Import Question JSON

Current Question (ID: 11172)

Question:

$\text{If } C_P \text{ and } C_V \text{ denote the specific heats (per unit mass) of an ideal gas of molecular weight } M \text{ (where } R \text{ is the molar gas constant), the correct relation is:}$

Options:

1. $C_P - C_V = R$
2. $C_P - C_V = \frac{R}{M}$ (Correct)
3. $C_P - C_V = MR$
4. $C_P - C_V = \frac{R}{M^2}$

Solution:

$\text{The correct relation is } C_P - C_V = \frac{R}{M}.$ $\text{This is derived from Mayer's relation for molar specific heats and the definition of specific heat per unit mass.}$ $\text{Mayer's Relation: For one mole of an ideal gas, the difference between the molar specific heat at constant pressure (} c_p \text{) and constant volume (} c_v \text{) is equal to the molar gas constant (} R \text{).}$ $c_p - c_v = R$ $\text{Specific Heat per Unit Mass: To get the specific heat per unit mass (} C \text{), you divide the molar specific heat (} c \text{) by the molecular weight (} M \text{).}$ $C = \frac{c}{M}$ $\text{Final Derivation: Substituting this into Mayer's relation gives us:}$ $MC_P - MC_V = R$ $M(C_P - C_V) = R$ $C_P - C_V = \frac{R}{M}$

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
}