Question

For gaseous state, if most probable speed is denoted by $C^{\ast }$, average speed by $$\bar{C}$$ and root mean square speed by C, then for large number of molecules the ratio of these speeds are:
a- $C^{\ast }$: $\bar{C}$ : C = 1.128 : 1.225 : 1
b- $C^{\ast }$: $\bar{C}$ : C = 1 : 1.128 : 1.225
c- $C^{\ast }$: $\bar{C}$ : C = 1 :1.125 : 1.128
d- $C^{\ast }$: $\bar{C}$ : C = 1.125 : 1.128 : 1

Answer 1

Hint: The given three representations of velocity are related to each other by following formula:
$C^{\ast }$: $\bar{C}$ : C = $\sqrt{{\displaystyle \frac{2RT}{M}}}$ : $\sqrt{{\displaystyle \frac{8RT}{\pi M}}}$: $\sqrt{{\displaystyle \frac{3RT}{M}}}$

Complete answer:
Kinetic Molecular Theory explains the macroscopic properties of gases and can be used to calculate the different kinds of gaseous speeds. Measuring the velocities of particles at a given time results in a large distribution of values; some particles may move very slowly, others very quickly, and because they are constantly moving in different directions, the velocity could equal zero.
-Most probable speed ($C^{\ast }$): The most probable speed is the speed most likely to be possessed by any molecule of the same mass m in the gaseous system and corresponds to the maximum value. And it is given by the following formula:
C*= $\sqrt{{\displaystyle \frac{2RT}{M}}}$
-Average speed ($$\bar{C}$$): Movement of gaseous molecules is in random speeds and in random directions. The Maxwell-Boltzmann Distribution describes the average speeds of collection gaseous particles at a given temperature. And it is given by:
$$\bar{C}$$=$\sqrt{{\displaystyle \frac{8RT}{\pi M}}}$
-Root mean square speed (C or $C_{rms}$): As the name represents roots of mean of squares of the velocities is known as the root-mean-square (RMS) velocity, and it is represented as follows:
C or $C_{rms}$=$\sqrt{{\displaystyle \frac{3RT}{M}}}$
On comparing these three velocities:
$C^{\ast }$: $\bar{C}$ : C = $\sqrt{{\displaystyle \frac{2RT}{M}}}$ : $\sqrt{{\displaystyle \frac{8RT}{\pi M}}}$: $\sqrt{{\displaystyle \frac{3RT}{M}}}$
=$\sqrt{2}:\sqrt{{\displaystyle \frac{8}{\pi }}}:\sqrt{3}$
=1 : 1.128 : 1.225

So, the correct option is (B) 1: 1.128: 1.225 .

Note:
Most probable speed, average speed, and root mean square speed are related to each other by the formula stated above.Their relation can be expressed as a graph as shown below: