Import Question JSON

Current Question (ID: 19630)

Question:

$\text{A cube is constructed from a metal sheet such that each of its sides has a length } a \text{ at room temperature } T. \text{ The coefficient of linear expansion of the metal is } \alpha. \text{ The entire cube is then heated uniformly so that its temperature increases by a small amount } \Delta T, \text{ making the new temperature } T + \Delta T. \text{ Assuming the expansion is small and isotropic, what is the cube's volume increase due to this heating?}$

Options:

1. $3a^3\alpha\Delta T$
2. $4a^3\alpha\Delta T$
3. $4\pi a^3\alpha\Delta T$
4. $\frac{4}{3}\pi a^3\alpha\Delta T$

Solution:

$\text{Hint: } \Delta V = V_0 \gamma \Delta T$ $\text{Step: Find the cube's volume increase due to this heating.}$ $\text{The coefficient of volume expansion is given by:}$ $\Rightarrow \gamma = \frac{\Delta V}{V \Delta T}$ $\Rightarrow \Delta V = \gamma V \Delta T$ $\text{The volume of a cube of side } 'a' \text{ is given as:}$ $\Rightarrow V = a^3$ $\Rightarrow \Delta V = \gamma a^3 \Delta T$ $\text{We know that the coefficient of linear expansion } \alpha \text{ and coefficient of volume expansion } \gamma \text{ are related as}$ $\Rightarrow \gamma = 3\alpha$ $\Rightarrow \Delta V = 3\alpha a^3 \Delta T$ $\text{Hence, option (1) is the correct answer.}$

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
}