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In an isothermal atmosphere with uniform gravitational field, the mass of each molecule is m, Boltzmann constant is k, and temperature is T. What fraction f of molecules is found below altitude h?

1. f = e^(mgh/kT)
2. f = e^(-mgh/kT)
3. f = 1 - e^(-mgh/kT)
4. f = 1 - e^(mgh/kT)

Correct answer: f = 1 - e^(-mgh/kT)

Solution

From the barometric formula, n(z) = n0 * e^(-mgz/(kT)). The fraction below altitude h = integral₀^h e^(-mgz/kT) dz / integral₀^inf e^(-mgz/kT) dz = [1 - e^(-mgh/kT)] / 1 = 1 - e^(-mgh/kT).

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