Step 1: Understanding the Question:
We need to find the internal (thermal) energy of one mole of a diatomic gas.
A "rigid rotator" diatomic gas has \(f = 5\) degrees of freedom (3 translational + 2 rotational).
Step 2: Key Formula or Approach:
1) Energy per mole \(U_{m} = \frac{f}{2} RT\).
2) From the ideal gas law \(PV = nRT\), for one mole (\(n=1\)), \(RT = PV_{m}\), where \(V_{m}\) is the molar volume.
3) Molar volume \(V_{m} = \frac{M}{\rho}\), where \(M\) is the molar mass and \(\rho\) is density.
Alternatively, energy per unit mass can be found, then multiplied by molar mass. Or use \(P = \frac{\rho RT}{M} \implies RT = \frac{PM}{\rho}\).
So, \(U_{m} = \frac{f}{2} \cdot \frac{PM}{\rho}\). Wait, the question asks for energy of one mole. Let's re-examine.
Actually, for any sample: Total Energy \(U = \frac{f}{2} PV\).
Energy per mole \(U_{m} = \frac{U}{n} = \frac{f}{2} \frac{PV}{n}\). From \(PV = nRT\), this is \(\frac{f}{2} RT\).
From \(P = \frac{\rho RT}{M} \implies RT = \frac{PM}{\rho}\).
Thus \(U_{m} = \frac{5}{2} \frac{P}{\rho}\) per unit mass? No, that's energy per unit mass if we ignore \(M\).
Energy of one mole is \(U_{m} = \frac{5}{2} \frac{PM}{\rho}\). We don't have \(M\).
Wait, let's look at the units and parameters. Pressure \(P = 10^{5}\), Density \(\rho = 5\).
Energy density (energy per unit volume) is \(\frac{f}{2} P\).
Energy per unit mass is \(\frac{\text{Energy density}}{\text{Mass density}} = \frac{f P / 2}{\rho} = \frac{5 \times 10^{5}}{2 \times 5} = 5 \times 10^{4} \text{ J/kg}\).
Usually, "energy of one mole" questions without molar mass mean they want you to find energy for the specific given mass and then perhaps it's a trick? Let's check the given answer.
If we calculate energy for the whole \(500 \text{ g}\) (\(0.5 \text{ kg}\)):
\(U = (\text{Energy per kg}) \times 0.5 \text{ kg} = 5 \times 10^{4} \times 0.5 = 2.5 \times 10^{4} \text{ J}\).
The question asks for "energy of one mole", but based on the options, it seems to be asking for the total energy of the given \(500 \text{ g}\) sample. Let's proceed with that.
Step 3: Detailed Explanation:
Volume of the gas \(V = \frac{\text{mass}}{\text{density}} = \frac{500 \times 10^{-3} \text{ kg}}{5 \text{ kg/m}^3} = 0.1 \text{ m}^{3}\).
Total thermal energy for a diatomic gas (rigid rotator, \(f=5\)):
\[ U = \frac{f}{2} PV \]
\[ U = \frac{5}{2} \times 10^{5} \times 0.1 \]
\[ U = \frac{5}{2} \times 10^{4} \]
\[ U = 2.5 \times 10^{4} \text{ J} \]
Step 4: Final Answer:
The energy of the given gas sample (which likely corresponds to the intended meaning of the question) is \(2.5 \times 10^{4} \text{ J}\).