Analysis
The internal energy of an ideal gas is given by $U = \frac{f}{2} PV$, where $f$ is the degrees of freedom.
Given equation: $U = 3PV$.
Comparing the two:
$$ \frac{f}{2} = 3 \implies f = 6 $$Interpretation
- Mono-atomic gas: $f=3$. (Incorrect)
- Di-atomic gas: Usually $f=5$ (rigid) or $f=7$ (vibrating).
- Tri-atomic / Polyatomic gas: A non-linear polyatomic gas (like methane or water vapor) has 3 translational and 3 rotational degrees of freedom, giving $f=6$. So, the gas can be tri-atomic.
Specific Heat
For $f=6$:
$$ C_V = \frac{f}{2}R = 3R $$ $$ C_P = C_V + R = 4R $$Thus, statement (d) is also correct.
Correct Conclusions: (a) Gas is not mono-atomic, (c) Gas can be tri-atomic, (d) Isobaric specific heat is 4R.