The average velocity of molecules of a gas of molecular weight (M) at temperature (T) is

Options

A. `sqrt (3 R T)/(M)`.
B. `sqrt (8 R T)/(pi M)`.
C. `sqrt (2 R T)/(M)`.
D. zero

Answer

To find the average velocity of molecules of a gas with molecular weight \( M \) at temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The average velocity \( V_{avg} \) of gas molecules can be expressed using the formula:
\[ V_{avg} = \sqrt{\frac{8kT}{\pi m}} \]
where:
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature,
- \( m \) is the mass of a single molecule of the gas.

2. **Relate Molecular Weight to Mass**: The molecular weight \( M \) is related to the mass of a single molecule \( m \) by the equation:
\[ m = \frac{M}{N_A} \]
where \( N_A \) is Avogadro's number.

3. **Substitute for Mass**: Substitute \( m \) in the average velocity formula:
\[ V_{avg} = \sqrt{\frac{8kT}{\pi \left(\frac{M}{N_A}\right)}} \]
This simplifies to:
\[ V_{avg} = \sqrt{\frac{8kTN_A}{\pi M}} \]

4. **Express Boltzmann Constant**: The Boltzmann constant \( k \) can be expressed as:
\[ k = \frac{R}{N_A} \]
where \( R \) is the universal gas constant.

5. **Substitute \( k \) into the Equation**: Now substitute \( k \) into the equation:
\[ V_{avg} = \sqrt{\frac{8\left(\frac{R}{N_A}\right)TN_A}{\pi M}} \]
The \( N_A \) cancels out:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

6. **Final Expression**: Thus, the average velocity of the molecules of the gas is given by:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

### Conclusion:
The average velocity of molecules of a gas of molecular weight \( M \) at temperature \( T \) is:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]