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Current Question (ID: 19845)

Question:

$\text{Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom. The gas is maintained at a temperature of } T. \text{ The total internal energy, } U \text{ of a mole of this gas, and the value of } \gamma \left( = \frac{C_P}{C_V} \right) \text{ are, respectively:}$

Options:

1. $U = 5RT \text{ and } \gamma = \frac{7}{5}$
2. $U = \frac{5}{2}RT \text{ and } \gamma = \frac{6}{5}$
3. $U = 5RT \text{ and } \gamma = \frac{6}{5}$
4. $U = \frac{5}{2}RT \text{ and } \gamma = \frac{7}{5}$

Solution:

$\text{Hint: } U = \frac{n f RT}{2}$ $\text{Step: Find the total internal energy and the value of } \gamma \left( = \frac{C_P}{C_V} \right)$ $\text{The total internal energy } (U) \text{ and the adiabatic index for a gas with three translational degrees of freedom and two rotational degrees of freedom.}$ $\text{Total degrees of freedom; } f = 3 + 2 = 5.$ $\text{The internal energy for one mole of gas is given by;}$ $\Rightarrow U = \frac{f}{2}RT$ $\Rightarrow U = \frac{5}{2}RT$ $\text{The adiabatic index } \gamma \text{ is given by;}$ $\Rightarrow \gamma = \frac{C_P}{C_V}$ $\Rightarrow \gamma = 1 + \frac{2}{f}$ $\Rightarrow \gamma = 1 + \frac{2}{5} = \frac{7}{5}$ $\text{Hence, option (4) is the correct answer.}$

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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
}