Maxwell distributions

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 ).

TASKS

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.

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

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

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

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

   

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