Substance A has atomic mass number 16 and half-life of 1 day. Ano… — Solution | JEE Physics

MCQEasyJEE 2023Radioactive Decay & Half-Life

Substance A has atomic mass number $16$ and half-life of $1 \, \text{day}$ . Another substance B has atomic mass number $32$ and half-life of $\frac{1}{2} \, \text{day}$ . If both A and B simultaneously start undergoing radioactivity at the same time with initial mass $320 \, \text{g}$ each, how many total atoms of A and B combined would be left after $2 \, \text{days}$ ?

A
$3.38 \times 10^{24}$
B
$6.76 \times 10^{24}$
C
$6.76 \times 10^{23}$
D
$1.69 \times 10^{24}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Substance A has mass number $16$ , half-life $1 \, \text{day}$ , and initial mass $320 \, \text{g}$ . Substance B has mass number $32$ , half-life $\frac{1}{2} \, \text{day}$ , and initial mass $320 \, \text{g}$ .

Find: Total number of atoms of A and B left after $2 \, \text{days}$ .

For substance A:

(N_0)_A = \frac{320}{16} = 20 \, \text{moles}

N_A = (N_0)_A \left(\frac{1}{2}\right)^2 = 20 \times \frac{1}{4} = 5 \, \text{moles}

For substance B:

(N_0)_B = \frac{320}{32} = 10 \, \text{moles}

N_B = (N_0)_B \left(\frac{1}{2}\right)^4 = 10 \times \frac{1}{16} = 0.625 \, \text{moles}

Total moles:

5 + 0.625 = 5.625

The total number of atoms:

5.625 \times 6.023 \times 10^{23} = 3.38 \times 10^{24}

Therefore, the total number of atoms left is $3.38 \times 10^{24}$ . The correct option is A.

Common mistakes

Using the same number of half-lives for both substances is incorrect because A and B have different half-lives. First calculate how many half-lives occur in $2 \, \text{days}$ for each substance separately.
Treating atomic mass number directly as the remaining number of atoms is wrong. Convert the given mass in grams to moles first using molar mass, then apply radioactive decay.
Adding remaining masses instead of remaining moles or atoms gives the wrong result. After decay calculation, combine the remaining moles and then multiply by Avogadro number to get total atoms.

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Answer

Step-by-step solution

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