Question

If the molecule could move upward without colliding with other molecules, then how high would it go before coming to rest? Give your answer in kilometers. [Hint: When the molecule comes to rest, the potential energy of the molecule will be m⋅g⋅h, where m is the molecular mass in kilograms, g=9.81m⋅s−2 is the acceleration due to gravity, and h is the height, in meters, above Earth's surface.]
Express your answer to three significant figs,
Note: A nitrogen molecule (N2) having the average kinetic energy at 330 K is released from Earth's surface to travel upward.

Answer 1

The maximum height at which nitrogen molecule will go before coming to rest is 14 kilometers.

Given:

The nitrogen gas molecule with a temperature of 330 Kelvins is released from Earth's surface to travel upward.

To find:

The maximum height of a nitrogen molecule when released from the Earth's surface before coming to rest.

Solution:

• The maximum height attained by nitrogen gas molecule = h
• The temperature of nitrogen gas particle = T = 330 K

The average kinetic energy of the gas particles is given by:

$K.E=\frac{3}{2}K_bT\\\\K.E=\frac{3}{2}\times 1.38\times 10^{-23} J/K\times 330 K\\\\K.E=6.381\times 10^{-21} J$

The nitrogen molecule at its maximum height will have zero kinetic energy as all the kinetic energy will get converted into potential energy

• The potential energy at height h = $P.E = 6.381\times 10^{-21} J$
• Molar mass of nitrogen gas =  28.0134 g/mol
• Mass of nitrogen gas molecule = m

$m= \frac{ 28.0134 g/mol}{6.022\times 10^{23} mol^{-1}}=4.652\times 10^{-23} g\\\\1g=0.001kg\\\\m=4.652\times 10^{-23}\times 0.001 kg\\\\=4.652\times 10^{-26} kg$

• The acceleration due to gravity = g = 9.8 m/s^2
• The maximum height attained by nitrogen gas molecule = h
• The potential energy is given by:

$P.E=mgh$

$6.381\times 10^{-21} J=4.652\times 10^{-26} kg\times 9.8 m/s^2\times h\\\\h=\frac{6.381\times 10^{-21} J}{4.652\times 10^{-26} kg\times 9.8 m/s^2}\\\\h=13,996.6 m\\\\1 m = 0.001 km\\\\h=13,996.6 m=h=13,996.6\times 0.001 k m\\\\=13.9966 km \approx 14 km$

The maximum height at which nitrogen molecule will go before coming to rest is 14 kilometers.

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