Import Question JSON

Current Question (ID: 18344)

Question:

$\text{The half-life for radioactive decay of } ^{14}\text{C is } 5730 \text{ y. An archaeological artifact containing wood had only } 80 \% \text{ of the } ^{14}\text{C found in a living tree. The age of the sample will be:}$

Options:

1. $1657.3 \text{ y}$
2. $1845.4 \text{ y}$
3. $1512.4 \text{ y}$
4. $1413.1 \text{ y}$

Solution:

$t_{1/2} = 5730 \text{ y}$ $\text{If } a \text{ is the initial } ^{14}\text{C activity in a living tree, then}$ $\text{Activity in the deadwood, } a_t = \frac{80}{100} \times a$ $\text{The nuclear / radioactive decays follow the first-order kinetics.}$ $\text{So, } t = \frac{2.303}{0.693/t_{1/2}} \log \frac{a}{a_t} \text{ and } k = \frac{0.693}{t_{1/2}}$ $\text{Therefore, } t = \frac{2.303}{0.693/t_{1/2}} \log \frac{a}{80/100 \times a}$ $= \frac{2.303 \times 5730 \text{ y}}{0.693} \log \frac{100}{80}$ $\text{So, } t = 1845.4 \text{ y}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}