For gaseous state, if most probable speed is denoted by $$C^*$$, average speed by $$\bar{C}$$ and root mean square speed by C, then for many molecules, what is the ratios of these speeds?

In the kinetic molecular theory of gases, three characteristic molecular speeds are widely used:

$$\begin{aligned} &\text{Most probable speed (denoted }C^*\text{)} \\ &\text{Average or mean speed (denoted }\bar{C}\text{)} \\ &\text{Root-mean-square speed (denoted }C\text{)} \end{aligned}$$

For a large collection of molecules that obey the Maxwell-Boltzmann distribution, the well-established formulae for these speeds are:

$$ \boxed{C^* = \sqrt{\dfrac{2RT}{M}}}, \qquad \boxed{\bar{C} = \sqrt{\dfrac{8RT}{\pi M}}}, \qquad \boxed{C = \sqrt{\dfrac{3RT}{M}}} $$

Here

$$R =$$ universal gas constant $$, \qquad T =$$ absolute temperature, $$\qquad M =$$ molar mass of the gas.

To compare the three speeds, we form their ratios. We shall express each speed relative to the most probable speed $$C^*$$, taking $$C^*$$ as the unit reference.

Step 1: Ratio of the average speed to the most probable speed.

We have

$$ \dfrac{\bar{C}}{C^*} = \dfrac{\sqrt{\dfrac{8RT}{\pi M}}}{\sqrt{\dfrac{2RT}{M}}}. $$

Inside the big square root we put every factor together:

$$ \dfrac{\bar{C}}{C^*} = \sqrt{\dfrac{8RT}{\pi M}\;\cdot\;\dfrac{M}{2RT}} = \sqrt{\dfrac{8}{\pi}\;\cdot\;\dfrac{1}{2}} = \sqrt{\dfrac{8}{2\pi}} = \sqrt{\dfrac{4}{\pi}}. $$

Now we convert the fraction under the root into a numerical value:

$$ \dfrac{4}{\pi} \approx \dfrac{4}{3.1416} \approx 1.2732. $$

Taking the square root of this value, we get

$$ \dfrac{\bar{C}}{C^*} \approx \sqrt{1.2732} \approx 1.128. $$

Step 2: Ratio of the root-mean-square speed to the most probable speed.

We start similarly:

$$ \dfrac{C}{C^*} = \dfrac{\sqrt{\dfrac{3RT}{M}}}{\sqrt{\dfrac{2RT}{M}}} = \sqrt{\dfrac{3RT}{M}\;\cdot\;\dfrac{M}{2RT}} = \sqrt{\dfrac{3}{2}}. $$

The fraction under the square root is

$$ \dfrac{3}{2} = 1.5. $$

Therefore

$$ \dfrac{C}{C^*} \approx \sqrt{1.5} \approx 1.225. $$

Step 3: Writing the complete ratio.

We have determined

$$ C^* : \bar{C} : C = 1 : 1.128 : 1.225. $$

Step 4: Matching with the given options.

The option that exactly reproduces the ratio $$1 : 1.128 : 1.225$$ in the order $$C^* : \bar{C} : C$$ is Option A.

Hence, the correct answer is Option A.