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Current Question (ID: 11128)

Question:
$\text{Two containers of equal volumes contain the same gas at pressures }\text{P}_1\text{ and }\text{P}_2\text{ and absolute temperatures }\text{T}_1\text{ and }\text{T}_2\text{, respectively. On joining the vessels, the gas reaches a common pressure }\text{P}\text{ and common temperature }\text{T}.\text{ The ratio }\frac{\text{P}}{\text{T}}\text{ is equal to:}$
Options:
  • 1. $\frac{\text{P}_1}{\text{T}_1} + \frac{\text{P}_2}{\text{T}_2}$
  • 2. $\frac{\text{P}_1\text{T}_1 + \text{P}_2\text{T}_2}{(\text{T}_1 + \text{T}_2)^{2}}$
  • 3. $\frac{\text{P}_1\text{T}_2 + \text{P}_2\text{T}_1}{(\text{T}_1 + \text{T}_2)^{2}}$
  • 4. $\frac{\text{P}_1}{2\text{T}_1} + \frac{\text{P}_2}{2\text{T}_2}$ (Correct)
Solution:
$\text{Let the volume of each container be }\text{V}.\text{Using the ideal gas equation, }\text{PV} = \text{nRT}.\text{The number of moles in the first container is }\text{n}_1 = \frac{\text{P}_1\text{V}}{\text{RT}_1}.\text{The number of moles in the second container is }\text{n}_2 = \frac{\text{P}_2\text{V}}{\text{RT}_2}.\text{When the two containers are joined, the total number of moles remains constant: }\text{n}_{\text{total}} = \text{n}_1 + \text{n}_2.\text{The final common pressure is }\text{P}\text{ and the final common temperature is }\text{T}.\text{The total volume is }\text{V}_{\text{total}} = \text{V} + \text{V} = 2\text{V}.\text{Applying the ideal gas equation to the final combined system: }\text{P}(2\text{V}) = (\text{n}_1 + \text{n}_2)\text{RT}.\text{Substituting the expressions for }\text{n}_1\text{ and }\text{n}_2\text{:}\text{P}(2\text{V}) = \left(\frac{\text{P}_1\text{V}}{\text{RT}_1} + \frac{\text{P}_2\text{V}}{\text{RT}_2}\right)\text{RT}.\text{Divide both sides by }\text{V}:\text{2P} = \left(\frac{\text{P}_1}{\text{RT}_1} + \frac{\text{P}_2}{\text{RT}_2}\right)\text{RT}.\text{Divide both sides by R:}\text{2P} = \left(\frac{\text{P}_1}{\text{T}_1} + \frac{\text{P}_2}{\text{T}_2}\right)\text{T}.\text{To find the ratio }\frac{\text{P}}{\text{T}}\text{, rearrange the equation:}\frac{\text{P}}{\text{T}} = \frac{1}{2}\left(\frac{\text{P}_1}{\text{T}_1} + \frac{\text{P}_2}{\text{T}_2}\right) = \frac{\text{P}_1}{2\text{T}_1} + \frac{\text{P}_2}{2\text{T}_2}.\text{This matches option 4.}$

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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
}