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Current Question (ID: 7501)

Question:

The wavelength of light emitted when the electron in a H atom undergoes the transition from an energy level with $n = 4$ to an energy level with $n = 2$, is :

Options:

1. $586 \, \text{mm}$
2. $486 \, \text{nm}$
3. $523 \, \text{nm}$
4. $416 \, \text{pm}$

Solution:

**Hint:** Find out the energy of the hydrogen atom for Balmer series ($n_f = 2$) and then calculate wavelength. **STEP 1:** Use $n_i$ and $n_f$ to calculate the energy The $n_i = 4$ to $n_f = 2$ transition will give rise to a spectral line of the Balmer series. The energy involved in the transition is given by the relation, \[E = 2.18 \times 10^{-18} \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right)\] \[= 2.18 \times 10^{-18} \left( \frac{1}{4^2} - \frac{1}{2^2} \right)\] \[= -(4.08 \times 10^{-19}) \, \text{J}\] The negative sign indicates the energy of emission. **STEP 2:** Calculate wavelength from energy formula We know that, \[E = \frac{hc}{\lambda}\] \[\lambda = \frac{hc}{E} = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{4.08 \times 10^{-19}} = 486.3 \times 10^{-9} \, \text{m}\] \[= 486.3 \, \text{nm}\]

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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
}