Internal Energy Calculation for Diatomic Gas

This solution calculates the internal energy of air within a specified room volume, treating air as a diatomic gas.

Calculate Room Volume

First, determine the total volume ($V$) of the room:

$ V = \text{length} \times \text{width} \times \text{height} $

$ V = 4 \, m \times 4 \, m \times 3 \, m = 48 \, m^3 $

Convert Pressure to Standard Units

The given pressure is 1 atmospheric pressure. Convert this to Pascals (Pa) for calculations:

$ P = 1 \, \text{atm} \approx 1.013 \times 10^5 \, \text{Pa} $

Internal Energy Formula for Diatomic Gas

The internal energy ($U$) of an ideal gas depends on its degrees of freedom. For a diatomic molecule, there are 5 degrees of freedom (3 translational, 2 rotational). The internal energy is given by:

$ U = \frac{f}{2} nRT $

Where $f=5$ for diatomic molecules. Using the ideal gas law, $PV = nRT$, we can express internal energy as:

$ U = \frac{5}{2} PV $

Compute Internal Energy

Substitute the volume and pressure values into the formula:

$ U = \frac{5}{2} \times (1.013 \times 10^5 \, \text{Pa}) \times (48 \, m^3) $

$ U = 2.5 \times 1.013 \times 48 \times 10^5 \, \text{J} $

$ U = 121.56 \times 10^5 \, \text{J} $

Final Result Formatting

Express the result in the required units ($\times 10^6$ J):

$ U = 12.156 \times 10^6 \, \text{J} $

The calculated value of approximately $12.156 \times 10^6$ J is consistent with the expected value.