## Boltzmann distribution law

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 https://ipood-kiev.academia.edu/ValentynaShvets/Books)

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.

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.

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.

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 .

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 $27^0C$ .