If internal energy of gas is `U= 3PV+4` then the gas can be

Options

A. monoatomic
B. diatomic
C. polyatomic
D. either mono or diatomic

Answer

To determine the type of gas based on the given internal energy equation \( U = 3PV + 4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Internal Energy Formula**:
The internal energy \( U \) of an ideal gas can be expressed as:
\[ U = \frac{f}{2} nRT \]
where \( f \) is the degrees of freedom of the gas, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

2. **Relate PV to nRT**:
From the ideal gas law, we know:
\[ PV = nRT \]
We can substitute \( nRT \) in the internal energy equation:
\[ U = \frac{f}{2} PV \]

3. **Equate the Two Expressions for Internal Energy**:
We have two expressions for internal energy:
\[ U = 3PV + 4 \quad \text{(given)} \]
\[ U = \frac{f}{2} PV \quad \text{(from ideal gas)} \]
Setting these equal to each other gives:
\[ \frac{f}{2} PV = 3PV + 4 \]

4. **Simplify the Equation**:
Divide both sides by \( PV \) (assuming \( PV \neq 0 \)):
\[ \frac{f}{2} = 3 + \frac{4}{PV} \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ f = 6 + \frac{8}{PV} \]

6. **Analyze the Degrees of Freedom**:
The term \( \frac{8}{PV} \) is always positive, which means:
\[ f > 6 \]
The degrees of freedom \( f \) for different types of gases are:
- Monatomic: \( f = 3 \)
- Diatomic: \( f = 5 \)
- Polyatomic: \( f \geq 6 \)

7. **Conclusion**:
Since \( f > 6 \), the gas must be polyatomic.

### Final Answer:
The gas can be classified as **polyatomic**.