At `27^@C`, the ratio of RMS velocities of ozone to oxygen is

Options

A. `sqrt(3//5)`
B. `sqrt(4//3)`
C. `sqrt(2//3)`
D. 0.25.

Explanation

To find the ratio of the root mean square (RMS) velocities of ozone (O₃) to oxygen (O₂) at 27°C, we can follow these steps:

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

### Step 2: Identify the molar masses
- The molar mass of ozone (O₃) is 48 g/mol.
- The molar mass of oxygen (O₂) is 32 g/mol.

### Step 3: Set up the ratio of RMS velocities
The ratio of the RMS velocities of ozone to oxygen can be expressed as:
\[ \frac{v_{RMS, O_3}}{v_{RMS, O_2}} = \frac{\sqrt{\frac{3RT}{M_{O_3}}}}{\sqrt{\frac{3RT}{M_{O_2}}}} \]
Since \( 3RT \) is a common factor in both the numerator and the denominator, it cancels out:
\[ \frac{v_{RMS, O_3}}{v_{RMS, O_2}} = \sqrt{\frac{M_{O_2}}{M_{O_3}}} \]

### Step 4: Substitute the molar masses into the equation
Now substituting the molar masses:
\[ \frac{v_{RMS, O_3}}{v_{RMS, O_2}} = \sqrt{\frac{32}{48}} \]

### Step 5: Simplify the ratio
We can simplify the fraction:
\[ \frac{32}{48} = \frac{2}{3} \]
Thus, we have:
\[ \frac{v_{RMS, O_3}}{v_{RMS, O_2}} = \sqrt{\frac{2}{3}} \]

### Step 6: Final result
Therefore, the ratio of the RMS velocities of ozone to oxygen is:
\[ \frac{v_{RMS, O_3}}{v_{RMS, O_2}} = \sqrt{\frac{2}{3}} \]