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 in terms of the average speed of the molecules \( U_{\text{avg}} \), we can follow these steps:

### Step 1: Understand the relationship between average speed and temperature
The average speed of the molecules in a gas can be expressed as:
\[ U_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 2: Square the average speed equation
To eliminate the square root, we square both sides:
\[ U_{\text{avg}}^2 = \frac{8RT}{\pi M} \]

### Step 3: Rearrange to find \( RT \)
From the squared equation, we can express \( RT \):
\[ RT = \frac{\pi M U_{\text{avg}}^2}{8} \]

### Step 4: Use the formula for average kinetic energy per mole
The average kinetic energy per mole of a monoatomic gas is given by:
\[ KE = \frac{3}{2} RT \]

### Step 5: Substitute \( RT \) into the kinetic energy formula
Now we substitute the expression for \( RT \) from Step 3 into the kinetic energy formula:
\[ KE = \frac{3}{2} \left( \frac{\pi M U_{\text{avg}}^2}{8} \right) \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ KE = \frac{3\pi M U_{\text{avg}}^2}{16} \]

### Final Expression
Thus, the average kinetic energy per mole of a monoatomic gas in terms of the average speed of the molecules \( U_{\text{avg}} \) is:
\[ KE = \frac{3\pi M U_{\text{avg}}^2}{16} \]