Nuclear reactions — Q-value calculation: Question bank

2.3.20 · D5 · Physics › Modern Physics › Nuclear reactions — Q-value calculation

Shuru karne se pehle, ek word jo hum baar baar use karenge: Q-value ek reaction $a + X \to Y + b$ ki leftover energy hai, define hoti hai $Q = (m_{before} - m_{after}) c^{2}$ — jo mass gayab hota hai ( $Q > 0$ ) woh kinetic energy ban jaata hai, jo mass appear hota hai ( $Q < 0$ ) usey kinetic energy se pay karna padta hai. Mnemonic BAM (Before Minus After) apne dimaag mein rakho jab tum aage bado.

True or false — justify karo

True or false: Ek reaction jisme

Q > 0

ho, woh tab bhi ho sakti hai jab projectile ki kinetic energy almost zero ho.

True — ek exothermic reaction ka koi threshold nahi hota; reaction khud apni energy supply karta hai, toh koi bhi choti si push (ya koi bhi nahi, spontaneous decay ke liye) kaafi hai.

True or false: Agar ek reaction ke products reactants se heavier hain, toh energy release hui.

False — heavier products matlab rest mass badhdi, toh

Q < 0

; energy ko absorb karna pada (dalna pada) us extra mass ko create karne ke liye.

True or false: Q-value us reference frame par depend karta hai jisme tum usse measure karte ho.

False —

Q

rest masses ka difference hai times

c^{2}

, aur rest masses frame-independent hote hain, isliye

Q

har frame mein same hota hai.

True or false: Spontaneous alpha decay mein Q-value positive honi chahiye.

True — koi external energy supply nahi hoti, toh process sirf "downhill" chal sakta hai;

Q > 0

matlab released rest mass alpha ki aur recoiling daughter ki kinetic energy ke roop mein appear hoti hai.

True or false:

Q

compute karte waqt tum hamesha nuclear masses ki jagah atomic masses substitute kar sakte ho.

False — yeh sirf tab kaam karta hai jab electron count balance ho (

Z

conserved ho); positron emission ya electron capture ke liye electrons mismatch hote hain aur tumhe correction terms chahiye.

True or false: Zyada mass defect

Δ m

hamesha zyada energy release matlab hota hai.

True — kyunki

Q = Δ m c^{2}

aur

c^{2}

ek fixed constant hai,

Q

directly scale karta hai us mass ke saath jo gayab hua.

True or false: Threshold kinetic energy ek endothermic reaction ke liye Q-value ke magnitude ke barabar hoti hai.

False — threshold energy

∣ Q ∣ (1 + m_{a} / m_{X})

hoti hai,

∣ Q ∣

se badi kyunki momentum products ko chalte rehne par majboor karta hai, aur kuch energy center-of-mass motion mein waste ho jaati hai.

True or false: Alpha decay mein emitted alpha particle pura Q-value le jaata hai.

False — momentum conservation daughter ko recoil karaata hai, toh

Q = K_{α} + K_{daughter}

; alpha ko bada share milta hai

Q \cdot m_{D} / (m_{D} + m_{α})

lekin poora nahi.

True or false: Kisi reaction ka

Q = 0

ho sakta hai.

True — yeh ek elastic collision hai jahan koi rest mass nahi badlata; pehle ki total kinetic energy baad ki total kinetic energy ke barabar hoti hai.

Error dhundo

Flaw dhundo: "

^{14}

+ α \to^{17}

+ p

ke liye jisme

Q = - 1.19

MeV hai, bilkul

1.19

MeV ka alpha reaction trigger kar dega."

Galat — tumhe threshold

∣ Q ∣ (1 + m_{a} / m_{X}) \approx 1.53

MeV chahiye;

1.19

MeV ka kuch hissa forward CM motion ke roop mein rehna padega aur usse nucleus rearrange karne ke liye use nahi kiya ja sakta.

Flaw dhundo: "

Q = m_{after} - m_{before}

times

c^{2}

Sign ulta hai — yeh Before minus After (BAM) hona chahiye; jaise likha hai yeh har exothermic reaction ko endothermic bana deta hai.

Flaw dhundo: "Kyunki daughter nucleus heavy aur slow hai, uski kinetic energy negligible hai, toh main usse

Q = K_{final}

mein ignore karta hoon."

Uski speed chhoti hai lekin uska momentum alpha ke momentum ke barabar hai;

K = p^{2} /2 m

, toh recoil energy real hai aur chhoti hai lekin zero nahi — isse drop karne se

Q

thoda kam ho jaata hai.

Flaw dhundo: "

1 u = 931.5

MeV, toh MeV mein

Q

sirf

Δ m

kilograms mein times

931.5

hai."

931.5

factor atomic mass units ko convert karta hai, kilograms ko nahi; tumhare paas pehle

Δ m

u mein hona chahiye (ya phir

E = Δ m c^{2}

directly SI mein use karo).

Flaw dhundo: "Yeh reaction

17.6

MeV release karti hai, toh har product particle ko

8.8

MeV milta hai."

Energy equally split nahi hoti — momentum conservation lighter fragment ko zyada kinetic energy deta hai, toh neutron aur helium

17.6

MeV ko unequally apne inverse mass ratio ke hisaab se share karte hain.

Flaw dhundo: "Reaction energy absorb karti hai, toh uski binding energy badh gayi."

Ulta hai — energy absorb karna matlab products kam tightly bound hain (zyada massive); energy tabhi release hoti hai jab products zyada tightly bound ho jaayein, binding energy ke according.

Why questions

Itni chhoti si mass change itni badi energy kyun produce karti hai?

Kyunki

Q = Δ m c^{2}

aur

c^{2} \approx 9 \times 1 0^{16}

bahut bada hai; yahan tak ki

0.02

u (

\sim 3 \times 1 0^{- 29}

kg) bhi [[Mass-energy equivalence E=mc^2|

E = m c^{2}

]] ke through tens of MeV ban jaata hai.

Q

sirf

K_{final}

kyun nahi balki

K_{final} - K_{initial}

ke barabar kyun hai?

Total energy (rest + kinetic) conserved hoti hai; rest-mass change

Q

, total kinetic energy ke change ke roop mein appear hota hai, toh koi bhi initial KE subtract karni padti hai.

Hum

Q

ko "final binding energy minus initial binding energy" kyun kehte hain?

Zyada tightly bound nucleus kam wazan karta hai, toh binding energy gain karne par mass lose hota hai; woh lost mass

\times c^{2}

exactly released

Q

hai.

Fusion of light nuclei aur fission of heavy nuclei dono energy kyun release karte hain?

Dono iron-peak of maximum binding energy per nucleon ki taraf move karte hain, toh products zyada tightly bound aur lighter ho jaate hain, jisse har direction mein

Q > 0

milta hai.

Threshold energy hamesha

∣ Q ∣

se zyada kyun hoti hai?

Projectile momentum carry karta hai jo conserved hona chahiye, toh products sabhi rest nahi ho sakte; unavoidable forward CM kinetic energy rearrangement ke liye available nahi hoti, isliye tumhe extra supply karni padti hai.

Aam case mein atomic masses use karte waqt electron masses kyun cancel ho jaate hain?

Charge conservation matlab same total protons — isliye electrons — pehle aur baad mein appear hote hain, toh unke masses

m_{before} - m_{after}

mein zero ho jaate hain.

Positron emission ko atomic masses ke saath

2 m_{e} c^{2}

correction kyun chahiye?

Daughter atom mein ek proton kam hota hai lekin jaise likha hua hai electron count same hota hai, toh har side par ek electron mass unmatched rehta hai; imbalance do electron masses ka hota hai jo subtract karni padti hain.

Edge cases

Edge case: Agar reactants aur products ka total rest mass identical ho toh

Q

kya hoga?

Q = 0

; collision perfectly elastic hai aur total kinetic energy unchanged hai, sirf redistribute hoti hai.

Edge case: Alpha decay ke liye jahan parent rest mein hai, measured energies se

Q

kaise nikaaloge?

Alpha ki kinetic energy aur daughter ki recoil energy add karo:

Q = K_{α} + K_{daughter}

; koi mass table ki zaroorat nahi.

Edge case: Kya ek endothermic reaction (

Q < 0

) kabhi zero projectile kinetic energy ke saath ho sakti hai?

Nahi —

∣ Q ∣

plus CM-motion tax supply karne ke liye kuch bhi nahi hai, us mass ke liye koi source nahi jo create hona chahiye, toh yeh proceed nahi kar sakti.

Edge case: Agar target projectile se bahut heavier hai (

m_{X} ≫ m_{a}

), toh threshold energy kis taraf approach karti hai?

K_{th} = ∣ Q ∣ (1 + m_{a} / m_{X}) \to ∣ Q ∣

; ek heavy stationary target barely recoil karta hai, toh almost koi energy CM motion ke roop mein waste nahi hoti.

Edge case: Agar projectile target se bahut heavier hai (

m_{a} ≫ m_{X}

), toh threshold ke saath kya hota hai?

Factor

(1 + m_{a} / m_{X})

bahut bada ho jaata hai, toh

K_{th} ≫ ∣ Q ∣

; incoming energy ka zyaadatar hissa bulk CM motion mein lock ho jaata hai aur thoda usable rehta hai.

Edge case: Ek gamma photon ek nucleus se takraata hai (photodisintegration). Threshold logic mein projectile "rest mass" ki jagah kya kaam karta hai?

Photon massless hota hai lekin momentum

E / c

carry karta hai; woh momentum phir bhi products ke saath share hona chahiye, toh photon energy ki zaroorat phir bhi

∣ Q ∣

se zyada hoti hai.

Recall Jin traps ko tumne abhi clear kiya unka fast recap

Sign ::: BAM — Before minus After; lighter products $Q > 0$ dete hain. Threshold ::: hamesha $∣ Q ∣ (1 + m_{a} / m_{X})$ , kabhi sirf $∣ Q ∣$ nahi. Energy split ::: kabhi equal nahi — momentum lighter fragment ko favour karta hai. Atomic masses ::: theek hain jab $Z$ balance ho, $β^{+}$ /capture ke liye corrected karo. Frame ::: $Q$ frame-independent hai; kinetic-energy splits nahi hote.