The rms speed of helium in `ms^(-1)` (atomic mass = 4.0g `mol^(-1)`) at 400K is

Options

A. 1580
B. 15.8
C. 28
D. 158

Answer

To calculate the root mean square (rms) speed of helium at a temperature of 400 K, we can use the formula for rms speed:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass in kg/mol.

### Step 1: Identify the values needed for the calculation
- Given:
- Atomic mass of helium = 4.0 g/mol
- Temperature \( T = 400 \) K
- Convert the molar mass from g/mol to kg/mol:
\[ M = 4.0 \, \text{g/mol} = 4.0 \times 10^{-3} \, \text{kg/mol} \]

### Step 2: Use the value of the universal gas constant
- The value of \( R \) in Joules per mole per Kelvin is:
\[ R = 8.31 \, \text{J/(mol K)} \]

### Step 3: Substitute the values into the rms speed formula
Now, substitute \( R \), \( T \), and \( M \) into the rms speed formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, \text{J/(mol K)} \times 400 \, \text{K}}{4.0 \times 10^{-3} \, \text{kg/mol}}} \]

### Step 4: Calculate the numerator
Calculate the numerator:
\[ 3 \times 8.31 \times 400 = 9996 \, \text{J/mol} \]

### Step 5: Calculate the denominator
The denominator is:
\[ 4.0 \times 10^{-3} \, \text{kg/mol} \]

### Step 6: Divide the numerator by the denominator
Now, divide the numerator by the denominator:
\[ \frac{9996}{4.0 \times 10^{-3}} = 2499000 \, \text{m}^2/\text{s}^2 \]

### Step 7: Take the square root
Finally, take the square root to find \( v_{rms} \):
\[ v_{rms} = \sqrt{2499000} \approx 1580 \, \text{m/s} \]

### Conclusion
The rms speed of helium at 400 K is approximately \( 1580 \, \text{m/s} \).

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