The internal energy of one mole mono-atomic gas is

Options

A. `(5RT)/2`
B. `(3RT)/2`
C. `(5RT)/3`
D. `(7RT)/3`

Explanation

To find the internal energy of one mole of a monoatomic gas, we can follow these steps:

### Step 1: Understand the Concept of Internal Energy
The internal energy (U) of a gas is related to the kinetic energy of its molecules. For an ideal gas, the internal energy is directly proportional to the temperature and the number of degrees of freedom of the gas molecules.

### Step 2: Use the Equipartition Theorem
According to the equipartition theorem, the energy associated with each degree of freedom is (1/2)kT, where k is the Boltzmann constant and T is the absolute temperature in Kelvin.

### Step 3: Determine the Degrees of Freedom for a Monoatomic Gas
A monoatomic gas has 3 translational degrees of freedom. Therefore, the total energy per molecule can be expressed as:
\[ E_{\text{molecule}} = \frac{3}{2} k T \]

### Step 4: Relate the Energy of One Mole of Gas
For one mole of gas, the total number of molecules is given by Avogadro's number (N_A = 6.022 x 10²³). Thus, the internal energy for one mole of a monoatomic gas can be calculated as:
\[ U = N_A \cdot E_{\text{molecule}} = N_A \cdot \frac{3}{2} k T \]

### Step 5: Substitute Boltzmann's Constant with Universal Gas Constant
The universal gas constant (R) is related to Boltzmann's constant by the equation:
\[ R = N_A \cdot k \]
Substituting this into the equation for internal energy gives:
\[ U = \frac{3}{2} N_A k T = \frac{3}{2} R T \]

### Step 6: Final Expression for Internal Energy
Thus, the internal energy of one mole of a monoatomic gas is given by:
\[ U = \frac{3}{2} R T \]

### Conclusion
The internal energy of one mole of a monoatomic gas is \( \frac{3}{2} R T \).

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