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

Explanation

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

### Step 1: Convert the Temperature to Kelvin
The temperature in Celsius needs to be converted to Kelvin. The formula for conversion is:
\[ T(K) = T(°C) + 273 \]
Given:
\[ T(°C) = 27 \]
So,
\[ T(K) = 27 + 273 = 300 \, K \]

### Step 2: Identify the Molecular Mass of Magnesium
The molecular mass of magnesium (Mg) is given as 24 g/mol. To use it in the RMS speed formula, we need to convert it to kilograms:
\[ \text{Molecular Mass (kg)} = 24 \, g/mol \times \frac{1 \, kg}{1000 \, g} = 0.024 \, kg/mol \]

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

### Step 4: Substitute the Values into the Formula
Substituting the values into the formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{0.024 \, kg/mol}} \]

### Step 5: Calculate the RMS Speed
Now, let's calculate:
1. Calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7488.9 \, J/mol \]
2. Divide by the molecular mass:
\[ \frac{7488.9}{0.024} = 311203.75 \, m^2/s^2 \]
3. Take the square root:
\[ v_{rms} = \sqrt{311203.75} \approx 558.23 \, m/s \]

### Final Answer
The RMS speed of magnesium molecules at a temperature of 27°C is approximately:
\[ v_{rms} \approx 558.23 \, m/s \]

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