Import Question JSON

Current Question (ID: 18928)

Question:

$\text{In a typical combustion engine the work done by a gas molecule is given } W = \alpha^2 \beta e^{-\frac{\beta x^2}{kT}} . \text{ where } x \text{ is the displacement, } k \text{ is the Boltzmann constant and } T \text{ is the temperature. If } \alpha \text{ and } \beta \text{ are constants, dimensions of } \alpha \text{ will be:}$

Options:

1. $[MLT^{-2}]$
2. $[M^0LT^0]$
3. $[M^2LT^{-2}]$
4. $[MLT^{-1}]$

Solution:

$\text{Hint: The argument of an exponential must be dimensionless.}$ $kT \text{ has a dimension of energy}$ $\frac{\beta x^2}{kT} \text{ is dimensionless}$ $[\beta][L^2] = [ML^2 T^{-2}]$ $[\beta] = [MT^{-2}]$ $\alpha^2 \beta \text{ has dimension of work}$ $[\alpha^2][MT^{-2}] = [ML^2 T^{-2}]$ $[\alpha] = [M^0 L^0 T^0]$

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
}