Calculate the root mean square velocity of ozone kept in a closed vessel at `20^(@)C` and `82 cm Hg` pressure.

Explanation

To calculate the root mean square (RMS) velocity of ozone gas in a closed vessel at a temperature of 20°C and a pressure of 82 cm Hg, we will follow these steps:

### Step 1: Convert the Temperature to Kelvin
The temperature in Celsius needs to be converted to Kelvin using the formula:
\[ T(K) = T(°C) + 273.15 \]
For 20°C:
\[ T = 20 + 273.15 = 293.15 \, K \]
**Hint:** Remember to always convert Celsius to Kelvin by adding 273.15.

### Step 2: Convert the Pressure from cm Hg to atm
To use the ideal gas constant R in appropriate units, we need to convert the pressure from cm Hg to atm. The conversion factor is:
\[ 1 \, atm = 76 \, cm \, Hg \]
Thus, for 82 cm Hg:
\[ P = \frac{82 \, cm \, Hg}{76 \, cm \, Hg/atm} = 1.0789 \, atm \]
**Hint:** Use the conversion factor to change pressure units when necessary.

### Step 3: Determine the Molar Mass of Ozone
The molar mass of ozone (O₃) is given as 48 g/mol. We need to convert this to kg/mol for consistency with SI units:
\[ M = 48 \, g/mol = 48 \times 10^{-3} \, kg/mol \]
**Hint:** Always convert mass to kg when working with the gas constant in SI units.

### Step 4: Use the RMS Velocity Formula
The formula for the root mean square velocity (u_rms) is:
\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]
Where:
- \( R = 8.314 \, J/(mol \cdot K) \)
- \( T = 293.15 \, K \)
- \( M = 48 \times 10^{-3} \, kg/mol \)

Substituting the values into the formula:
\[ u_{rms} = \sqrt{\frac{3 \times 8.314 \, J/(mol \cdot K) \times 293.15 \, K}{48 \times 10^{-3} \, kg/mol}} \]

### Step 5: Calculate the RMS Velocity
Calculating the numerator:
\[ 3 \times 8.314 \times 293.15 = 7300.85 \, J/mol \]
Now, divide by the molar mass:
\[ \frac{7300.85}{48 \times 10^{-3}} = 152.1 \times 10^{3} \, m^2/s^2 \]
Taking the square root:
\[ u_{rms} = \sqrt{152100} \approx 390.2 \, m/s \]

### Final Answer
The root mean square velocity of ozone is approximately:
\[ u_{rms} \approx 390.2 \, m/s \]