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In an isothermal atmosphere under uniform gravity, molecules of mass m are distributed with altitude. Using Boltzmann constant k_B and temperature T, what fraction f of molecules lies below altitude h?

1. f = e^(-m*g*h / (k_B*T))
2. f = e^(-m*g*h / (k_B*T))
3. f = 1 - e^(-m*g*h / (k_B*T))
4. f = 1 - e^(m*g*h / (k_B*T))

Correct answer: f = 1 - e^(-m*g*h / (k_B*T))

Solution

Integrating the Boltzmann distribution from 0 to h gives n0*k_BT/mg * (1 - e^(-mgh/k_BT)); dividing by the total (n0*k_BT/mg) yields f = 1 - e^(-mgh/k_BT).

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