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Current Question (ID: 18923)

Question:

$\text{The work done by a gas molecule in an isolated system is given by,}$ $\text{W} = \alpha \beta^2 e^{-\frac{x^2}{\alpha k T}}, \text{ where } x \text{ is the displacement, } k \text{ is the Boltzmann}$ $\text{constant and } T \text{ is the temperature, } \alpha \text{ and } \beta \text{ are constants. Then the}$ $\text{dimensions of } \beta \text{ will be:}$

Options:

1. $[ML^2 T^{-2}]$
2. $[MLT^{-2}]$
3. $[M^2 L T^2]$
4. $[M^0 L^0 T^0]$

Solution:

$\text{Hint: The argument of an exponential must be dimensionless.}$ $\text{Step: Find the dimensions of } \beta.$ $\text{As, The argument of an exponential must be dimensionless.}$ $\Rightarrow \frac{x^2}{\alpha k T} \rightarrow \text{dimensionless}$ $\text{Then, } [\alpha] = \frac{x^2}{k T} = \frac{[L^2]}{[ML^2 T^{-2}]} = [M^{-1}T^2]$ $\text{also, } [W] = [\alpha][\beta^2]$ $\text{Therefore, } [\beta] = \sqrt{\frac{[ML^2 T^{-2}]}{[M^{-1}T^2]}} = [M^1L^1T^{-2}]$ $\text{Hence, option (2) is the correct answer.}$

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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
}