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

Options

A. 731m/s
B. 831m/s
C. 781m/s
D. 631m/s

Answer

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

### Step 1: Convert the temperature to Kelvin
The temperature in Kelvin is calculated by adding 273 to the Celsius temperature.

\[ T = 27 + 273 = 300 \, K \]

**Hint:** Remember that to convert Celsius to Kelvin, you add 273.

### Step 2: Identify the molecular mass of nitrogen
Nitrogen (N₂) has a molecular mass of 28 g/mol (since each nitrogen atom has an atomic mass of approximately 14 g/mol).

To use the formula, we need to convert this mass into kilograms:

\[ m = 28 \, g/mol = 28 \times 10^{-3} \, kg/mol \]

**Hint:** Always convert grams to kilograms when using SI units.

### 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, which is \(8.31 \, J/(K \cdot mol)\)
- \(T\) is the temperature in Kelvin
- \(m\) is the molecular mass in kg/mol

### Step 4: Substitute the values into the formula
Now, we can substitute the values into the rms speed formula:

\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \times 300}{28 \times 10^{-3}}} \]

### Step 5: Calculate the value
First, calculate the numerator:

\[ 3 \times 8.31 \times 300 = 7473 \]

Now, calculate the denominator:

\[ 28 \times 10^{-3} = 0.028 \]

Now substitute these values back into the formula:

\[ v_{rms} = \sqrt{\frac{7473}{0.028}} \approx \sqrt{266,125} \approx 516.0 \, m/s \]

### Step 6: Final calculation
After calculating the square root:

\[ v_{rms} \approx 516.0 \, m/s \]

### Conclusion
The rms speed of nitrogen molecules at \(27^\circ C\) is approximately \(516.0 \, m/s\).

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