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](https://tex.z-dn.net/?f=K.E%3D%5Cfrac%7B3%7D%7B2%7DK_bT%5C%5C%5C%5CK.E%3D%5Cfrac%7B3%7D%7B2%7D%5Ctimes%201.38%5Ctimes%2010%5E%7B-23%7D%20J%2FK%5Ctimes%20330%20K%5C%5C%5C%5CK.E%3D6.381%5Ctimes%2010%5E%7B-21%7D%20J)
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](https://tex.z-dn.net/?f=P.E%20%3D%206.381%5Ctimes%2010%5E%7B-21%7D%20J)
- 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](https://tex.z-dn.net/?f=m%3D%20%5Cfrac%7B%2028.0134%20g%2Fmol%7D%7B6.022%5Ctimes%2010%5E%7B23%7D%20mol%5E%7B-1%7D%7D%3D4.652%5Ctimes%2010%5E%7B-23%7D%20g%5C%5C%5C%5C1g%3D0.001kg%5C%5C%5C%5Cm%3D4.652%5Ctimes%2010%5E%7B-23%7D%5Ctimes%200.001%20kg%5C%5C%5C%5C%3D4.652%5Ctimes%2010%5E%7B-26%7D%20kg)
- 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](https://tex.z-dn.net/?f=P.E%3Dmgh)
![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](https://tex.z-dn.net/?f=6.381%5Ctimes%2010%5E%7B-21%7D%20J%3D4.652%5Ctimes%2010%5E%7B-26%7D%20kg%5Ctimes%209.8%20m%2Fs%5E2%5Ctimes%20h%5C%5C%5C%5Ch%3D%5Cfrac%7B6.381%5Ctimes%2010%5E%7B-21%7D%20J%7D%7B4.652%5Ctimes%2010%5E%7B-26%7D%20kg%5Ctimes%209.8%20m%2Fs%5E2%7D%5C%5C%5C%5Ch%3D13%2C996.6%20m%5C%5C%5C%5C1%20m%20%3D%200.001%20km%5C%5C%5C%5Ch%3D13%2C996.6%20m%3Dh%3D13%2C996.6%5Ctimes%200.001%20k%20m%5C%5C%5C%5C%3D13.9966%20km%20%5Capprox%2014%20km)
The maximum height at which nitrogen molecule will go before coming to rest is 14 kilometers.
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