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

Explanation

To find the root mean square (rms) speed of sodium molecules at a temperature of 27°C, we can follow these steps:

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

### Step 2: Identify the molecular mass of sodium
The molecular mass of sodium (Na) is approximately 23 g/mol. To use it in the rms speed formula, we need to convert it to kilograms per mole:
\[ m = 23 \, g/mol = 23 \times 10^{-3} \, kg/mol \]

### Step 3: Use the rms speed formula
The rms speed (\( v_{rms} \)) can be calculated using the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{m}} \]
Where:
- \( R \) is the gas constant, approximately \( 8.31 \, J/(K \cdot mol) \)
- \( T \) is the temperature in Kelvin (300 K)
- \( m \) is the molecular mass in kg/mol (23 x 10^-3 kg/mol)

### Step 4: Substitute the values into the formula
Now we substitute the values into the rms speed formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{23 \times 10^{-3} \, kg/mol}} \]

### Step 5: Calculate the value inside the square root
First, calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7473 \, J/mol \]

Now calculate the denominator:
\[ 23 \times 10^{-3} = 0.023 \, kg/mol \]

Now divide the numerator by the denominator:
\[ \frac{7473}{0.023} \approx 324,826.087 \]

### Step 6: Take the square root
Now take the square root of the result:
\[ v_{rms} = \sqrt{324,826.087} \approx 570.24 \, m/s \]

### Final Answer
The rms speed of sodium molecules at a temperature of 27°C is approximately **570.24 m/s**.

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