Calculation of atomic mass from isotopic abundance

1.2.6 · Chemistry › Atomic Structure (Classical)

1. Hum average kya kar rahe hain?

WHAT hai fractional abundance aur percentage mein fark? $75.77%$ percentage abundance aur fraction $f = 0.7577$ ek hi information hai. Formula ke fraction form mein use karne se pehle hamesha percentage ko 100 se divide karo.

2. Formula scratch se derive karna

HOW hum formula banate hain? Imagine karo ek bade bucket mein $N$ atoms hain us element ke.

Step 1 — Har isotope ko count karo. Agar isotope $i$ ki fractional abundance $f_{i}$ hai, to us isotope ke atoms ki count hai $N_{i} = f_{i} N$ . Why this step? Fraction × total = count. Yahi to "fraction" ka matlab hai.

Step 2 — Bucket ki total mass. Har isotope contribute karta hai (uski mass) × (uski count): $M_{total} = \sum_{i} m_{i} N_{i} = \sum_{i} m_{i} f_{i} N = N \sum_{i} m_{i} f_{i}$ Why this step? Mass additive hoti hai: total mass, saari individual masses ka sum hoti hai.

Step 3 — Average mass per atom = total mass ÷ number of atoms: $A_{r} = \frac{M _{total}}{N} = \frac{N \sum _{i} m _{i} f _{i}}{N} = \sum_{i} m_{i} f_{i}$ Why this step? Unknown $N$ cancel ho jaata hai — answer is par depend nahin karta ki tumhara sample kitna bada hai, sirf proportions pe karta hai. Yahi cancellation hai jis wajah se weighted average kaam karta hai.

Figure — Calculation of atomic mass from isotopic abundance

3. Worked examples

4. Common mistakes (Steel-man + fix)

5. Active recall

Recall Pehle khud try karo, phir reveal karo

Atomic mass whole number kyun nahin hoti?
Silver: $^{107}$ Ag (106.905 u, 51.84%), $^{109}$ Ag (108.905 u, 48.16%). $A_{r}$ nikalo.
Agar $A_{r}$ halke isotope ki mass ke bahut nazdeek hai, to yeh kya batata hai?

Answers: 1. Yeh alag-alag mass ke isotopes ka weighted average hai, isliye generally non-integer hota hai. 2. $(106.905 \times 51.84 + 108.905 \times 48.16) /100 = 107.87$ u. 3. Halka isotope zyaada abundant hai.

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

Imagine karo ek bag mein marbles hain. Zyaadatar marbles ka weight 35 grams hai, lekin kuch ka 37 grams. Agar koi puche "is bag se ek typical marble kitna bhaari hoga?", to tum sirf 36 (middle value) nahin keh sakte — kyunki halke wale bahut zyaada hain. Tumhe count karna padega ki kitne kitne hain aur common waalon ko zyaada "vote" dena padega. Chlorine ki atomic mass 35.45 exactly wohi vote hai: mostly halke atoms hain, isliye average 35 ki taraf jhuka hua hai.

Flashcards

What is an isotope?

Ek hi element ke atoms (same

Z

) jinmein neutrons ki alag-alag sankhya hoti hai, isliye alag mass numbers

A

Why is relative atomic mass usually not a whole number?

Yeh ek naturally occurring mixture ke isotopic masses ka abundance-weighted average hota hai.

Formula for relative atomic mass (fractions)?

A_{r} = \sum_{i} m_{i} f_{i}

with

\sum f_{i} = 1

Formula for relative atomic mass (percentages)?

A_{r} = \frac{\sum _{i} m _{i} p _{i}}{100}

In the derivation, why does the total number of atoms

N

cancel?

Kyunki hum total mass ko

N

se divide karte hain; result sirf proportions pe depend karta hai, sample size pe nahin.

Cl has

^{35}

Cl (75.77%) and

^{37}

Cl (24.23%). Why is

A_{r}

closer to 35 than to 37?

Kyunki

^{35}

Cl lagbhag 3× zyaada abundant hai, isliye woh weighted average pe dominate karta hai.

Fix for the "just average the masses" error?

Har mass ko uski abundance se weight karo; simple mean sirf tab kaam karta hai jab abundances equal hon.

Given two isotopes and

A_{r}

, how do you find the abundance

f

of one?

A_{r} = m_{1} f + m_{2} (1 - f)

set karo aur

f

ke liye solve karo.

Difference between mass number

A

and isotopic mass?

A

protons+neutrons ki whole-number count hai; isotopic mass u mein measured mass hai (thodi kam, mass defect ki wajah se).

Connections

Isotopes and Mass Number — woh raw ingredients jo average kiye ja rahe hain.
Atomic Mass Unit (u) and the Carbon-12 standard — $m_{i}$ ki unit define karta hai.
Mole Concept and Molar Mass — $A_{r}$ grams mein = ek mole ki mass.
Mass Spectrometry — woh experiment jo isotopic masses aur abundances measure karta hai.
Weighted Average (mathematics) — woh general statistical tool jo yahan use ho raha hai.
Nuclear Binding Energy and Mass Defect — kyun isotopic mass ≠ integer mass number.