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

Explanation

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

### Step-by-Step Solution:

1. **Convert Temperature to Kelvin**:
- The given temperature is 27°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
- Therefore:
\[ T = 27 + 273.15 = 300.15 \approx 300 \, K \]

2. **Identify the Molecular Mass of Potassium**:
- The molecular mass of potassium (K) is given as 39 g/mol. To use this in our calculations, we need to convert it to kilograms:
\[ M_K = 39 \, g/mol = 39 \times 10^{-3} \, kg/mol = 0.039 \, kg/mol \]

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

4. **Substitute the Values into the Formula**:
- Now, substituting the values into the rms speed formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{0.039 \, kg/mol}} \]

5. **Calculate the Numerator and Denominator**:
- Calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7473 \, J/mol \]
- Now, substitute this back into the formula:
\[ v_{rms} = \sqrt{\frac{7473}{0.039}} \]

6. **Calculate the Final Value**:
- Calculate the division:
\[ \frac{7473}{0.039} \approx 191,153.85 \]
- Now, take the square root:
\[ v_{rms} \approx \sqrt{191,153.85} \approx 437.91 \, m/s \]

### Final Answer:
The rms speed of potassium molecules at a temperature of 27°C is approximately **437.91 m/s**.

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