Boltzmann distribution

November 10th, 2016

Let’s consider a gas that is in the gravity field (fig.1).


Fig. 1. Ideal gas in the gravity field.

Let’s  we write that pressure has a gas  at altitudes z (1) and z-dz (2):

p=\rho g z      (1)

p+dp = \rho g (z-dz)      (2)

Let’s find dp from the equations (1) and (2):

dp=\rho g (z-dz) - \rho g z =-\rho g dz (3)

Let’s substitute to the equation (3) the expression for \rho obtained from the ideal gas law \rho=\frac{p \mu}{RT}:

dp=-\frac{p \mu}{RT}gdz (4)

Let’s perform the separation of variables:

\frac{dp}{p}=-\frac{\mu g}{RT}dz (5)

Let’s  integrate the equation (5):

\int \frac{dp}{p}=-\frac{\mu g}{RT}\int dz (6)

We receive from the equation (6):

lnp=-\frac{\mu g z}{RT}+const (7)

Let’s represent a constant in the form lnp_0, then we obtain the barometric formula:

p=p_0 exp (-\frac{\mu g z }{RT}) (8)

Given that p=nkT and p_0=n_kT we obtaine the Boltzmann distribution:

n=n_0 exp (-\frac{\mu g z }{RT})  (9)

(The information has been taken from e-book «Fundamentals of classical physics” (author V.D. Shvets), which was published in 2007 by “1C-Multimedia Ukraine” publishing company. Copyright is reserved by certificate Number 55955 from 06.08.2014.More information can be found at


Task 1. Barometer in the cabin of the helicopter, flying at the height h, shows the pressure p =90 kPa. At what height the helicopter is flying if the barometer showed pressure of 100 kPa at the runway? Consider that the air temperature T is equal to 290 K and does not change with height.

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Task 2. Rotor of a centrifuge is rotating with angular velocity \omega. Using the Boltzmann distribution function, set the distribution of concentration of particles as a function of a distance from the axis of rotation.

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Task 3. Mass of each of the dust particles suspended in the air, equals to m . Find the Avogadro number, if the relation of concentrations of the dust at its height h , on the surface of the Earth, and the air temperature T are known.

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Task 4. Find the height h , corresponding to the change of pressure \Delta p, if you know the pressure at the surface of the Earth p_0 and air temperature T .

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Task 5. At what height air density is 53.2% of the density at sea level? The temperature should be considered as constant and equal to 27^0C .

(The tasks has been taken from book «Elements of Statistical Physics” (author V.D. Shvets), which was published in 2002 by “University Ukraine” publishing house. This book has Recommendation of Ministry Education of Ukraine № 44 from 14.01.1999. More information can be found at


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