Find rms speed of Chlorine molecules at temperature `27^(@)C`.

Options

A. 459m/s
B. 849m/s
C. 900m/s
D. 478m/s

Explanation

To find the root mean square (RMS) speed of chlorine molecules at a temperature of \(27^\circ C\), we can follow these steps:

### Step 1: Convert the temperature from Celsius to Kelvin
The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]
For \(27^\circ C\):
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the molecular mass of chlorine
The molecular mass of chlorine (Cl) is approximately \(35.5 \, g/mol\). Since we need it in kilograms for our calculations, we convert it:
\[ M = 35.5 \, g/mol = 35.5 \times 10^{-3} \, kg/mol \]

### Step 3: Use the RMS speed formula
The formula for the RMS speed (\(v_{rms}\)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \(R\) is the gas constant, \(8.31 \, J/(K \cdot mol)\)
- \(T\) is the temperature in Kelvin
- \(M\) is the molar mass in kg/mol

### Step 4: Substitute the values into the formula
Now, substituting the values we have:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{35.5 \times 10^{-3} \, kg/mol}} \]

### Step 5: Calculate the value
Calculating the numerator:
\[ 3 \times 8.31 \times 300 = 7479 \, J \]
Now, calculating the denominator:
\[ 35.5 \times 10^{-3} = 0.0355 \, kg/mol \]
Now substituting these into the RMS speed formula:
\[ v_{rms} = \sqrt{\frac{7479}{0.0355}} \approx \sqrt{210,000} \approx 458.99 \, m/s \]

### Step 6: Final result
Thus, the RMS speed of chlorine molecules at \(27^\circ C\) is approximately:
\[ v_{rms} \approx 459 \, m/s \]