The average energy per molecule of a gas at a given temperature, T, is given by

Options

A. `(3)/(2)RT`
B. `sqrt((3RT)/(M))`
C. `sqrt((8(R//N_(A))T)/(piM))`
D. `(3)/(2)((R )/(N_(A)))T`

Explanation

To find the average energy per molecule of a gas at a given temperature \( T \), we can follow these steps:

### Step 1: Understand the Average Energy of a Gas
The average energy of one mole of an ideal gas at temperature \( T \) is given by the formula:
\[ E = \frac{3}{2} RT \]
where \( R \) is the gas constant.

**Hint:** Remember that this formula is derived from the kinetic theory of gases, which relates temperature to energy.

### Step 2: Relate Energy to Molecules
Since the question asks for the average energy per molecule, we need to convert the energy for one mole to energy per molecule. We know that one mole of any substance contains Avogadro's number (\( N_A \)) of molecules, which is approximately \( 6.022 \times 10^{23} \).

**Hint:** Avogadro's number is key to converting from moles to individual molecules.

### Step 3: Calculate Average Energy per Molecule
To find the average energy per molecule, we divide the total energy for one mole by Avogadro's number:
\[ E_{\text{per molecule}} = \frac{E}{N_A} = \frac{\frac{3}{2} RT}{N_A} \]

**Hint:** This step involves simple division, so ensure you keep track of the units.

### Step 4: Simplify the Expression
We can simplify the expression further:
\[ E_{\text{per molecule}} = \frac{3}{2} \frac{R}{N_A} T \]
Here, \( \frac{R}{N_A} \) is defined as the Boltzmann constant (\( k \)):
\[ k = \frac{R}{N_A} \]

**Hint:** Recognizing the Boltzmann constant will help in simplifying the expression.

### Step 5: Final Expression
Thus, the average energy per molecule of a gas at temperature \( T \) can be expressed as:
\[ E_{\text{per molecule}} = \frac{3}{2} k T \]

**Hint:** The final expression shows how energy relates to temperature and the Boltzmann constant.

### Conclusion
The average energy per molecule of a gas at a given temperature \( T \) is given by:
\[ E_{\text{per molecule}} = \frac{3}{2} k T \]