Express the average kinetic energy per mole of a monoatomic gas of molar mass M, at temperature T K in terms of the average speed of the molecules `U_(avg)` :

Options

A. `(8M)/(3pi) U_(avg)^(2)`
B. `(4M)/(3pi) U_(avg)^(2)`
C. `((2M)/(pi)) U_(avg)^(2)`
D. `((3pi M)/(16)) U_(avg)^(2)`

Explanation

To express the average kinetic energy per mole of a monoatomic gas of molar mass \( M \) at temperature \( T \) in terms of the average speed of the molecules \( U_{\text{avg}} \), we can follow these steps:

### Step 1: Relate Average Speed to Temperature
According to the Maxwell distribution of energy, the average speed \( U_{\text{avg}} \) of the molecules in a gas is given by the formula:

\[ U_{\text{avg}} = \sqrt{\frac{8RT}{\pi m}} \]

where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule of the gas.

### Step 2: Square the Average Speed
To eliminate the square root, we square both sides of the equation:

\[ U_{\text{avg}}^2 = \frac{8RT}{\pi m} \]

### Step 3: Solve for \( RT \)
Rearranging the equation to express \( RT \) in terms of \( U_{\text{avg}} \):

\[ RT = \frac{U_{\text{avg}}^2 \cdot \pi m}{8} \]

### Step 4: Kinetic Energy of a Monoatomic Gas
For a monoatomic ideal gas, the average kinetic energy \( KE \) per mole is given by:

\[ KE = \frac{3}{2} RT \]

### Step 5: Substitute \( RT \) into the Kinetic Energy Equation
Now, we substitute the expression for \( RT \) from Step 3 into the kinetic energy formula:

\[ KE = \frac{3}{2} \left( \frac{U_{\text{avg}}^2 \cdot \pi m}{8} \right) \]

### Step 6: Simplify the Expression
Simplifying this expression gives:

\[ KE = \frac{3 \pi m}{16} U_{\text{avg}}^2 \]

### Final Result
Thus, the average kinetic energy per mole of a monoatomic gas of molar mass \( M \) at temperature \( T \) in terms of the average speed of the molecules \( U_{\text{avg}} \) is:

\[ KE = \frac{3 \pi M}{16} U_{\text{avg}}^2 \]