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28 September, 06:55

If it has enough kinetic energy, a moleculeat the surface of the Earth can escape the Earth's gravitation. The acceleration of gravity is 9.8 m/s2, and the Boltzmanns' constant is 1.38066*10-23 J/K. Using energy conservation, a) determine theminimum kinetic energy needed to escape interms of the mass of the molecule m, the free-fall acceleration at the surface g, and the radius of the Earth R

b) Calculate the temperature for which the min - imum escape energy is 11 times the average kinetic energy of an oxygen molecule.

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  1. 28 September, 07:08
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    a. mgR b. 483.6 * 10³ K

    Explanation:

    a. Using the law of conservation of energy,

    K₁ + U₁ = K₂ + U₂ where K₁ and K₂ are the initial and final kinetic energies of the molecule and U₁ and U₂ are the initial and final potential energies of the molecule.

    K₁ + - GMm/R = 0 + 0

    K₁ = GMm/R since GM = gR²,

    K₁ = mgR

    So its minimum kinetic energy is mgR

    b. The average kinetic energy energy of a gas molecule is K = 3/2kT. If K₁ = 11K, then

    mgR = 11 * 3/2kT = 33/2kT.

    T = 2mgR/k

    For oxygen molecule, m = 32 g/6.02 * 10²³ = 5.32 * 10⁻²⁶ kg/molecule

    T = 2 * 5.32 * 10⁻²⁶ kg * 9.8 m/s² 6.4 * 10⁶ m/1.38 10⁻²³ J/K = 483.6 * 10³ K
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