Import Question JSON

Current Question (ID: 7497)

Question:

The ratio of the wavelengths of the last lines of the Balmer to Lyman series is

Options:

1. 4:1
2. 27:5
3. 3:1
4. 9:4

Solution:

### **HINT:** Use the Rydberg formula for wavelength calculation **Step 1:** The wavelength of spectral lines is given by: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \times Z^2 \] where $R$ is the Rydberg constant, $n_1$ is the lower energy level, and $n_2$ is the upper energy level. **Step 2:** Calculate wavelengths for both series with $n_2 \to \infty$: For Lyman series ($n_1 = 1$): \[ \frac{1}{\lambda_L} = R \left( 1 - 0 \right) = R \] \[ \lambda_L = \frac{1}{R} \] For Balmer series ($n_1 = 2$): \[ \frac{1}{\lambda_B} = R \left( \frac{1}{4} - 0 \right) = \frac{R}{4} \] \[ \lambda_B = \frac{4}{R} \] **Step 3:** Compute the ratio: \[ \frac{\lambda_B}{\lambda_L} = \frac{4/R}{1/R} = 4 \] Thus, the ratio of wavelengths is 4:1.

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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
}