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Current Question (ID: 19537)

Question:

$\text{The normal density of a material is } \rho \text{ and its bulk modulus of elasticity is } K. \text{ The magnitude of the increase in the density of material when a pressure } P \text{ is applied uniformly on all sides, will be:}$

Options:

1. $\frac{\rho K}{P}$
2. $\frac{\rho P}{K}$
3. $\frac{K}{P \rho}$
4. $\frac{P K}{\rho}$

Solution:

$\text{Hint: } K = -\frac{\Delta P}{\Delta V / V}$ $\text{Step: Find the magnitude of the increase in density of the liquid.}$ $\text{The density of the liquid is given by:}$ $\rho = \frac{M}{V}$ $\Rightarrow \Delta \rho = -\frac{M}{V^2} \Delta V = -\frac{M}{V} \frac{\Delta V}{V}$ $\Rightarrow \Delta \rho = -\rho \frac{\Delta V}{V}$ $\Rightarrow \frac{\Delta \rho}{\rho} = -\frac{\Delta V}{V}$ $\text{The bulk modulus of the liquid is given by:}$ $K = -\frac{P}{\Delta V / V}$ $\Rightarrow K = -\frac{P}{-\Delta \rho / \rho} \left[ \frac{\Delta V}{V} = -\frac{\Delta \rho}{\rho} \right]$ $\Rightarrow \frac{\Delta \rho}{\rho} = \frac{P}{K}$ $\Rightarrow \Delta \rho = \frac{\rho P}{K}$ $\text{Therefore, the increase in density of the liquid is } \frac{\rho P}{K}. \text{ Hence, option (2) is the correct answer.}$

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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
}