Import Question JSON

Current Question (ID: 8285)

Question:

$\text{The factor } \left(\frac{\partial Q}{\partial T}\right)_P - \left(\frac{\partial Q}{\partial T}\right)_V \text{ is equal to:}$

Options:

1. $\gamma$
2. $\text{R}$
3. $\frac{R}{M}$
4. $\Delta \text{nRT}$

Solution:

$\text{Hint: } \left(\frac{\partial Q}{\partial T}\right)_P = C_P$ $\text{Step 1: Identify the partial derivatives in terms of heat capacities}$ $\left(\frac{\partial Q}{\partial T}\right)_P \text{ represents the heat capacity at constant pressure, } C_P$ $\left(\frac{\partial Q}{\partial T}\right)_V \text{ represents the heat capacity at constant volume, } C_V$ $\text{Step 2: Express the difference in terms of heat capacities}$ $\left(\frac{\partial Q}{\partial T}\right)_P - \left(\frac{\partial Q}{\partial T}\right)_V = C_P - C_V$ $\text{Step 3: Apply the relationship between heat capacities}$ $\text{From thermodynamics, for an ideal gas: } C_P - C_V = R$ $\text{Where R is the universal gas constant (8.314 J/mol·K)}$ $\text{Therefore, } \left(\frac{\partial Q}{\partial T}\right)_P - \left(\frac{\partial Q}{\partial T}\right)_V = C_P - C_V = R$ $\text{The answer is R, corresponding to option 2.}$

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
}