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Current Question (ID: 17924)

Question:

$\text{The relation between the atomic mass of unknown metal with density and the dimension of its unit cell is given by:}$ $\text{(Given: All the cell edge lengths are equal)}$

Options:

1. $m = \frac{\rho a^3 N_A}{Z}$
2. $m = \rho a^3 N_A Z$
3. $m = \frac{Z}{\rho a^3 N_A}$
4. $m = \frac{\rho a^3 N_A}{Z N_A a^3 \rho}$

Solution:

$\text{HINT: Density} = \frac{Z}{N_A} a^3$ $\text{STEP 1:}$ $\text{Let 'a' be the edge length of a unit cell of a crystal, '} ho\text{' be the density of the metal, 'm' be the atomic mass of the metal and 'Z' be the number of atoms in the unit cell.}$ $\text{Now, the density of the unit cell} = \frac{\text{mass of the unit cell}}{\text{volume of the unit cell}}$ $\Rightarrow \rho = \frac{Z}{m a^3} \ldots (i)$ $\text{STEP 2:}$ $\text{[Since the mass of the unit cell = Number of atoms in the unit cell} \times \text{Atomic mass]}$ $\text{[Volume of the unit cell = (Edge length of the cubic unit cell)}^3]$ $\text{From equation (i), we have: } m = \frac{\rho}{a^3 Z} \ldots (ii)$ $\text{STEP 3:}$ $\text{Now, the mass of the metal (m) = Atomic mass (M) Avogadro's number (} N_A \text{)}$ $\text{Thus, } m = \rho a^3 N_A Z \ldots (iii)$

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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
}