Question

What does the area under Maxwell distribution curve represent?

Answer 1

Hint : Maxwell Boltzmann distribution curve is used to determine the speed of gaseous molecules at a given temperature. With the help of this curve, we can determine the most probable speed, average velocity and root mean square velocity of the gaseous molecule.

Complete Step By Step Answer:
The Maxwell Boltzmann curve shows the variation of speed of gaseous molecules with the variation in temperature.
The curve is sketched according to Maxwell Boltzmann distribution law which states that the fraction of gaseous molecules is the function of its velocity, mass and temperature. The expression is as follows:
\dfrac{{dN}}{N} = {\left( {\dfrac{m}{{2\pi {k_B}T}}} \right)^{{\raise0.7ex\hbox{ 1 } \\!\mathord{\left/ {\vphantom {1 2}}\right.} \\!\lower0.7ex\hbox{ 2 }}}}{e^{\left( {\dfrac{{ - m{v^2}}}{{2{k_B}T}}} \right)}}dv
For the given expression:
$\dfrac{{dN}}{N} \Rightarrow$ fraction of gaseous molecules
$m \Rightarrow$ mass of the gaseous molecules
${k_b} \Rightarrow$ Boltzmann constant
$T \Rightarrow$ Temperature
Therefore, it is used to determine various forms of velocities for a gaseous molecule which are as follows:
Most probable velocity: It is the velocity when a fraction of molecules are maximum at a given temperature.
most probable velocity $= \sqrt {\dfrac{{2RT}}{M}}$
Average velocity: It is the sum of velocities of gaseous molecules divided by the number of gaseous molecules present.
average velocity $= \sqrt {\dfrac{{8RT}}{{\pi M}}}$
Root means square velocity: It is the square root of the sum of square velocities of gaseous molecules divided by the number of gaseous molecules present.
r.m.s velocity $= \sqrt {\dfrac{{3RT}}{M}}$
As the curve is sketched between a fraction of molecules and temperature, the value on the curve represents velocity. Hence the area under the Maxwell distribution curve represents the number of molecules per unit speed.

Note :
The number of molecules decreases on increasing temperature but the velocity increases. Therefore, the correct order of velocities is r.m.s velocity $>$ average velocity $>$ most probable velocity.