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Molecular Speeds & Maxwell Distribution question from NEET 2018

Q83NEET 2018Medium

Molecular Speeds & Maxwell Distribution

At what temperature will the rms speed of oxygen molecules become just sufficient for escaping from the Earth’s atmosphere?

(Given: Mass of oxygen molecule (m) = 2.76 × 10–26 kg, Boltzmann’s constant kB = 1.38 × 10–23 J K–1)

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Answer: (b)

  1. Set vrms=3kBTmv_{rms} = \sqrt{\frac{3k_BT}{m}} equal to escape speed ve=11.2 km/sv_e = 11.2\text{ km/s}
  2. T=mve23kB=2.76×10−26×(1.12×104)23×1.38×10−23T = \frac{m v_e^2}{3k_B} = \frac{2.76\times10^{-26}\times(1.12\times10^4)^2}{3\times1.38\times10^{-23}}
  3. T=3.46×10−184.14×10−23≈8.36×104 KT = \frac{3.46\times10^{-18}}{4.14\times10^{-23}} \approx 8.36\times10^4\text{ K}
  4. So the answer is (b) 8.360×104 K8.360\times10^4\text{ K}.
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