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

Options

A. 1366m/s
B. 1166m/s
C. 13.66m/s
D. 1.366m/s

Explanation

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

### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Kelvin can be calculated using the formula:
\[ T(K) = T(°C) + 273 \]
Given \(T = 27^\circ C\):
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the molar mass of helium
The molar mass of helium is given as \(4 \, g/mol\). To use it in our calculations, we need to convert it to kilograms:
\[ m = 4 \, g/mol = 4 \times 10^{-3} \, kg/mol \]

### 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, which we found to be \(300 \, K\)
- \(M\) is the molar mass in kg/mol, which we found to be \(4 \times 10^{-3} \, kg/mol\)

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

### Step 5: Calculate the value inside the square root
First, calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7479 \, J/mol \]
Now, divide by the molar mass:
\[ \frac{7479}{4 \times 10^{-3}} = 1869750 \, m^2/s^2 \]

### Step 6: Take the square root
Now, take the square root to find the rms speed:
\[ v_{rms} = \sqrt{1869750} \approx 1366.5 \, m/s \]

### Final Answer
The rms speed of helium molecules at \(27^\circ C\) is approximately \(1366.5 \, m/s\).

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