower ki height hamesha 4 ke multiples se differ karti hai — isliye sirf 4 different families of towers hoti hain!
Mnemonic Series labels aur endpoints yaad karne ke liye"Uncle Tom Acts Naturally" → U ranium (4 n + 2 206 T horium (4 n 208 A ctinium (4 n + 3 207 N eptunium (4 n + 1 209 U-Ac-Th padhte waqt 2-7-8 order mein jaate hain — aur jo odd one out hai (Np) woh Bismuth par end hota hai.
Alpha decay — woh aakela step jo A − 4 4 n + k Beta decay — Z A + 2 x Half-life and decay constant — kyun 237 238 Group displacement law (Soddy–Fajans) — in chains ke peeche har step ka bookkeeping.Radioactive dating — U–Pb aur Th–Pb clocks in series ke stable endpoints use karte hain.Nuclear stability and band of stability — heavy nuclei ko mass shed karna hi kyun padta hai.
Exactly four radioactive series kyun hote hain? Kyunki α A β A mod 4
Decay chain mein kaunsi quantity conserved rehti hai? A mod 4 A = 4 n + k k
Uranium series ka parent aur end-product kya hai? 92 238 82 206 4 n + 2
Thorium series ka parent aur end-product kya hai? 90 232 82 208 4 n
Actinium series ka parent aur end-product kya hai? 92 235 82 207 4 n + 3
Neptunium series ka parent aur end-product kya hai? 93 237 83 209 4 n + 1
Neptunium series nature mein missing kyun hai? Iska sabse zyada lived member 237 t 1/2 ∼ 2.1 × 1 0 6
Alpha particles emit hone ki formula kya hai? x = ( A 1 − A 2 ) /4
Beta particles emit hone ki formula kya hai? y = Z 2 − Z 1 + 2 x
238 206 α β 8 α 6 β −
232 208 α β 6 α 4 β −
Cannot reach stable in one jump
