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Current Question (ID: 10783)

Question:

\text{Which of the following graphs correctly represents the relation between } \ln E \text{ and } \ln T \text{ where } E \text{ is the amount of radiation emitted per unit time from a unit area of a body and } T \text{ is the absolute temperature?} \text{(Take } \sigma = 5.67 \times 10^{-8} \text{ W m}^{-2}\text{K}^{-4} \text{ and } 0 < \epsilon < 1\text{)}

Options:

1. $\text{Graph 1: } \ln E \text{ vs } \ln T \text{ showing a straight line with positive slope passing through negative y-intercept}$
2. $\text{Graph 2: } \ln E \text{ vs } \ln T \text{ showing a curved line}$
3. $\text{Graph 3: } \ln E \text{ vs } \ln T \text{ showing a straight line with positive slope passing through negative y-intercept}$
4. $\text{Both 1 and 3}$ (Correct)

Solution:

\text{According to Stefan's law: } E = \epsilon \sigma T^4 \text{Therefore: } \ln E = \ln \epsilon + \ln \sigma + 4 \ln T \text{Here, ln epsilon and ln sigma are negative.} \text{On comparing this equation with } y = mx + c \text{We conclude that the graph between ln E and ln T will be a straight line,} \text{having a positive slope (m = 4) and intercept on ln E axis equal to} \text{(ln epsilon + ln sigma) which is negative.}

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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
}