Question

For gaseous state, if most probable speed is denoted by $c *$, average speed by $\overline{C}$ and root square speed by $c$, then for a large number of molecules, the ratios of these speeds are

• A. $c^* : \overline{c} : c = 1.225 : 1.128 : 1$
• B. $c^* : \overline{c} : c = 1.128 : 1.225 : 1$
• C. $c^* : \overline{c} : c = 1 : 1.128 : 1.225$
• D. $c^* : \overline{c} : c = 1 : 1.225 : 1.128$

Answer 1

The Correct Option is C

In the study of the kinetic theory of gases, three different types of molecular speeds are important: the most probable speed $c^*$, the average speed $\overline{c}$, and the root mean square speed $c$. These speeds are defined as follows:

1. Most Probable Speed ($c^*$): This is the speed at which the highest number of molecules in a gas move. It is given by the equation: $$c^* = \sqrt{\frac{2kT}{m}}$$ where $k$ is the Boltzmann constant, $T$ is the temperature, and $m$ is the mass of a gas molecule.
2. Average Speed ($\overline{c}$): This is the arithmetic mean speed of all the molecules in a gas, given by: $$\overline{c} = \sqrt{\frac{8kT}{\pi m}}$$
3. Root Mean Square Speed ($c$): This is the square root of the average of the squares of the speeds of all molecules, given by: $$c = \sqrt{\frac{3kT}{m}}$$

To find the relationship among these speeds, one can calculate the ratios:

1. $$\frac{\overline{c}}{c^*} = \sqrt{\frac{8}{\pi}} \approx 1.128$$
2. $$\frac{c}{c^*} = \sqrt{\frac{3}{2}} \approx 1.225$$

Therefore, comparing these ratios leads to the conclusion that:

$$c^* : \overline{c} : c = 1 : 1.128 : 1.225$$

Hence, the correct answer is:

$c^* : \overline{c} : c = 1 : 1.128 : 1.225$