For gaseous state, if most probable speed is denoted by `C^(**)` average speed by `barC` and root square speed by `C`, then for a large number of molecules, the ratios of these speeds are

Options

A. `C^(**):overset(-)C:C=1:1.225:1.128`
B. `C^(**):overset(-)C:C=1.128:1.225:1`
C. `C^(**):overset(-)C:C=1:1.128:1.225`
D. `C^(**):overset(-)C:C=1.225:1.128:1`

Answer

To solve the problem regarding the ratios of the most probable speed, average speed, and root mean square speed for a gaseous state, we will follow these steps:

### Step 1: Define the Speeds
We need to recall the formulas for the three types of speeds in a gas:
1. Most Probable Speed (\(C^{**}\)):
\[ C^{**} = \sqrt{\frac{2RT}{M}} \]
2. Average Speed (\(\bar{C}\)):
\[ \bar{C} = \sqrt{\frac{8RT}{\pi M}} \]
3. Root Mean Square Speed (\(C\)):
\[ C = \sqrt{\frac{3RT}{M}} \]

### Step 2: Set Up the Ratios
We need to find the ratio of these speeds:
\[ C^{**} : \bar{C} : C \]
This translates to:
\[ \sqrt{\frac{2RT}{M}} : \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{3RT}{M}} \]

### Step 3: Simplify the Ratios
To simplify the ratios, we can express them in a common format. We can factor out \(\sqrt{RT/M}\):
\[ \sqrt{RT/M} \left( \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \right) \]
Now we will focus on simplifying the ratios inside the parentheses:
\[ \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

### Step 4: Calculate Each Component
1. \(\sqrt{2} \approx 1.414\)
2. \(\sqrt{8} = 2\sqrt{2} \approx 2.828\), thus \(\sqrt{\frac{8}{\pi}} \approx \frac{2.828}{\sqrt{3.14}} \approx 1.607\)
3. \(\sqrt{3} \approx 1.732\)

### Step 5: Form the Final Ratios
Now we can express the ratios numerically:
\[ C^{**} : \bar{C} : C \approx 1.414 : 1.607 : 1.732 \]
To make it easier to interpret, we can divide each term by the smallest value (1.414):
\[ 1 : \frac{1.607}{1.414} : \frac{1.732}{1.414} \approx 1 : 1.136 : 1.225 \]

### Final Answer
Thus, the ratios of the most probable speed, average speed, and root mean square speed are approximately:
\[ 1 : 1.136 : 1.225 \]