At `27^(@)C` , the ratio of rms speed of ozone to that of oxygen is :

Options

A. `sqrt(3//5)`
B. `sqrt(4//3)`
C. `sqrt(2//3)`
D. `sqrt(1//16)`

Explanation

To solve the problem of finding the ratio of the root mean square (RMS) speed of ozone (O₃) to that of oxygen (O₂) at 27°C, we will follow these steps:

### Step 1: Understand the formula for RMS speed
The RMS speed (U_RMS) of a gas is given by the formula:
\[ U_{RMS} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 2: Identify the molar masses
- The molar mass of oxygen (O₂) is:
\[ M_{O_2} = 2 \times 16 \, \text{g/mol} = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol} \]

- The molar mass of ozone (O₃) is:
\[ M_{O_3} = 3 \times 16 \, \text{g/mol} = 48 \, \text{g/mol} = 0.048 \, \text{kg/mol} \]

### Step 3: Write the ratio of RMS speeds
The ratio of the RMS speeds of ozone to oxygen can be expressed as:
\[ \frac{U_{RMS, O_3}}{U_{RMS, O_2}} = \frac{\sqrt{\frac{3RT}{M_{O_3}}}}{\sqrt{\frac{3RT}{M_{O_2}}}} \]

### Step 4: Simplify the ratio
Since \( R \) and \( T \) are constant for both gases, they cancel out:
\[ \frac{U_{RMS, O_3}}{U_{RMS, O_2}} = \sqrt{\frac{M_{O_2}}{M_{O_3}}} \]

### Step 5: Substitute the molar masses
Now substituting the values of \( M_{O_2} \) and \( M_{O_3} \):
\[ \frac{U_{RMS, O_3}}{U_{RMS, O_2}} = \sqrt{\frac{0.032}{0.048}} = \sqrt{\frac{32}{48}} = \sqrt{\frac{2}{3}} \]

### Step 6: Final answer
Thus, the ratio of the RMS speed of ozone to that of oxygen is:
\[ \frac{U_{RMS, O_3}}{U_{RMS, O_2}} = \sqrt{\frac{2}{3}} \]