Question

Choose the correct arrangement, where the symbols have their usual meanings
A. \[\overline u \] >\[{u_p}\] >\[{u_{rms}}\]
B. \[{u_{rms}}\]>\[\overline u \]>\[{u_p}\]
C. \[{u_p}\]>\[\overline u \]>\[{u_{rms}}\]
D. \[{u_p}\]>\[{u_{rms}}\]>\[\overline u \]>

Answer 1

Hint: Here, we have first to understand each of the velocities namely, rms velocity, probable velocity and average velocity. Then, we have to compare them using their equations. After that, we got to know which is the highest and lowest.

Complete Step by Step Solution:
Let’s first understand rms velocity.
Root mean square (rms) velocity defines square root of average velocity. The formula used for calculation of this velocity is \[{u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] , where, R stands for gas constant, T stands for temperature, M stands for molar mass of the gas.
Now we discuss average velocity. Average velocity defines the arithmetic mean calculating velocities of different gaseous molecules at a particular temperature. The average velocity can be find out by the formula \[\bar u = \sqrt {\dfrac{{8RT}}{{\pi M}}} \]

The velocity which is possessed by the most number of molecules at any given temperature is called the most probable velocity. Its mathematical expression is, \[{u_p} = \sqrt {\dfrac{{2RT}}{M}} \]
Now, we have to compare all the velocities.
\[{u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] \[\bar u = \sqrt {\dfrac{{8RT}}{{\pi M}}} \] \[{u_p} = \sqrt {\dfrac{{2RT}}{{\pi M}}} \]
  \[\sqrt 3 :\sqrt {\dfrac{8}{\pi }} :\sqrt 2 \]
And we know, \[\sqrt 3 = 1.73\],\[\sqrt 2 = 1.41\] and \[\sqrt {\dfrac{8}{\pi }} = 1.59\]
Therefore, \[{u_{rms}} > \bar u > {u_p}\]
Hence, option B is right.

Note: The laws of an ideal gas originated from the kinetic theory of gases. This theory states that the particles of gases are in constant motion. All particles move at different speeds. They undergo collisions with each other and keep changing their directions. All three velocities help us to know the molecular speed. The RMS velocity is always greater than the average velocity and probable velocity.