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Current Question (ID: 11127)

Question:

$\text{How does the pressure of an ideal gas change during the process shown in the diagram?}$

Options:

1. $\text{pressure increases continuously.}$ (Correct)
2. $\text{pressure decreases continuously.}$
3. $\text{pressure first increases and then decreases.}$
4. $\text{pressure first decreases and then increases.}$

Solution:

$\text{We can analyze the change in pressure by examining the relationship between volume (V) and temperature (T) for an ideal gas.}\text{The ideal gas equation is: }\text{PV} = \text{nRT}.\text{We can rearrange this equation to express V as a function of T: }\text{V} = \left(\frac{\text{nR}}{\text{P}}\right)\text{T}.\text{This equation is in the form of a straight line, }\text{y} = \text{mx}\text{, where the volume }\text{V}\text{ is on the y-axis, the temperature }\text{T}\text{ is on the x-axis, and the slope }\text{m} = \frac{\text{nR}}{\text{P}}\text{.}\text{Since }\text{n}\text{ (number of moles) and }\text{R}\text{ (gas constant) are constants, the slope of the line is inversely proportional to the pressure P: }\text{slope} \propto \frac{1}{\text{P}}.\text{This means that as the slope of the V-T graph increases, the pressure decreases. Conversely, as the slope decreases, the pressure increases.}\text{Looking at the given diagram, the graph is a straight line from point 'a' to 'b'. This means the slope of the graph is constant.}\text{Since the slope }\left(\frac{\text{nR}}{\text{P}}\right)\text{ is constant and n and R are constants, the pressure }\text{P}\text{ must also be constant throughout the process.}\text{Therefore, the pressure neither increases nor decreases; it remains constant.}\text{However, since none of the given options state that the pressure is constant, we need to re-evaluate the problem based on the provided solution. The solution image states 'As the slope increases, the pressure decreases or vice-versa. So, pressure increases continuously.' This is a contradiction, as the graph is a straight line, meaning the slope is constant. The logic in the provided solution seems to have an error. A straight line segment from a to b on a V vs T graph indicates a constant slope, which means the pressure (P) is constant. There is no change in pressure.}\text{Let's re-examine the provided solution's conclusion: 'So, pressure increases continuously.' This is the correct answer according to the source, despite the apparent contradiction in the explanation. The source's explanation is flawed, but the correct answer is option 1, as deduced by the source. This is a common issue with some question banks. The provided explanation's final conclusion is used to determine the correct option.}$

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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
}