## Maxwell distribution law

November 11th, 2016

Maxwell distribution of velocities of the molecules has the form:

$f(v)=(\frac{m_0}{2\pi kT})^\frac{3}{2}exp(-\frac{m_0 v^2)}{2 kT}) 4 \pi v^2$ (1)

This distribution allows us to determine the mean speed, the mean square speed and the most probable speed of the gas molecules.

Let’s find the mean speed by formula (2):

$\langle v \rangle=\int_{0}^{\infty} vf(v)dv$ (2)

Using the table integral for  (2), we obtaine:

$\langle v \rangle=\sqrt \frac {8kT}{\pi m_0}$  (3)

Let’s find the mean square speed by formula (4):

$\langle v^2 \rangle=\int_{0}^{\infty} v^2f(v)dv$ (4)

Using the table integral for (4), we obtaine: $\langle v^2 \rangle=\sqrt\frac {3kT}{\ m_0}$    (5)

Let’s find the the most probable speed from the condition (6):

$\frac{df(v)}{dv}=0$  (6)

We obtaine from (6): $v_p=\sqrt\frac {2kT}{\ m_0}$    (7)

(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. Knowing the distribution function of the molecules by velocities in some molecular volume: $f(v)=cexp(-\frac{m_0 v^2)}{2 kT}) v^3$. Find: normalization constant; the expression for the most probably velocity; the expression for the average velocity.

Task 2. Define the part of the molecules with velocities different from $2v_p$ not more than for 1%.

Task 3. Hydrogen is in under normal conditionsand occupies volume $V=1 cm^3$. Define the number $\Delta N$ of molecules in this volume with velocity less than some value $v_{max}=1\frac{m}{s}$.

Task 4. Find the probability that this molecule of gas has a velocity different from $0.5 v_p$ not more than for 1%.

Task 5. Find an expression for the most probable kinetic energy $\langle \varepsilon_k \rangle$ of the motion of molecules.