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Current Question (ID: 8174)

Question:

Find the mass of payload of a balloon of radius 10 m, if a mass of 100 kg is filled with helium at 1.66 bar and 27°C temperature. (Density of air = 1.2 kg/m³ and R = 0.083 bar·dm³·K⁻¹·mol⁻¹).

Options:

1. 2905 Kg
2. 3811 Kg
3. 2721 Kg
4. 4201 Kg

Solution:

**Step 1:** Calculate the volume of the balloon. - Radius (r) = 10 m - Volume (V) = (4/3) × π × r³ = (4/3) × (22/7) × 10³ = 4190.5 m³ **Step 2:** Determine the mass of the displaced air. - Density of air = 1.2 kg/m³ - Mass of displaced air = 4190.5 × 1.2 = 5028.6 kg **Step 3:** Calculate the mass of helium in the balloon using the ideal gas equation. - Molar mass of helium (M) = 4 g/mol = 4 × 10⁻³ kg/mol - Pressure (p) = 1.66 bar - Volume (V) = 4190.5 m³ = 4190.5 × 10³ dm³ - Temperature (T) = 27°C = 300 K - Gas constant (R) = 0.083 bar·dm³·K⁻¹·mol⁻¹ m_He = (M × p × V)/(R × T) = (4 × 10⁻³ × 1.66 × 4190.5 × 10³)/(0.083 × 300) ≈ 1117.5 kg **Step 4:** Compute the total mass of the balloon (including helium and the balloon mass). - Mass of balloon = 100 kg - Total mass = 100 + 1117.5 = 1217.5 kg **Step 5:** Determine the payload. - Payload = Mass of displaced air - Total mass of balloon = 5028.6 - 1217.5 = 3811.1 kg Hence, the payload of the balloon is **3811 kg**.

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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
}